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                                                                    X. Boyen
     S/MIME Working Group                                          L. Martin
     Internet Draft                                         Voltage Security
     Expires: May 2007                                         November 2006
     
     
     
         Identity-Based Cryptography Standard (IBCS) #1: Supersingular Curve
                   Implementations of the BF and BB1 Cryptosystems
     
     
                           <draft-martin-ibcs-01.txt>
     
     
     Status of this Memo
     
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     Abstract
     
        This document describes the algorithms that implement Boneh-Franklin
        and Boneh-Boyen Identity-based Encryption. This document is in part
        based on IBCS #1 v2 of Voltage Security's Identity-based Cryptography
        Standards (IBCS) documents.
     
     
     
     
     
     
     
     
     

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     Table of Contents
     
     
        1. Introduction...................................................3
        2. Notation and definitions.......................................4
           2.1. Notation..................................................4
           2.2. Definitions...............................................6
        3. Basic elliptic curve algorithms................................7
           3.1. The group action in affine coordinates....................7
              3.1.1. Implementation for type-1 curves.....................7
           3.2. Point multiplication......................................9
           3.3. Special operations in projective coordinates.............11
              3.3.1. Implementation for type-1 curves....................11
           3.4. Divisors on elliptic curves..............................13
              3.4.1. Implementation in F_p^2 for type-1 curves...........13
           3.5. The Tate pairing.........................................15
              3.5.1. The Miller algorithm for type-1 curves..............16
        4. Supporting algorithms.........................................19
           4.1. Integer range hashing....................................19
           4.2. Pseudo-random generation by hashing......................20
           4.3. Canonical encodings of extension field elements..........21
              4.3.1. Type-1 curve implementation.........................21
           4.4. Hashing onto a subgroup of an elliptic curve.............22
              4.4.1. Type-1 curve implementation.........................23
           4.5. Bilinear pairing.........................................23
              4.5.1. Type-1 curve implementation.........................24
           4.6. Ratio of bilinear pairings...............................25
              4.6.1. Type-1 curve implementation.........................25
        5. The Boneh-Franklin BF cryptosystem............................26
           5.1. Setup....................................................26
              5.1.1. Type-1 curve implementation.........................27
           5.2. Public key derivation....................................28
           5.3. Private key extraction...................................28
           5.4. Encryption...............................................29
           5.5. Decryption...............................................30
        6. Wrapper methods for the BF system.............................32
           6.1. Private key generator (PKG) setup........................32
           6.2. Private key extraction by the PKG........................32
           6.3. Session key encryption...................................33
        7. Concrete encoding guidelines for BF...........................35
           7.1. Encoding of points on a curve............................35
           7.2. Public parameters blocks.................................36
              7.2.1. Type-1 implementation...............................36
           7.3. Master secret blocks.....................................38
           7.4. Private key blocks.......................................38
     
     
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           7.5. Ciphertext blocks........................................39
        8. The Boneh-Boyen BB1 cryptosystem..............................40
           8.1. Setup....................................................40
              8.1.1. Type-1 curve implementation.........................41
           8.2. Public key derivation....................................42
           8.3. Private key extraction...................................43
           8.4. Encryption...............................................44
           8.5. Decryption...............................................46
        9. Wrapper methods for the BB1 system............................48
           9.1. Private key generator (PKG) setup........................49
           9.2. Private key extraction by the PKG........................49
           9.3. Session key encryption...................................50
        10. Concrete encoding guidelines for BB1.........................52
           10.1. Encoding of points on a curve...........................52
           10.2. Public parameters blocks................................52
              10.2.1. Type-1 implementation..............................53
           10.3. Master secret blocks....................................54
           10.4. Private key blocks......................................55
           10.5. Ciphertext blocks.......................................56
        11. Test vectors.................................................58
           11.1. Algorithm 3.2.2 (PointMultiply).........................58
           11.2. Algorithm 4.1.1 (HashToRange)...........................58
           11.3. Algorithm 4.5.1 (Pairing)...............................59
           11.4. Algorithm 5.2.1 (BFderivePubl)..........................59
           11.5. Algorithm 5.3.1 (BFextractPriv).........................59
           11.6. Algorithm 5.4.1 (BFencrypt).............................60
           11.7. Algorithm 8.3.1 (BBextractPriv).........................61
           11.8. Algorithm 8.4.1 (BBencrypt).............................62
        12. ASN.1 module.................................................63
        13. Security considerations......................................67
        14. IANA considerations..........................................68
        15. Acknowledgments..............................................68
        16. References...................................................69
           16.1. Normative references....................................69
           16.2. Informative references..................................69
        Authors' Addresses...............................................69
        Intellectual Property Statement..................................69
        Disclaimer of Validity...........................................70
        Copyright Statement..............................................70
        Acknowledgment...................................................70
     
     1. Introduction
     
        This document provides a set of specifications for implementing
        identity-based encryption (IBE) systems based on bilinear pairings.
        Two cryptosystems are described: the IBE system proposed by Boneh and
        Franklin (BF) [BF], and the first IBE system proposed by Boneh and
     
     
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        Boyen (BB1) [BB1]. Fully secure and practical implementations are
        described for each system, comprising the core IBE algorithms as well
        as ancillary hybrid components used to achieve security against
        active attacks. These specifications are restricted to a family of
        supersingular elliptic curves over finite fields of large prime
        characteristic, referred to as "type-1" curves (see Section 2.1).
        Implementations based on other types of curves currently fall outside
        the scope of this document.
     
     2. Notation and definitions
     
     2.1. Notation
     
        This section summarizes the essential notions and definitions
        regarding identity-based cryptosystems on elliptic curves. The reader
        is referred to [ECC] for the mathematical background and to [2, 3]
        regarding all notions pertaining to identity-based encryption.
     
        Let F_p be a finite field of large prime characteristic p, and let
        F_p^2 denote its extension field of degree 2. Denote by F*_p^2 the
        multiplicative subgroup of F_p^2.
     
        Let E/F_p: y^2 = x^3 + a * x + b be an elliptic curve over F_p. For
        any extension degree k >= 1, the curve E/F_p defines a group
        (E(F_p^2), +), which is the additive group of points of affine
        coordinates (x, y) in (F_p^2)^2 satisfying the curve equation over
        F_p^2, with null element, or point at infinity, denoted 0. Let
        #E(F_p^2) be the size of E(F_p^2).
     
        Let q be a prime such that E(F_p) has a cyclic subgroup G1' of order
        q. Let G1'' be a cyclic subgroup of E(F_p^2) of order q, and G2 be a
        cyclic subgroup of F*_p^2 of order p.
     
        Under these conditions, two mathematical constructions known as the
        Weil pairing and the Tate pairing, each provide an efficiently
        computable map e: G1' x G1'' -> G2 that is linear in both arguments
        and believed hard to invert. If an efficiently computable isomorphism
        phi: G1' -> G1'' is available for the selected elliptic curve on
        which the Tate pairing is computed, then one can construct a function
        e': G1' x G1'' -> G2, defined as e'(A, B) = e(A, phi(B)), called the
        modified Tate pairing. We generically call a pairing either the Tate
        pairing e or the modified Tate pairing e', depending on the chosen
        elliptic curve used in a particular implementation.
     
        The following additional notation is used throughout this document.
     
     
     
     
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        P - A 512-bit to 1536-bit prime, being the order of the finite field
        F_p.
     
        F_p - The base finite field of size p over which the elliptic curve
        of interest E/F_p is defined.
     
        #G - The size of G, where G is a finite group.
     
        G* - The multiplicative group of the invertible elements in G; e.g.,
        (F_p)* is the multiplicative group of the finite field F_p.
     
        E/F_p - The equation of an elliptic curve over the field F_p, which,
        when p is neither 2 nor 3, is of the form E/F_p: y^2 = x^3 + a * x +
        b, for specified a, b in F_p.
     
        0 - The conventional null element of any additive group of points on
        an elliptic curve, also called the point at infinity.
     
        E(F_p) - The additive group of points of affine coordinates (x, y),
        with x, y in F_p, that satisfy the curve equation E/F_p, including
        the point at infinity 0.
     
        q - A 160-bit to 256-bit prime, being the order of the cyclic
        subgroup of interest in E(F_p).
     
        k - The embedding degree, or security multiplier, of the cyclic
        subgroup of order q in E(F_p).
     
        F_p^2 - The extension field of degree 2 of the field F_p.
     
        E(F_p^2) - The group of points of affine coordinates in F_p^2
        satisfying the curve equation E/F_p, including the  point at infinity
        0.
     
        Z_p - The additive group Z / pZ.
     
        The following conventions are assumed for curve operations.
     
        Point addition: If A and B are two points on a curve E, their sum is
        denoted A + B.
     
        Point multiplication: If A is a point on a curve, and n an integer,
        the result of adding A to itself a total of n times is denoted [n]A.
     
        The following class of elliptic curves is exclusively considered for
        pairing operations in the present version of the IBCS#1 standard,
        referred to as "type-1."
     
     
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        Type-1 curves: The class of curves of type 1 is defined as the class
        of all elliptic curves of equation E/F_p: y^2 = x^3 + 1 for all
        primes p congruent to 11 modulo 12. This class forms a subclass of
        the class of supersingular curves. These curves satisfy #E(F_p) = p +
        1, so that the p pairs of (x, y) coordinates corresponding to the p
        non-zero points E(F_p) \ {0} satisfy a useful bijective relation x <-
        > y, with x = (y^2 - 1)^(1/3) (mod p) and y = (x^3 + 1)^(1/2) (mod
        p). Type-1 curves always lead to a security multiplier k = 2, where
        f(x) = (x^2 + 1) is always irreducible, allowing the uniform
        representation of F_p^2 = F[x] / (x^2 + 1). Type-1 curves are
        plentiful and easy to construct by random selection of a prime p of
        the appropriate form. Therefore, rather than to standardize upon a
        small set of common values of p, it is henceforth assumed that all
        type-1 curves are freshly generated at random for the given
        cryptographic application (an example of such generation will be
        given in Algorithm 5.1.2 (BFsetup1) or Algorithm 8.1.2 (BBsetup1)).
        Implementations based on different classes of curves are currently
        unsupported.
     
        We assume that the following concrete representations of mathematical
        objects are used.
     
        Base field elements: The p elements of the base field F_p are
        represented directly using the integers from 0 to p - 1.
     
        Extension field elements: The p^2 elements of the extension field
        F_p^2 are represented as ordered pairs of elements of F_p. An ordered
        pair (a_0, a_1) is interpreted as the polynomial a_1 * x + a_0 in
        F_p[x] / (x^2 + 1).
     
        Type-1 curves: For type-1 curves, which are supersingular curves of
        equation E/F_p: y^2 = x^3 + 1 with p congruent to 11 modulo 12, and
        elements of F_p^2 are represented as polynomials a_1 *_x + a_0 in
        F_p[x] / (x^2 + 1).
     
        Elliptic curve points: Points in E(F_p^2) with the point P = (x, y)
        in F_p^2 x F_p^2 satisfying the curve equation E/F_p. Points not
        equal to 0 are internally represented using the affine coordinates
        (x, y), where x and y are elements of F_p^2.
     
     2.2. Definitions
     
        The following terminology is used to describe an IBE system.
     
        Public parameters: The public parameters are set of common system-
        wide parameters generated and published by the private key server
        (PKG).
     
     
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        Master secret: The master secret is the master key generated and
        privately kept by the key server, and used to generate the private
        keys of the users.
     
        Identity: An identity an arbitrary string, usually a human-readable
        unambiguous designator of a system user, possibly augmented with a
        time stamp and other attributes.
     
        Public key: A public key is a string that is algorithmically derived
        from an identity. The derivation may be performed by anyone,
        autonomously.
     
        Private key: A private key is issued by the key server to correspond
        to a given identity (and the public key that derives from it), under
        the published set of public parameters.
     
        Plaintext: A plaintext is an unencrypted representation, or in the
        clear, of any block of data to be transmitted securely. For the
        present purposes, plaintexts are typically session keys, or sets of
        session keys, for further symmetric encryption and authentication
        purposes.
     
        Ciphertext: A ciphertext is an encrypted representation of any block
        of data, including a plaintext, to be transmitted securely.
     
     3. Basic elliptic curve algorithms
     
        This section describes algorithms for performing all needed basic
        arithmetic operations on elliptic curves. The presentation is
        specialized to the type of curves under consideration for simplicity
        of implementation. General algorithms may be found in [ECC].
     
     3.1.  The group action in affine coordinates
     
     3.1.1. Implementation for type-1 curves
     
        Algorithm 3.1.1 (PointDouble1): adds a point to itself on a type-1
        elliptic curve.
     
        Input:
     
        o  A point A in E(F_p^2), with A = (x, y) or 0.
     
        o  An elliptic curve E/F_p: y^2 = x^3 + 1.
     
        Output:
     
     
     
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        o  The point [2]A = A + A.
     
        Method:
     
        1. If A = 0 or y = 0, then return 0.
     
        2. Let lambda = (3 * x^2) / (2 * y).
     
        3. Let x' = lambda^2 - 2 * x.
     
        4. Let y' = (x - x') * lambda - y.
     
        5. Return (x', y').
     
        Algorithm 3.1.2 (PointAdd1): adds two points on a type-1 elliptic
        curve.
     
        Input:
     
        o  A point A in E(F_p^2), with A = (x_A, y_A) or 0,
     
        o  A point B in E(F_p^2), with B = (x_B, y_B) or 0,
     
        o  An elliptic curve E/F_p: y^2 = x^3 + 1.
     
        Output:
     
        o  The point A + B.
     
        Method:
     
        1. If A = 0, return B.
     
        2. If B = 0, return A.
     
        3. If x_A = x_B:
     
           (a) If y_A = -y_B, return 0.
     
           (b) Else return [2]A computed using Algorithm 2.1.1
        (PointDouble1).
     
        4. Otherwise:
     
           (a) Let lambda = (y_B - y_A) / (x_B - x_A).
     
           (b) Let x' = lambda^2 - x_A - x_B.
     
     
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           (c) Let y' = (x_A - x') * lambda - y_A.
     
           (d) Return (x', y').
     
     3.2. Point multiplication
     
        Algorithm 3.2.1 (SignedWindowDecomposition): computes the signed m-
        ary window representation of a positive integer.
     
        Input:
     
        o  An integer l > 0,
     
        o  An integer window bit-size r > 0.
     
        Output:
     
        o  The unique d-element sequence {(b_i, e_i), for i = 0 to d - 1}
           such that l = {Sum(b_i * 2^(e_i), for i = 0 to d - 1} and b_i =
           +/- 2^j for some 0 <= j <= r - 1.
     
        Method:
     
        1. Let d = 0.
     
        2. Let j = 0.
     
        3. While j <= l, do:
     
           (a) If l_k = 0 then:
     
              i. Let j = j + 1.
     
           (b) Else:
     
              i. Let t = min{j + r - 1, l}.
     
              ii. Let h_d = (l_t, l_(t - 1), ..., l_j) (base 2).
     
              iii. If h_d > 2^(r - 1) then:
     
                 A. Let b_d = h_d - 2^r.
     
                 B. Let l' = l' + 2^(t + 1).
     
              iv. Else:
     
     
     
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                 A. Let b_d = h_d.
     
              v. Let e_d = j.
     
              vi. Let d  = d + 1.
     
              vii. Let j  = t + 1.
     
        4. Return d and the sequence {(b_0, e_0), ..., (b_(d - 1), e_(d -
        1))}.
     
        Algorithm 3.2.2 (PointMultiply): scalar multiplication on an elliptic
        curve using the signed m-ary window method.
     
        Input:
     
        o  A point A in E(F_p^2),
     
        o  An integer l > 0,
     
        o  An elliptic curve E/F_p: y^2 = x^3 + a * x + b.
     
        Output:
     
        o  The point [l]A.
     
        Method:
     
        1. (Window decomposition)
     
           (a) Let r > 0 be an integer (fixed) bit-wise window size, e.g., r
        = 5.
     
           (b) Let l' = l where l = {Sum(l_j * 2^j), for j = 0 to l} is the
        binary expansion of l.
     
           (c) Compute (d, {(b_i, e_i), for i = 0 to d - 1} =
        SignedWindowDecomposition(l, r), the signed 2^r-ary window
        representation of l using Algorithm 3.2.1
        (SignedWindowDecomposition).
     
        2. (Precomputation)
     
           (a) Let A_1 = A.
     
           (b) Let A_2 = [2]A, using Algorithm 3.1.1 (PointDouble1).
     
     
     
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           (c) For i = 1 to 2^(r - 2) - 1, do:
     
              i. Let A_(2 * i + 1) = A_(2 * i - 1) + A_2 using Algorithm
        3.1.2 (PointAdd1).
     
           (d) Let Q = A_(b_(d - 1)).
     
        3. Main loop
     
           (a) For i = d - 2 to 0 by -1, do:
     
              i. Let Q = [2^(e_(i + 1) - e_i)]Q, using repeated applications
        of Algorithm 3.1.1 (PointDouble1) e_(i + 1) - e_i times.
     
              ii. If b_i > 0 then:
     
                 A. Let Q = Q + A_(b_i) using Algorithm 3.1.2 (PointAdd1).
     
              iii. Else:
     
                 A. Let Q = Q - A_(-b_i) using Algorithm 3.1.2 (PointAdd1).
     
           (b) Calculate Q = [2^(e_0)]Q using repeated applications of
        Algorithm 3.1.1 (PointDouble1) e_0 times.
     
        4. Return Q.
     
     3.3. Special operations in projective coordinates
     
     3.3.1. Implementation for type-1 curves
     
        Algorithm 3.3.1 (ProjectivePointDouble1): adds a point to itself in
        projective coordinates for type-1 curves.
     
        Input:
     
        o  A point (x, y, z) = A in E(F_p^k) in projective coordinates,
     
        o  An elliptic curve E/F_p: y^2 = x^3 + 1.
     
        Output:
     
        o  The point [2]A in projective coordinates.
     
        Method:
     
        1. If z = 0 or y = 0, return (0, 1, 0) = 0. Otherwise:
     
     
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        2. Let lambda_1 = 3 * x^2.
     
        3. Let z' = 2 * y * z.
     
        4. Let lambda_2 = y^2.
     
        5. Let lambda_3 = 4 * lambda_2 * x.
     
        6. Let x' = lambda_1^2 - 2 * lambda_3.
     
        7. Let lambda_4 = 8 * lambda_2^2.
     
        8. Let y' = lambda_1 * (lambda_3 - x) - lambda_4.
     
        9. Return (x', y', z').
     
        Algorithm 3.3.2 (ProjectivePointAccumulate1): adds a point in affine
        coordinates to an accumulator in projective coordinates, for type-1
        curves.
     
        Input:
     
        o  A point (x_A, y_A, z_A) = A in E(F_p^2) in projective coordinates,
     
        o  A point (x_B, y_B) = B in E(F_p^2) \ {0} in affine coordinates,
     
        o  An elliptic curve E/F_p: y^2 = x^3 + 1.
     
        Output:
     
        o  The point A + B in projective coordinates.
     
        Method:
     
        1. If z_A = 0 return (x_B, y_B, 1) = B. Otherwise:
     
        2. Let lambda_1 = z_A^2
     
        3. Let lambda_2 = lambda_1 * x_B.
     
        4. Let lambda_3 = x_A - lambda_2.
     
        5. If lambda_3 = 0 then return (0, 1, 0) = 0. Otherwise:
     
        6. Let lambda_4 = lambda_3^2.
     
        7. Let lambda_5 = lambda_1 * y_B * z_A.
     
     
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        8. Let lambda_6 = lambda_4 - lambda_5.
     
        9. Let lambda_7 = x_A + lambda_2.
     
        10. Let lambda_8 = y_A + lambda_5.
     
        11. Let x' = lambda_6^2 - lambda_7 * lambda_4.
     
        12. Let lambda_9 = lambda_7 * lambda_4 - 2 * x'.
     
        13. Let y' = (lambda_9 * lambda_6 - lambda_8 * lambda_3 * lambda_4) /
        2.
     
        14. Let z' = lambda_3 * z_A.
     
        15. Return (x', y', z').
     
     3.4. Divisors on elliptic curves
     
     3.4.1. Implementation in F_p^2 for type-1 curves
     
        Algorithm 3.4.1 (EvalVertical1): evaluates the divisor of a vertical
        line on a type-1 elliptic curve.
     
        Input:
     
        o  A point B in E(F_p^2) with B != 0.
     
        o  A point A in E(F_p).
     
        o  A description of a type-1 elliptic curve E/F_p.
     
        Output:
     
        o  An element of F_p^2 that is the divisor of the vertical line going
           through A evaluated at B.
     
        Method:
     
        1. Let r = x_B - x_A.
     
        2. Return r.
     
        Algorithm 2.4.2 (EvalTangent1): evaluates the divisor of a tangent on
        a type-1 elliptic curve.
     
        Input:
     
     
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        o  A point B in E(F_p^2) with B != 0.
     
        o  A point A in E(F_p).
     
        o  A description of a type-1 elliptic curve E/F_p.
     
        Output:
     
        o  An element of F_p^2 that is the divisor of the line tangent to A
           evaluated at B.
     
        Method:
     
        1. (Special cases)
     
           (a) If A = 0 return 1 = 1 + 0 * i.
     
           (b) If y_A = 0 return EvalVertical1(B, A) using Algorithm 3.4.1
        (EvalVertical1).
     
        2. (Line computation)
     
           (a) Let a = -3 * (x_A)^2.
     
           (b) Let b = 2 * y_A.
     
           (c) Let c = -b * y_A - a * x_A.
     
        3. (Evaluation at B)
     
        (a) Let r = a * x_B + b * y_B) + c.
     
        4. Return r.
     
        Algorithm 3.4.3 (EvalLine1): evaluates the divisor of a line on a
        type-1 elliptic curve.
     
        Input:
     
        o  A point B in E(F_p^2) with B != 0.
     
        o  Two points A', A'' in E(F_p).
     
        o  A description of a type-1 elliptic curve E/F_p.
     
        Output:
     
     
     
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        o  An element of F_p^2 that is the divisor of the line going through
           A' and A'' evaluated at B.
     
        Method:
     
        1. (Special cases)
     
           (a) If A' = 0 return EvalVertical1(B, A'') using Algorithm 3.4.1
        (EvalVertical1).
     
           (b) If A'' = 0 return EvalVertical1(B, A') using Algorithm 3.4.1
        (EvalVertical1).
     
           (c) If A' = -A'' return EvalVertical1(B, A') using Algorithm 3.4.1
        (EvalVertical1).
     
           (d) If A' = A'' return EvalTangent1(B, A') using Algorithm 3.4.2
        (EvalTangent1).
     
        2. (Line computation)
     
           (a) Let a = y_A' - y_A''.
     
           (b) Let b = x_A'' - x_A'.
     
           (c) Let c = -b * y_A' - a * x_A'.
     
        3. (Evaluation at B)
     
           (a) Let r = a * x_B + b * y_B + c.
     
        4. Return r.
     
     3.5. The Tate pairing
     
        Algorithm 3.5.1 (Tate): computes the Tate pairing on an elliptic
        curve.
     
        Input:
     
        o  A point A of order q in E(F_p),
     
        o  A point B of order q in E(F_p^2),
     
        o  A description of an elliptic curve E/F_p such that E(F_p) and
           E(F_p^2) have a subgroup of order q.
     
     
     
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        Output:
     
        o  The value e(A, B) in F_p^2, computed using the Miller algorithm.
     
        Method:
     
        1. For type-1 curve E, proceed with Algorithm 3.5.2
        (TateMillerSolinas).
     
     3.5.1. The Miller algorithm for type-1 curves
     
        Algorithm 3.5.2 (TateMillerSolinas): computes the Tate pairing on a
        type-1 elliptic curve.
     
        Input:
     
        o  A point A of order q in E(F_p),
     
        o  A point B of order q in E(F_p^2),
     
        o  A description of a type-1 supersingular elliptic curve E/F_p such
           that E(F_p) and E(F_p^2) have a subgroup of prime order q, where q
           = 2^a + s * 2^b + c with c and s equal to either 1 or -1.
     
        Output:
     
        o  The value e(A, B) in F_p^2, computed using the Miller algorithm.
     
        The following description assumes that F_p^2 = F_p[i], where i^2 = -
        1.
     
        Elements x in F_p^2 may be explicitly represented as a + i * b, with
        a, b in F_p.
     
        Points in E(F_p) may also be represented as coordinate pairs (x, y)
        with x, y in F_p.
     
        Points in E(F_p^2) may be represented either as (x, y), with x, y in
        F_p^2 or as (a + i * b, c + i * d), with a, b, c, d in F_p.
     
        Method:
     
        1. (Initialization)
     
           (a) Let v_num = 1 in F_p^2.
     
           (b) Let v_den = 1 in F_p^2.
     
     
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           (c) Let V = (x_V , y_V , z_V ) = (x_A, y_A, 1) in (F_p)^3, being
        the representation of (x_A, y_A) = A using projective coordinates.
     
           (d) Let t_num = 1 in F_p^2.
     
           (e) Let t_den = 1 in F_p^2.
     
        2. (Calculation of the (s * 2^b) contribution)
     
           (a) (Repeated doublings) For n = 0 to b - 1:
     
              i. Let t_num = t_num^2.
     
              ii. Let t_den = t_den^2.
     
              iii. Let t_num = t_num * EvalTangent1(B, V) using Algorithm
        3.4.2 (EvalTangent1).
     
              iv. Let V = (x_V , y_V , z_V ) = [2]V  using Algorithm 3.3.1
        (ProjectivePointDouble1).
     
              v. Let t_den = t_den * EvalVertical1(B, V) using Algorithm
        3.4.1 (EvalVertical1).
     
           (b) (Normalization)
     
              i. Let V_b = (x_(V_b) , y_(V_b)) = (x_V / z_V^2, s * y_V /
        z_V^3) in (F_p)^2, resulting in a point V_b in E(F_p).
     
           (c) (Accumulation) Selecting on s:
     
              i. If s = -1:
     
                 A. Let v_num = v_num * t_den.
     
                 B. Let v_den = v_den * t_num * EvalVertical1(B, V) using
        Algorithm 3.4.1 (EvalVertical1).
     
              ii. If s = 1:
     
                 A. Let v_num = v_num * t_num.
     
                 B. Let v_den = v_den * t_den.
     
        3. (Calculation of the 2^a contribution)
     
           (a) (Repeated doublings) For n = b to a - 1:
     
     
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              i. Let t_num = t_num^2.
     
              ii. Let t_den = t_den^2.
     
              iii. Let t_num = t_num * EvalTangent1(B, V) using Algorithm
        3.4.2 (EvalTangent1).
     
              iv. Let V = (x_V , y_V , z_V) = [2]V  using Algorithm 3.3.1
        (ProjectivePointDouble1).
     
              v. Let t_den = t_den * EvalVertical1(B, V) using Algorithm
        3.4.1 (EvalVertical1).
     
           (b) (Normalization)
     
              i. Let V_a = (x_(V_a) , y_(V_a)) = (x_V /z_V^2, s * x_V /
        z_V^3) in (F_p)^2, resulting in a point V_a in E(F_p).
     
           (c) (Accumulation)
     
              i. Let v_num = v_num * t_num.
     
              ii. Let v_den = v_den * t_den.
     
        4. (Correction for the (s * 2^b) and (c) contributions)
     
           (a) Let v_num = v_num * EvalLine1(B, V_a, V_b) using Algorithm
        3.4.3 (EvalLine1).
     
           (b) Let v_den = v_den * EvalVertical1(B, V_a + V_b) using
        Algorithm 3.4.1 (EvalVertical1).
     
           (c) If c = -1 then:
     
              i. Let v_den = v_den * EvalVertical1(B,A) using Algorithm 3.4.1
        (EvalVertical1).
     
        5. (Correcting exponent)
     
           (a) Let eta = (p^2 - 1) / q (an integer).
     
        6. (Final result)
     
           (a) Return (v_num / v_den)^eta in F_p^2.
     
     
     
     
     
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     4. Supporting algorithms
     
        This section describes a number of supporting algorithms for encoding
        and hashing.
     
     4.1. Integer range hashing
     
        HashToRangen(s, n) takes a string s and an integer n as input, and
        returns an integer in the range 0 to n - 1 by cryptographic hashing.
        The function performs a number l of SHA1 applications, with l chosen
        in function of n so that, for random input, the output has an almost
        uniform distribution in the entire range 0 to n - 1 with a
        statistical relative non-uniformity no greater than 1/sqrt(n). I.e.,
        for arbitrarily large n, for all v in 0 to n - 1, the probability
        that HashToRangen(s, n) = v lies in the interval [(1 - n^(-1/2)) / n,
        (1 + n^(-1/2)) / n].
     
        Algorithm 4.1.1 (HashToRange): cryptographically hashes strings to
        integers in a range.
     
        Input:
     
        o  A string s of length |s| bytes,
     
        o  A positive integer n represented as Ceiling(8 * lg(n)) bytes.
     
        Output:
     
        o  A positive integer v in the range 0 to n - 1.
     
        Method:
     
        1. Let v_0 = 0.
     
        2. Let h_0 = 0x0000000000000000000000000000000000000000, a string of
        20 null bytes.
     
        3. Let l = Ceiling((3 / 5) * lg(n)).
     
        4. For each i in 1 to l, do:
     
           (a) Let t_i = h_(i - 1) || s, which is the (|s| + 20)-byte string
        concatenation of the strings h_(i - 1) and s.
     
           (b) Let h_i = SHA1(t_i), which is a 20-byte string resulting from
        the SHA1 algorithm on input t_i.
     
     
     
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           (c) Let a_i = Value(h_i) be the integer in the range 0 to 256^20 -
        1 denoted by the raw byte string h_i interpreted in the unsigned big
        endian convention.
     
           (d) Let v_i = 256^20 * v_(i - 1) + a_i.
     
        5. Let v = v_l (mod n).
     
     4.2. Pseudo-random generation by hashing
     
        HashStream(b, p) takes an integer b and a string p as input, and
        returns a b-byte pseudo-random string r as output. This function
        relies on the SHA1 cryptographic hashing algorithm, and has a 160-bit
        internal effective key space equal to the range of SHA1.
     
        Algorithm 4.2.1 (HashStream): keyed cryptographic pseudo-random
        stream generator.
     
        Input:
     
        o  An integer b,
     
        o  A string p.
     
        Output:
     
        o  A string r of size b bytes.
     
        Method:
     
        1. Let K = SHA1(p).
     
        2. Let h_0 = 0x0000000000000000000000000000000000000000, a string of
        20 null bytes.
     
        3. Let l = Ceiling(b / 20).
     
        4. For each i in 1 to l do:
     
           (a) Let h_i = SHA1(h_(i - 1)).
     
           (b) Let r_i = SHA1(h_i || K), where h_i || K is the 40-byte
        concatenation of h_i and K.
     
        5. Let r = LeftmostBytes(b, r_1 || ... || r_l), i.e., r is formed as
        the concatenation of the r_i, truncated to the desired number of
        bytes.
     
     
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     4.3. Canonical encodings of extension field elements
     
        Canonical(p, k, o, v) takes an element v in F_p^2, and returns a
        canonical byte-string of fixed length representing v. The parameter o
        must be either 0 or 1, and specifies the ordering of the encoding.
     
        Algorithm 4.3.1 (Canonical): encodes elements of an extension field
        F_p^2 as strings.
     
        Input:
     
        o  An element v in F_p^2,
     
        o  A description of F_p^2 ,
     
        o  A ordering parameter o, either 0 or 1.
     
        Output:
     
        o  A fixed-length string s representing v.
     
        Method:
     
        1. For a type-1 curve, execute Algorithm 4.3.2 (Canonical1).
     
     4.3.1. Type-1 curve implementation
     
        Canonical1(p, o, v) takes an element v in F_p^2 and returns a
        canonical representation of v as a byte-string s of fixed size. The
        parameter o must be either 0 or 1, and specifies the ordering of the
        encoding.
     
        Algorithm 4.3.2 (Canonical1): canonically represents elements of an
        extension field F_p^2.
     
        Input:
     
        o  An element v in F_p^2,
     
        o  A description of p, where p is congruent to 3 modulo 4,
     
        o  A ordering parameter o, either 0 or 1.
     
        Output:
     
        o  A string s of size Ceiling(16 * lg(p)) bytes.
     
     
     
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        Method:
     
        1. Let l = 8 * Ceiling(lg(p)), the number of bytes needed to
        represent integers in Z_p.
     
        2. Let (a, b) = v, where (a, b) in (Z_p)^2 is the canonical
        representation of v in F_p^2 = F_p / (x^2 + 1) as a polynomial a +i *
        b with i^2 = -1.
     
        3. Let a_(256^l) be the big-endian zero-padded fixed-length byte-
        string representation of a in Z_p.
     
        4. Let b_(256^l) be the big-endian zero-padded fixed-length byte-
        string representation of b in Z_p.
     
        5. Depending on the choice of ordering o:
     
           (a) If o = 0, then let s = a_(256^l) || b_(256^l), which is the
        concatenation of a_(256^l) followed by b_(256^l).
     
           (b) If o = 1, then let s = b_(256^l) || a_(256^l), which is the
        concatenation of b_(256^l) followed by a_(256^l).
     
        6. The fixed-length encoding of v is output as the string s.
     
     4.4. Hashing onto a subgroup of an elliptic curve
     
        HashToPoint(E, p, q, id) takes an identity string id and the
        description of a subgroup of prime order q in E(F_p) or E(F_p^2) and
        returns a point Q_id of order q in E(F_p) or E(F_p^2).
     
        Algorithm 4.4.1 (HashToPoint): cryptographically hashes strings to
        points on elliptic curves.
     
        Input:
     
        o  A string id,
     
        o  A description of a subgroup of prime order q on a curve E/F_p.
     
        Output:
     
        o  A point Q_id = (x, y) of order q on E.
     
        Method:
     
        1. For a type-1 curve E, execute Algorithm 4.4.2 (HashToPoint1).
     
     
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     4.4.1. Type-1 curve implementation
     
        HashToPoint1(E, p, q, id) takes an identity string id and the
        description of a subgroup of order q in E(F_p) where E: y^2 = x^3 + 1
        with p congruent to 11 modulo 12, and returns a point Q_id of order q
        in E/F_p. This algorithm exploits the bijective mapping between the x
        and y coordinates of non-zero points on such supersingular curves.
     
        Algorithm 4.4.2 (HashToPoint1). Cryptographically hashes strings to
        points on type-1 curves.
     
        Input:
     
        o  A string id,
     
        o  A description of a subgroup of prime order q on a curve E/F_p: y^2
           = x^3 + 1 where p is congruent to 11 modulo 12.
     
        Output:
     
        o  A point Q_id of order q on E(F_p).
     
        Method:
     
        1. Let y = HashToRangen(p, id), using Algorithm 4.1.1 (HashToRange).
     
        2. Let x = (y^2 - 1)^(1/3) = (y^2 - 1)^((2 * p - 1) / 3).
     
        3. Let Q' = (x, y), a non-zero point in E(F_p).
     
        4. Let Q = [(p + 1) / q ]Q', a point of order q in E(F_p).
     
     4.5. Bilinear pairing
     
        Pairing(E, p, q, A, B) takes two points A and B, both of order q,
        and, in the type-1 case, returns the modified pairing e'(A, phi(B))
        in F_p^2 where A and B are both in E(F_p).
     
        Algorithm 4.5.1 (Pairing): computes the regular or modified Tate
        pairing depending on the curve type.
     
        Input:
     
        o  A description of an elliptic curve E/F_p such that E(F_p) and
           E(F_p^2) have a subgroup of order q,
     
        o  Two points A and B of order q in E(F_p) or E(F_p^2).
     
     
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        Output:
     
        o  An supersingular curves, the value of e'(A, B) in F_p^2 where A
           and B are both in E(F_p).
     
        Method:
     
        1. If E is a type-1 curve, execute Algorithm 4.5.2 (Pairing1).
     
     4.5.1. Type-1 curve implementation
     
        Algorithm 4.5.2 (Pairing1): computes the modified Tate pairing on
        type-1 curves.
     
        Input:
     
        o  A curve E/F_p: y^2 = x^3 + 1 where p is congruent to 11 modulo 12
           and E(F_p) has a subgroup of order q,
     
        o  Two points A and B of order q in E(F_p),
     
        Output:
     
        o  The value of e'(A, B) = e(A, phi(B)) in F_p^2.
     
        Method:
     
        1. Compute B' = phi(B), as follows:
     
           (a) Let (x, y) in F_p x F_p be the coordinates of B in E(F_p).
     
           (b) Let zeta = 1^(1/3) in F_p^2 , with zeta != 1. Specifically, as
        p is congruent to 3 modulo 4, and representing the elements of F_p^2
        = F_p[x] / (x^2 + 1) as polynomials a + bx with x = (-1)^(1/2), the
        representation of zeta = (a_zeta , b_zeta) is obtained as:
     
              i. Let a_zeta = (p - 1) / 2.
     
              ii. Let b_zeta = 3^((p + 1) / 4) (mod p).
     
           (c) Let x' =  x * x_zeta in F_p^2,
     
           (d) Let B' = (x', y) in F_p^2 x F_p.
     
        2. Compute the Tate pairing e(A, B') = e(A, phi(B)) in F_p^2 using
        the Miller method, as in Algorithm 4.5.1 (Tate) described in Section
        4.5.
     
     
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     4.6. Ratio of bilinear pairings
     
        PairingRatio(E, p, q, A, B, C, D) takes four points as input, and
        computes the ratio of the two bilinear pairings, Pairing(E, p, q, A,
        B) / Pairing(E, p, q, C, D), or, equivalently, the product,
        Pairing(E, p, q, A, B) * Pairing(E, p, q, C, -D).
     
        On type-1 curves, all four points are of order q in E(F_p), and the
        result is an element of order q in the extension field F_p^2 .
     
        The motivation for this algorithm is that the ratio of two pairings
        can be calculated more efficiently than by computing each pairing
        separately and dividing one into the other, since certain
        calculations that would normally appear in each of the two pairings
        can be combined and carried out at once. Such calculations include
        the repeated doublings in steps 2(a)i, 2(a)ii, 3(a)i, and 3(a)ii of
        Algorithm 4.5.2 (TateMillerSolinas), as well as the final
        exponentiation in step 6(a) of Algorithm 4.5.2 (TateMillerSolinas).
     
        Algorithm 4.6.1 (PairingRatio): computes the ratio of two regular or
        modified Tate pairings depending on the curve type.
     
        Input:
     
        o  A description of an elliptic curve E/F_p such that E(F_p) and
           E(F_p^2) have a subgroup of order q,
     
        o  Four points A, B, C, and D, of order q in E(F_p) or E(F_p^2).
     
        Output:
     
        o  On supersingular curves, the value of e'(A, B) / e'(C, D) in F_p^2
           where A, B, C, D are all in E(F_p);
     
        Method:
     
        1. If E is a type-1 curve, execute Algorithm 4.6.2 (PairingRatio1).
     
     4.6.1. Type-1 curve implementation
     
        Algorithm 4.6.2 (PairingRatio1). Computes the ratio of two modified
        Tate pairings on type-1 curves.
     
        Input:
     
        o  A curve E/F_p: y^2 = x^3 + 1, where p is congruent to 11 modulo 12
           and E(F_p) has a subgroup of order q,
     
     
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        o  Four points A, B, C, and D, of order q in E(F_p),
     
        Output:
     
        o  The value of e'(A, B) / e'(C, D) = e(A, phi(B)) / e(C, phi(D)) =
           e(A, phi(B)) * e(-C, phi(D)), in F_p^2.
     
        Method:
     
        1. The step-by-step description of the optimized algorithm is omitted
        in this normative specification.
     
        The correct result can always be obtained, albeit more slowly, by
        computing the product of pairings Pairing1(E, p, q, A, B) *
        Pairing1(E, p, q, -C, D) by using two invocations of Algorithm 4.5.2
        (Pairing1).
     
     5. The Boneh-Franklin BF cryptosystem
     
        This chapter describes the algorithms constituting the Boneh-Franklin
        identity-based cryptosystem as described in [BF].
     
     5.1. Setup
     
        Algorithm 5.1.1 (BFsetup): randomly selects a master secret and the
        associated public parameters.
     
        Input:
     
        o  A curve type t (currently required to be fixed to t = 1),
     
        o  A security parameter n (currently required to take values n >=
           1024).
     
        Output:
     
        o  A set of common public parameters,
     
        o  A corresponding master secret.
     
        Method:
     
        1. Depending on the selected type t:
     
           (a) If t = 1, then Algorithm 5.1.2 (BFsetup1) is executed.
     
     
     
     
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        2. The resulting master secret and public parameters are separately
        encoded as per the application protocol requirements.
     
     5.1.1. Type-1 curve implementation
     
        BFsetup1 takes a security parameter n as input. For type-1 curves,
        the scale of n corresponds to the modulus bit-size believed of
        comparable security in the classical Diffie-Hellman or RSA public-key
        cryptosystems. For this implementation, the allowed value of n is
        limited to 1024, which corresponds to 80 bits of symmetric key
        security.
     
        Algorithm 5.1.2 (BFsetup1): randomly establishes a master secret and
        public parameters for type-1 curves.
     
        Input:
     
        o  A security parameter n, assumed to be equal to 1024.
     
        Output:
     
        o  A set of common public parameters (t, p, q, P, Ppub),
     
        o  A corresponding master secret s.
     
        Method:
     
        1. Determine the subordinate security parameters n_p and n_q as
        follows:
     
           (a) Let n_p = 512, which will determine the size of the field F_p.
     
           (b) Let n_q = 160, which will determine the size of the subgroup
        order q.
     
        2. Construct the elliptic curve and its subgroup of interest, as
        follows:
     
           (a) Select an arbitrary n_q-bit prime q, i.e., such that
        Ceiling(lg(q)) = n_q. For better performance, q is chosen as a
        Solinas prime, i.e., a prime of the form q = 2^a +/- 2^b +/- 1 where
        0 < b < a.
     
           (b) Select a random integer r such that p = 12 * r * q - 1 is an
        n_p-bit prime, i.e., such that Floor(lg(p)) = n_p.
     
        3. Select a point P of order q in E(F_p), as follows:
     
     
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           (a) Select a random point P' of coordinates (x', y') on the curve
        E/F_p: y^2 = x^3 + 1 (mod p).
     
           (b) Let P = [12 * r]P'.
     
           (c) If P = 0, then start over in step 3a.
     
        4. Determine the master secret and the public parameters as follows:
     
           (a) Select a random integer s in the range 2 to q - 1.
     
           (b) Let P_pub = [s]P.
     
        5. (t, E, p, q, P, P_pub) are the common public parameters, where E:
        y^2 = x^3 + 1.
     
        6. The integer s is the master secret.
     
     5.2. Public key derivation
     
        BFderivePubl takes an identity string id and a set of public
        parameters, and returns a point Q_id.
     
        Algorithm 5.2.1 (BFderivePubl): derives the public key corresponding
        to an identity string.
     
        Input:
     
        o  An identity string id,
     
        o  A set of common public parameters (t, E, p, q, P, P_pub).
     
        Output:
     
        o  A point Q_id of order q in E(F_p) or E(F_p^2).
     
        Method:
     
        1. Q_id = HashToPoint(E, p, q, id), using Algorithm 4.4.1
        (HashToPoint).
     
     5.3. Private key extraction
     
        BFextractPriv takes an identity string id, and a set of public
        parameters and corresponding master secret, and returns a point S_id.
     
     
     
     
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        Algorithm 5.3.1 (BFextractPriv): extracts the private key
        corresponding to an identity string.
     
        Input:
     
        o  An identity string id,
     
        o  A set of common public parameters (t, E, p, q, P, P_pub).
     
        Output:
     
        o  A point S_id or order q in E(F_p).
     
        Method:
     
        1. Let Q_id = HashToPoint(E, p, q, id) using Algorithm 4.4.1
        (HashToPoint).
     
        2. Let S_id = [s]Q_id.
     
     5.4. Encryption
     
        BFencrypt takes three inputs: a public parameter block, an identity
        id, and a plaintext m. The plaintext is intended to be a symmetric
        session key, although variable-sized short messages are allowed.
     
        Algorithm 5.4.1 (BFencrypt): encrypts a short message or session key
        for an identity string.
     
        Input:
     
        o  A plaintext string m of size |m| bytes,
     
        o  A recipient identity string id,
     
        o  A set of public parameters.
     
        Output:
     
        o  A ciphertext tuple (U, V, W) in E(F_p) x {0, ... , 255}^20 x {0,
           ... , 255}^|m|.
     
        Method:
     
        1. Let the public parameter set be comprised of a prime p, a curve
        E/F_p, the order q of a large prime subgroup of E(F_p), and two
        points P and P_pub of order q in E(F_p).
     
     
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        2. Q_id = HashToPoint(E, p, q, id), using Algorithm 4.4.1
        (HashToPoint), which results in a point of order q in E(F_p) or
        E(F_p^2).
     
        3. Select s random 160-bit vector rho, represented as 20-byte string
        in big-endian convention.
     
        4. Let t = SHA1(m), a 20-byte string resulting from the SHA1
        algorithm.
     
        5. Let l = HashToRangeq(rho || t), an integer in the range 0 to q - 1
        resulting from applying Algorithm 4.1.1 (HashToRange) to the 40-byte
        concatenation of rho and t.
     
        6. Let U = [l]P, which is a point of order q in E(F_p).
     
        7. Let Theta = Pairing(E, p, q, P_pub, Q_id), which is an element of
        the extension field F_p^2 obtained using the modified Tate pairing of
        Algorithm 4.5.1 (Pairing).
     
        8. Let theta' = theta^l, which is theta raised to the power of l in
        F_p^2.
     
        9. Let z = Canonical(p, k, 0, theta'), using Algorithm 4.3.1
        (Canonical), the result of which is a canonical string representation
        of theta'.
     
        10. Let w = SHA1(z) using the SHA1 hashing algorithm, the result of
        which is a 20-byte string.
     
        11. Let V = w XOR rho, which is the 20-byte long bit-wise exclusive-
        OR of w and rho.
     
        12. Let W = HashStream(|m|, rho XOR m), which is the bit-wise
        exclusive-OR of m with the first |m| bytes of the pseudo-random
        stream produced by Algorithm 4.2.1 (HashStream) with seed rho.
     
        13. The ciphertext is the triple (U, V, W).
     
     5.5. Decryption
     
        BFdecrypt takes three inputs: a public parameter block, a private key
        block key, and a ciphertext parsed as (U' , V' , W').
     
        Algorithm 5.5.1 (BFdecrypt): decrypts a short message or session key
        using a private key.
     
     
     
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        Input:
     
        o  A private key point S_id of order q in E(F_p),
     
        o  A ciphertext triple (U', V', W') in E(F_p) x {0, . . . , 255}^20 x
           {0, . . . , 255}*.
     
        o  A set of public parameters.
     
        Output:
     
        o  A decrypted plaintext m', or an invalid ciphertext flag.
     
        Method:
     
        1. Let the public parameter set be comprised of a prime p, a curve
        E/F_p, the order q of a large prime subgroup of E(F_p), and two
        points P and P_pub of order q in E(F_p).
     
        2. Let theta' = Pairing(E, p ,q, U', S_id) by applying the modified
        Tate pairing of Algorithm 4.5.1 (Pairing).
     
        3. Let z = Canonical(p, k, 0, theta') using Algorithm 4.3.1
        (Canonical), the result of which is a canonical string representation
        of theta'.
     
        4. Let w' = SHA1(z), using the SHA1 hashing algorithm, the result of
        which is a 20-byte string.
     
        5. Let rho = w XOR V, the bit-wise XOR of w and V.
     
        6. Let m = HashStream(|W|, rho) XOR W, which is the bit-wise
        exclusive-OR of m with the first |W| bytes of the pseudo-random
        stream produced by Algorithm 4.2.1 (HashStream) with seed rho.
     
        7. Let t = SHA1(m) using the SHA1 algorithm.
     
        8. Let l = HashToRange(q, rho || t) using Algorithm 4.1.1
        (HashToRange) on the 40-byte concatenation of rho and t.
     
        9. Verify that U' = [l]P:
     
           (a) If this is the case, then the decrypted plaintext m is
        returned.
     
           (b) Otherwise, the ciphertext is rejected and no plaintext is
        returned.
     
     
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     6. Wrapper methods for the BF system
     
        This chapter describes a number of wrapper methods providing the
        identity-based cryptosystem functionalities using concrete encodings.
        The following functions are presently given based on the Boneh-
        Franklin algorithms.
     
     6.1. Private key generator (PKG) setup
     
        Algorithm 6.1.1 (BFwrapperPKGSetup): randomly selects a PKG master
        secret and a set of public parameters.
     
        Input:
     
        o  A curve type t,
     
        o  A security parameter n.
     
        Output:
     
        o  A common public parameter block pi,
     
        o  A corresponding master secret block sigma.
     
        Method:
     
        1. Perform Algorithm 5.1.1 (BFsetup) on parameters t and n, producing
        a public parameter set and a master secret.
     
        2. Apply Algorithm 7.2.1 (BFencodeParams) on the public parameter set
        obtained in step 1 to get the public parameter block pi.
     
        3. Apply Algorithm 7.3.1 (BFencodeMaster) on the master secret
        obtained in step 1 to get the master secret block sigma.
     
     6.2. Private key extraction by the PKG
     
        Algorithm 6.2.1 (BFwrapperPrivateKeyExtract): extraction by the PKG
        of a private key corresponding to an identity.
     
        Input:
     
        o  A master secret block sigma,
     
        o  A corresponding public parameter block pi,
     
        o  An identity string id.
     
     
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        Output:
     
        o  A private key block kappa_id
     
        Method:
     
        1. Apply Algorithm 7.2.2 (BFdecodeParams) to the public parameter
        block pi to obtain the public parameters, comprising a prime p, a
        curve E/F_p, the order q of a large prime subgroup of E(F_p), and two
        points P and P_pub of order q in E(F_p).
     
        2. Apply Algorithm 7.3.2 (BFdecodeMaster) on the master secret block
        sigma to obtain the master secret s.
     
        3. Perform Algorithm 5.3.1 (BFextractPriv) on the identity id, using
        the decoded parameters and secret, to produce a private key point
        S_id.
     
        4. Apply Algorithm 7.4.1 (BFencodePrivate) to S_id to produce a
        private key block kid.
     
     6.3. Session key encryption
     
        Algorithm 6.3.1 (BFwrapperSessionKeyEncrypt): encrypts a short
        message or session key for an identity.
     
        Input:
     
        o  A public parameter block pi,
     
        o  A recipient identity string id,
     
        o  A plaintext string m (possibly comprising the concatenation of a
           pair of random session keys for symmetric encryption and message
           authentication purposes on a larger plaintext).
     
        Output:
     
        o  A ciphertext block
     
        Method:
     
        1. Apply Algorithm 7.2.2 (BFdecodeParams) on the public parameter
        block pi to obtain a set of public parameters, comprising a prime p,
        a curve E/F_p, the order q of a large prime subgroup of E(F_p), and
        two points P and P_pub of order q in E(F_p).
     
     
     
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        2. Perform Algorithm 5.4.1 (BFencrypt) on the plaintext m for
        identity id using the decoded set of parameters, to obtain a
        ciphertext tuple (U, V, W).
     
        3. Apply Algorithm 7.5.1 (BFencodeCiphertext) on (U, V, W) to obtain
        a serialized ciphertext string
     
        Algorithm 6.3.2 (BFwrapperSessionKeyDecrypt): decrypts a short
        message or session key using a private key.
     
        Input:
     
        o  A public parameter block pi,
     
        o  A private key block kappa,
     
        o  A ciphertext block gamma.
     
        Output:
     
        o  A decrypted plaintext string m, or an error flag signaling an
           invalid ciphertext.
     
        Method:
     
        1. Apply Algorithm 7.2.2 (BFdecodeParams) on the public parameter
        block pi to obtain the public parameters, comprising a prime p, a
        curve E/F_p, the order q of a large prime subgroup of E(F_p), and two
        points P and P_pub of order q in E(F_p).
     
        2. Apply Algorithm 7.4.2 (BFdecodePrivate) to kappa to obtain a
        private key point S_id.
     
        3. Apply Algorithm 7.5.2 (BFdecodeCiphertext) to gamma to obtain a
        ciphertext triple (U', V', W').
     
        4. Perform Algorithm 5.5.1 (BFdecrypt) on (U', V', W') using the
        private key S_id and the decoded set of public parameters, to obtain
        decrypted plaintext m, or an invalid ciphertext flag.
     
           (a) If the decryption was successful, return the plaintext m.
     
           (b) Otherwise, raise an error condition.
     
     
     
     
     
     
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     7. Concrete encoding guidelines for BF
     
        This section specifies a set of concrete encoding schemes for the
        inputs and outputs of the previously described algorithms. ASN.1
        encodings are specified in Section 11 of this document.
     
     7.1. Encoding of points on a curve
     
        Algorithm 7.1.1 (EncodePoint): encodes a point in E(F_p) in an
        exportable format.
     
        Input:
     
        o  A non-zero point Q in E(F_p).
     
        Output:
     
        o  A fixed-length (for given p) byte-string encoding of Q.
     
        Method:
     
        1. Let (x, y) in F_p x F_p be the coordinates of P, where (x, y)
        satisfy the equation of E.
     
        2. The point P is then encoded as a FpPoint using the ASN.1 rules
        given in the ASN.1 module given in Section 11 of this document.
     
        Algorithm 6.1.2 (DecodePoint): decodes a point in E(F_p) from an
        exportable format.
     
        Input:
     
        o  A byte-string encoding of a non-zero point Q in E(F_p).
     
        Output:
     
        o  Q = (x, y).
     
        Method:
     
        1. The string is parsed and decoded as a pair (x, y), where x and y
        are integers in Z_p.
     
        2. Q is reconstructed as (x, y).
     
     
     
     
     
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     7.2. Public parameters blocks
     
        Algorithm 7.2.1 (BFencodeParams): encodes a BF public parameter set
        in an exportable format.
     
        Input:
     
        o  A set of public parameters (t, E, p, q, P, P_pub).
     
        Output:
     
        o  A public parameter block pi, represented as a byte string.
     
        Method:
     
        1. Separate encodings for E, p, q, P, P_pub are obtained as follows:
     
           (a) If t = 1, execute Algorithm 7.2.3 (BFencodeParams1).
     
        2. The separate encodings as well as a type indicator flag for t are
        then serialized in any suitable manner as dictated by the
        application.
     
        Algorithm 7.2.2 (BFdecodeParams): imports a BF public parameter block
        from a serialized format.
     
        Input:
     
        o  A public parameter block pi, represented as a byte string.
     
        Output:
     
        o  A set of public parameters (t, E, p, q, P, P_pub).
     
        Method:
     
        1. Identify from the appropriate flag the type t of curve upon which
        the parameter block is based.
     
        2. Then:
     
           (a) If t = 1, execute Algorithm 7.2.4 (BFdecodeParams1).
     
     7.2.1. Type-1 implementation
     
        Algorithm 7.2.3 (BFencodeParams1): encodes a BF type-1 public
        parameter set in an exportable format.
     
     
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        Input:
     
        o  A set of public parameters (t, E, p, q, P, P_pub) with t = 1.
     
        Output:
     
        o  Separate encodings for each of the E, p, q, P, P_pub components.
     
        Method:
     
        1. E: y^2 = x^3 + a * x + b is represented as a constant string, such
        as the empty string, since a and b are invariant for type-1 curves.
     
        2. p = 12 * r * q - 1 is represented as the smaller integer r,
        encoded, e.g., using a big-endian byte-string representation.
     
        3. q = 2^a + s * 2^b + c, where a, b are small and c and s are either
        1 or -1, is compactly represented as the 4-tuple (a, b, c, s).
     
        4. P = (x_P , y_P ) in F_p x F_p is represented using the point
        compression technique of Algorithm 7.1.1 (EncodePoint).
     
        5. P_pub is similarly encoded using Algorithm 7.1.1 (EncodePoint).
     
        Algorithm 7.2.4 (BFdecodeParams1): decodes the components of a BF
        type-1 public parameter block.
     
        Input:
     
        o  Separate encodings for each one of E, p, q, P, P_pub.
     
        Output:
     
        o  A set of public parameters (t, E, p, q, P, P_pub) with t = 1.
     
        Method:
     
        1. The equation of E is set to E = E : y^2 = x^3 + 1, as is always
        the case for type-1 curves. The actual encoding of E is ignored.
     
        2. The encoding of q is parsed as (a, b, c, s), and its value set to
        q = 2^a + s * 2^b + c.
     
        3. The encoding of p is parsed as the integer r, from which p is
        given by p = 12 * r * q - 1.
     
     
     
     
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        4. P is reconstructed from its encoding (x, y') using the point
        decompression technique of Algorithm 7.1.2 (DecodePoint).
     
        5. P_pub is similarly reconstructed from its encoding using Algorithm
        7.1.2 (DecodePoint).
     
     7.3. Master secret blocks
     
        Algorithm 7.3.1 (BFencodeMaster): encodes a BF master secret in an
        exportable format.
     
        Input:
     
        o  A master secret integer s between 2 and q - 1.
     
        Output:
     
        o  A master secret block sigma, represented as a byte string.
     
        Method:
     
        1. Sigma is constructed as the unsigned big-endian byte-string
        encoding of s of length 8 * Ceiling(lg(p)).
     
        Algorithm 7.3.2 (BFdecodeMaster): decodes a BF master secret from a
        block in exportable format.
     
        Input:
     
        o  A master secret block sigma, represented as a byte string.
     
        Output:
     
        o  A master secret integer s in between 2 and q - 1 .
     
        Method:
     
        1. Let s = Value(sigma), where sigma is interpreted in the unsigned
        big endian convention.
     
     7.4. Private key blocks
     
        Algorithm 7.4.1 (BFencodePrivate): encodes a BF private key point in
        an exportable format.
     
        Input:
     
     
     
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        o  A private key point S_id in E(F_p).
     
        Output:
     
        o  A private key block kappa, represented as a byte string.
     
        Method:
     
        1. kappa is obtained by applying Algorithm 7.1.1 (EncodePoint) to
        S_id.
     
        Algorithm 7.4.2 (BFdecodePrivate): decodes a BF private key point
        from an exportable format.
     
        Input:
     
        o  A private key block kappa, represented as a byte string.
     
        Output:
     
        o  A private key point S_id in E(F_p).
     
        Method:
     
        1. Kappa is parsed and decoded into a point S_id in E(F_p) using
        Algorithm 7.1.2 (DecodePoint).
     
     7.5. Ciphertext blocks
     
        Algorithm 7.5.1 (BFencodeCiphertext): encodes a BF ciphertext tuple
        in an exportable format.
     
        Input:
     
        o  A ciphertext tuple (U, V, W) in E(F_p) x {0, . . . , 255}^20 x {0,
           . . . , 255}*.
     
        Output:
     
        o  A ciphertext block gamma, represented as a byte string.
     
        Method:
     
        1. U = (x, y) is first encoded as a fixed-length string using
        Algorithm 7.1.1 (EncodePoint).
     
     
     
     
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        2.  The output gamma is obtained as the encoding of U, concatenated
        with the fixed-length string V, and the variable length string W,
        both already in byte-string format.
     
        Algorithm 7.5.2 (BFdecodeCiphertext): decodes a BF ciphertext tuple
        from an exportable format.
     
        Input:
     
        o  A ciphertext block gamma, represented as a byte string.
     
        Output:
     
        o  A ciphertext tuple (U, V, W) in E(F_p) x {0, . . . , 255}^20 x {0,
           . . . , 255}*.
     
        Method:
     
        1. The ciphertext block gamma is parsed as a 3-tuple comprising a
        fixed-length encoding of U, followed by a 20-byte string V, followed
        by an arbitrary-length string W.
     
        2. U in E(F_p) is then recovered by applying Algorithm 7.1.2
        (DecodePoint) on its encoding.
     
     8. The Boneh-Boyen BB1 cryptosystem
     
        This chapter describes the algorithms constituting the first of the
        two Boneh-Boyen identity-based cryptosystems proposed in [BB1]. The
        description follows the practical implementation given in [BB1].
     
     8.1. Setup
     
        Algorithm 8.1.1 (BBsetup). Randomly selects a set of master secrets
        and the associated public parameters.
     
        Input:
     
        o  A curve type t (currently required to be fixed to t = 1),
     
        o  A security parameter n (currently required to take values n >=
           1024).
     
        Output:
     
        o  A set of common public parameters,
     
     
     
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        o  A corresponding master secret.
     
        Method:
     
        1. Depending on the selected type t:
     
        (a) If t = 1, then Algorithm 8.1.2 (BBsetup1) is executed.
     
        2. The resulting master secret and public parameters are separately
        encoded as per the application protocol requirements.
     
     8.1.1. Type-1 curve implementation
     
        BBsetup1 takes a security parameter n as input. For type-1 curves,
        the scale of n corresponds to the modulus bit-size believed of
        comparable security in the classical Diffie-Hellman or RSA public-key
        cryptosystems. For this implementation, allowed values of n are
        limited to 1024, 2048, and 3072, which correspond to the equivalent
        security level ranging from 80-, 112- and 128-bit symmetric keys
        respectively.
     
        Algorithm 8.1.2 (BBsetup1): randomly establishes a master secret and
        public parameters for type-1 curves.
     
        Input:
     
        o  A security parameter n, either 1024, 2048 or 3072.
     
        Output:
     
        o  A set of common public parameters (t, k, E, p, q, P, P_1, P_2,
           P_3, v),
     
        o  A corresponding triple of master secrets (alpha, beta, gamma).
     
        Method:
     
        1. Determine the subordinate security parameters n_p and n_q as
        follows:
     
           (a) Let n_p = n / 2, which will determine the size of the field
        F_p.
     
           (b) If n = 1024, n_q = 160; if n = 2048, n_q = 224; if n = 3072,
        n_q = 256, which will determine the size of the subgroup order q.
     
     
     
     
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        2. Construct the elliptic curve and its subgroup of interest as
        follows:
     
           (a) Select an arbitrary n_q-bit prime q, i.e., such that
        Ceiling(lg(p)) = n_q. For better performance, q is chosen as a
        Solinas prime, i.e., a prime of the form q = 2^a +/- 2^b +/- 1 where
        0 < b < a.
     
           (b) Select a random integer r such that p = 12 * r * q - 1 is an
        n_p-bit prime, i.e., such that Ceiling(lg(p)) = n_p.
     
        3. Select a point P of order q in E(F_p), as follows:
     
           (a) Select a random point P' of coordinates (x', y') on the curve
        E/F_p: y2 = x3 + 1 (mod p).
     
           (b) Let P = [12 * r]P'.
     
           (c) If P = 1, then start over in step 3a.
     
        4. Determine the master secret and the public parameters as follows:
     
           (a) Select three random integers alpha, beta, gamma, each of them
        in the range 1 to q - 1.
     
           (b) Let P_1 = [alpha]P.
     
           (c) Let P_2 = [beta]P.
     
           (d) Let P_3 = [gamma]P.
     
           (e) Let v = Pairing(E, p, q, P_1, P_2), which is an element of the
        extension field F_p^2 obtained using the modified Tate pairing of
        Algorithm 3.5.1 (Pairing).
     
        5. (t, k, E, p, q, P, P_1, P_2, P_3, v) are the common public
        parameters, where t = 1, k = 2, and E: y^2 = x^3 + 1.
     
        6. (alpha, beta, gamma) constitute the master secret.
     
     8.2. Public key derivation
     
        takes an identity string id and a set of public parameters, and
        returns an integer h_id.
     
        Algorithm 8.2.1 (BBderivePubl): derives the public key corresponding
        to an identity string.
     
     
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        Input:
     
        o  An identity string id,
     
        o  A set of common public parameters (t, k, E, p, q, P, P_1, P_2,
           P_3, v).
     
        Output:
     
        o  An integer h_id modulo q.
     
        Method:
     
        1. Let h_id = HashToRangeq(id), using Algorithm 3.1.1 (HashToRange).
     
     8.3. Private key extraction
     
        BBextractPriv takes an identity string id, and a set of public
        parameters and corresponding master secrets, and returns a private
        key consisting of two points D_0 and D_1.
     
        Algorithm 8.3.1 (BBextractPriv): extracts the private key
        corresponding to an identity string.
     
        Input:
     
        o  An identity string id,
     
        o  A set of common public parameters (t, k, E, p, q, P, P_1, P_2,
           P_3, v).
     
        Output:
     
        o  A pair of points (D_0, D_1), each of which has order q in E(F_p).
     
        Method:
     
        1. Select a random integer r in the range 1 to q - 1.
     
        2. Calculate the point D_0 as follows:
     
           (a) Let hid = HashToRange(q, id), using Algorithm 3.1.1
        (HashToRange).
     
           (b) Let y = alpha * beta + r * (alpha * h_id * gamma) in F_q.
     
           (c) Let D_0 = [y]P.
     
     
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        3. Calculate the point D_1 as follows:
     
           (a) Let D_1 = [r]P.
     
        4. The pair of points (D_0, D_1) constitutes the private key for id.
     
     8.4. Encryption
     
        BBencrypt takes three inputs: a set of public parameters, an identity
        id, and a plaintext m. The plaintext is intended to be a short random
        session key, although messages of arbitrary size are in principle
        allowed.
     
        Algorithm 8.4.1 (BBencrypt): encrypts a short message or session key
        for an identity string.
     
        Input:
     
        o  A plaintext string m of size |m| bytes,
     
        o  A recipient identity string id,
     
        o  A set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v).
     
        Output:
     
        o  A ciphertext tuple (u, C_0, C_1, y) in F_q x E(F_p) x E(F_p) x {0,
           . . . , 255}^|m|.
     
        Method:
     
        1. Let the public parameter set be comprised of a prime p, a curve
        E/F_p, the order q of a large prime subgroup of E(F_p), four points
        P, P_1, P_2, P_3, of order q in E(F_p), and an extension field
        element v of order q in F_p^2 .
     
        2. Select a random integer s in the range 1 to q - 1.
     
        3. Let w = v^s, which is v raised to the power of s in F_p^2, the
        result is an element of order q in F_p^2 .
     
        4. Calculate the point C_0 as follows:
     
           (a) Let C_0 = [s]P.
     
        5. Calculate the point C_1 as follows:
     
     
     
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           (a) Let _hid = HashToRangeq(id), using Algorithm 3.1.1
        (HashToRange).
     
        (b) Let y = s * h_id in F_q.
     
        (c) Let C_1 = [y]P_1 + [s]P_3.
     
        6. Obtain canonical string representations of certain elements:
     
           (a) Let psi = Canonical(p, k, 1, w) using Algorithm 3.3.1
        (Canonical), the result of which is a canonical byte-string
        representation of w.
     
           (b) Let l = Ceiling(8 * lg(p)), the number of bytes needed to
        represent integers in F_p, and represent each of these F_p elements
        as a big-endian zero-padded byte-string of fixed length l:
     
           (x_0)_(256^l) to represent the x coordinate of C_0.
     
           (y_0)_(256^l) to represent the y coordinate of C_0.
     
           (x_1)_(256^l) to represent the x coordinate of C_1.
     
           (y_1)_(256^l) to represent the y coordinate of C_1.
     
        7. Encrypt the message m into the string y as follows:
     
           (a) Compute an encryption key h_0 as a dual-pass hash of w via its
        representation psi:
     
              i. Let zeta = SHA1(psi), using the SHA1 hashing algorithm; the
        result is a 20-byte string.
     
              ii. Let xi = SHA1(zeta || psi), using the SHA1 hashing
        algorithm; the result is a 20-byte string.
     
              iii. Let h' = xi || zeta, the 40-byte concatenation of the
        previous two SHA1 outputs.
     
           (b) Let y = HashStream(|m|, h') XOR m, which is the bit-wise
        exclusive-OR of m with the first |m| bytes of the pseudo-random
        stream produced by Algorithm 3.2.1 (HashStream) with seed h'.
     
        8. Create the integrity check tag u as follows:
     
           (a) Compute a one-time pad h'' as a dual-pass hash of the
        representation of (w, C_0, C_1, y):
     
     
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              i. Let sigma = (y_1)_(256^l) || (x_1)_(256^l) || (y_0)_(256^l)
        || (x_0)_(256^l) || y || psi be the concatenation of y and the five
        indicated strings in the specified order.
     
              ii. Let eta = SHA1(sigma), using the SHA1 hashing algorithm to
        get a 20-byte string.
     
              iii. Let mu = SHA1(eta || sigma), using the SHA1 hashing
        algorithm to get a 20-byte string.
     
              iv. Let h'' = mu || eta, the 40-byte concatenation of the
        previous two SHA1 outputs.
     
           (b) Build the tag u as the encryption of the integer s with the
        one-time pad h'':
     
              i. Let rho = HashToRangeq(h'') to get an integer in Z_q.
     
              ii. Let u = s + rho (mod q).
     
        9. The complete ciphertext is given by the quadruple (u, C_0, C_1,
        y).
     
     8.5. Decryption
     
        BBdecrypt takes three inputs: a set of public parameters, a private
        key (D_0, D_1), and a ciphertext parsed as (u, C_0, C_1, y). It
        outputs a message m, or signals an error if the ciphertext is invalid
        for the given key.
     
        Algorithm 7.5.1 (BBdecrypt): decrypts a short message or session key
        using a private key.
     
        Input:
     
        o  A private key given as a pair of points (D_0, D_1) of order q in
           E(F_p),
     
        o  A ciphertext quadruple (u, C_0, C_1, y) in Z_q x E(F_p) x E(F_p) x
           {0, . . . , 255}*.
     
        o  A set of public parameters.
     
        Output:
     
        o  A decrypted plaintext m, or an invalid ciphertext flag.
     
     
     
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        Method:
     
        1. Let the public parameter set be comprised of a prime p, a curve
        E/F_p, the order q of a large prime subgroup of E(F_p), four points
        P, P_1, P_2, P_3, of order q in E(F_p), and an extension field
        element v of order q in F_p^2 .
     
        2. Let w = PairingRatio(E, p, q, C_0, D_0, C_1, D_1), which computes
        the ratio of two Tate pairings (modified, for type-1 curves) as
        specified in Algorithm 4.6.1 (PairingRatio).
     
        3. Obtain canonical string representations of certain elements:
     
           (a) Let psi = Canonical(p, k, 1, w), using Algorithm 4.3.1
        (Canonical); the result is a canonical byte-string representation of
        w.
     
           (b) Let l = Ceiling(8 * lg(p)), the number of bytes needed to
        represent integers in F_p, and represent each of these F_p elements
        as a big-endian zero-padded byte-string of fixed length l:
     
           (x_0)_(256^l) to represent the x coordinate of C_0.
     
           (y_0)_(256^l) to represent the y coordinate of C_0.
     
           (x_1)_(256^l) to represent the x coordinate of C_1.
     
           (y_1)_(256^l) to represent the y coordinate of C_1.
     
        4. Decrypt the message m from the string y as follows:
     
           (a) Compute the decryption key h' as a dual-pass hash of w via its
        representation psi:
     
              i. Let zeta = SHA1(psi), using the SHA1 hashing algorithm to
        get a 20-byte string.
     
              ii. Let xi = SHA1(zeta || psi), using the SHA1 hashing
        algorithm to get a 20-byte string.
     
              iii. Let h' = xi || zeta, the 40-byte concatenation of the
        previous two SHA1 outputs.
     
           (b) Let m = HashStream(|y|, h')_XOR y, which is the bit-wise
        exclusive-OR of y with the first |y| bytes of the pseudo-random
        stream produced by Algorithm 3.2.1 (HashStream) with seed h'.
     
     
     
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        5. Obtain the integrity check tag u as follows:
     
           (a) Recover the one-time pad h'' as a dual-pass hash of the
        representation of (w, C_0, C_1, y):
     
              i. Let sigma = (y_1)_(256^l) || (x_1)_(256^l) || (y_0)_(256^l)
        || (x_0)_(256^l) || y || psi be the concatenation of y and the five
        indicated strings in the specified order.
     
              ii. Let eta = SHA1(sigma) using the SHA1 hashing algorithm to
        get a 20-byte string.
     
              iii. Let mu = SHA1(eta || sigma), using the SHA1 hashing
        algorithm to get a 20-byte string.
     
              iv. Let h'' = mu || eta, the 40-byte concatenation of the
        previous two SHA1 outputs.
     
           (b) Unblind the encryption randomization integer s from the tag u
        using h'':
     
              i. Let rho = HashToRangeq(h'') to get an integer in Z_q.
     
              ii. Let s = u - rho (mod q).
     
        6. Verify the ciphertext consistency according to the decrypted
        values:
     
           (a) Test whether the equality w = v^s holds in F_p^2 .
     
           (b) Test whether the equality C_0 = [s]P holds in E(F_p).
     
        7. Adjudication and final output:
     
           (a) If either of the tests performed in step 6 fails, the
        ciphertext is rejected, and no decryption is output.
     
           (b) Otherwise, i.e., when both tests performed in step 6 succeed,
        the decrypted message is output.
     
     9. Wrapper methods for the BB1 system
     
        This section describes a number of wrapper methods providing the
        identity-based cryptosystem functionalities using concrete encodings.
        The following functions are presently given based on the Boneh-
        Franklin algorithms.
     
     
     
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     9.1. Private key generator (PKG) setup
     
        Algorithm 9.1.1 (BBwrapperPKGSetup): randomly selects a PKG master
        secret and a set of public parameters.
     
        Input:
     
        o  A curve type t,
     
        o  A security parameter n.
     
        Output:
     
        o  A common public parameter block pi,
     
        o  A corresponding master secret block sigma.
     
        Method:
     
        1. Perform Algorithm 8.1.1 (BBsetup) on parameters t and n, producing
        a set of public parameters and master secret.
     
        2. Apply Algorithm 10.2.1 (BBencodeParams) on the public parameters
        obtained in step 1 to get the public parameter block pi.
     
        3. Apply Algorithm 10.3.1 (BBencodeMaster) on the master secrets
        obtained in step 1 to get the master secret block sigma.
     
     9.2. Private key extraction by the PKG
     
        Algorithm 9.2.1 (BBwrapperPrivateKeyExtract): extraction by the PKG
        of a private key corresponding to an identity.
     
        Input:
     
        o  A master secret block sigma,
     
        o  A corresponding public parameter block pi,
     
        o  An identity string id.
     
        Output:
     
        o  A private key block kappa_id.
     
        Method:
     
     
     
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        1. Apply Algorithm 10.2.2 (BBdecodeParams) on the public parameter
        block pi to obtain the public parameters, comprising a prime p, the
        parameters of a curve E/F_p with embedding degree k = 2, the order q
        of a large prime subgroup of E(F_p), four points P, P_1, P_2, P_3, of
        order q in E(F_p), and an element v of order q in the extension field
        F_p^k of degree k.
     
        2. Apply Algorithm 10.3.2 (BBdecodeMaster) on the master secret block
        sigma to obtain the master secret (alpha, beta, gamma).
     
        3. Perform Algorithm 8.3.1 (BBextractPriv) on the identity id, using
        the decoded public parameters and master secret, to produce a private
        key (D_0, D_1).
     
        4. Apply Algorithm 10.4.1 (BBencodePrivate) on the private key to
        produce a private key block kappa_id.
     
     9.3. Session key encryption
     
        Algorithm 9.3.1 (BBwrapperSessionKeyEncrypt): encrypts a short
        message or session key for an identity.
     
        Input:
     
        o  A public parameter block pi,
     
        o  A recipient identity string id,
     
        o  A plaintext string m (possibly comprising the concatenation of a
           pair of random session keys for symmetric encryption and message
           authentication purposes on a larger plaintext).
     
        Output:
     
        o  A ciphertext block omega.
     
        Method:
     
        1. Apply Algorithm 10.2.2 (BBdecodeParams) on the public parameter
        block pi to obtain the public parameters, comprising a prime p, the
        parameters of a curve E/F_p with embedding degree k = 2, the order q
        of a large prime subgroup of E(F_p), four points P, P_1, P_2, P_3, of
        order q in E(F_p), and an element v of order q in the extension field
        F_p^k .
     
     
     
     
     
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        2. Perform Algorithm 8.4.1 (BBencrypt) on the plaintext m for
        identity id using the decoded set of parameters, to obtain a
        ciphertext quadruple (u, C_0, C_1, y).
     
        3. Apply Algorithm 10.5.1 (BBencodeCiphertext) on the ciphertext (u,
        C_0, C_1, y) to obtain a string representation of omega.
     
        Algorithm 9.3.2 (BBwrapperSessionKeyDecrypt): decrypts a short
        message or session key using a private key.
     
        Input:
     
        o  A public parameter block pi,
     
        o  A private key block kappa,
     
        o  A ciphertext block omega.
     
        Output:
     
        o  A decrypted plaintext string m, or an error flag signaling an
           invalid ciphertext.
     
        Method:
     
        1. Apply Algorithm 10.2.2 (BBdecodeParams) on the public parameter
        block pi to obtain the public parameters, comprising a prime p, the
        parameters of a curve E/F_p with embedding degree k = 2, the order q
        of a large prime subgroup of E(F_p), four points P, P_1, P_2, P_3, of
        order q in E(F_p), and an element v of order q in the extension field
        F_p^2.
     
        2. Apply Algorithm 10.4.2 (BBdecodePrivate) on kappa to obtain the
        private key points (D_0, D_1).
     
        3. Apply Algorithm 10.5.2 (BBdecodeCiphertext) on omega to obtain a
        ciphertext quadruple (u, C_0, C_1, y).
     
        4. Perform Algorithm 8.5.1 (BBdecrypt) on (u, C_0, C_1, y) using the
        private key (D_0, D_1) and the decoded set of public parameters, to
        obtain decrypted plaintext m, or an invalid ciphertext flag.
     
           (a) If the decryption was successful, return the plaintext string
        m.
     
           (b) Otherwise, raise an error condition.
     
     
     
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     10. Concrete encoding guidelines for BB1
     
        This section specifies a set of concrete encoding schemes for the
        inputs and outputs of the previously described algorithms. ASN.1
        encodings are specified in Section 11 of this document.
     
     10.1. Encoding of points on a curve
     
        We refer to the description of Algorithm 7.1.1 (EncodePoint) and
        Algorithm 7.1.2 (DecodePoint).
     
     10.2. Public parameters blocks
     
        Algorithm 10.2.1 (BBencodeParams): encodes a BB1 public parameter set
        in an exportable format.
     
        Input:
     
        o  A set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v).
     
        Output:
     
        o  A public parameter block pi, represented as a byte string.
     
        Method:
     
        1. Separate encodings for k, E, p, q, P, P_1, P_2, P_3 are obtained
        as follows:
     
           (a) If t = 1, execute Algorithm 10.2.3 (BBencodeParams1).
     
        2. The separate encodings as well as a type indicator flag for t are
        then serialized in any suitable manner as dictated by the
        application.
     
        Algorithm 10.2.2 (BBdecodeParams): imports a BB1 public parameter
        block from a serialized format.
     
        Input:
     
        o  A public parameter block pi, represented as a byte string.
     
        Output:
     
        o  A set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v).
     
        Method:
     
     
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        1. Identify from the appropriate flag the type t of curve upon which
        the parameter block is based.
     
        2. Then:
     
           (a) If t = 1, execute Algorithm 10.2.4 (BBdecodeParams1).
     
     10.2.1. Type-1 implementation
     
        Algorithm 10.2.3 (BBencodeParams1): encodes a BB1 type-1 public
        parameter set in an exportable format.
     
        Input:
     
        o  A set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v)
           with t = 1.
     
        Output:
     
        o  Separate encodings for each of the k, E, p, q, P, P_1, P_2, P_3
           components (v is redundant and omitted).
     
        Method:
     
        1. The elliptic curve E: y^2 = x^3 + a * x + b and the embedding
        degree k = 2 are represented as a constant string, such as the empty
        string, since the coefficients a and b and the embedding degree are
        invariant for type-1 curves.
     
        2. The integer p = 12 * r * q - 1 is represented as the smaller
        integer r, encoded, e.g., using a big-endian byte-string
        representation.
     
        3. The integer q = 2^a + s* 2^b + c, where a, b are small and both c
        and s are either 1 or -1 is compactly represented as the 4-tuple (a,
        b, c, s).
     
        4. The point P = (x_P , y_P) in F_p x F_p is represented using the
        point compression technique of Algorithm 7.1.1 (EncodePoint).
     
        5. Each of P_1, P_2, and P_3 is similarly encoded using Algorithm
        7.1.1 (EncodePoint).
     
        Algorithm 10.2.4 (BBdecodeParams1): decodes the components of a BB1
        type-1 public parameter block.
     
        Input:
     
     
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        o  Separate encodings for each one of k, E, p, q, P, P_1, P_2, P_3.
     
        Output:
     
        o  A set of public parameters (t, k, E, p, q, P, P_1, P_2, P_3, v)
           with t = 1.
     
        Method:
     
        1. The equation of E is set to E: y^2 = x^3 + 1, as is always the
        case for type-1 curves.
     
        2. The embedding degree is set to k = 2 for type-1 curves.
     
        3. The encoding of q is parsed as (a, b, c, s), and its value set to
        q = 2^a + s * 2^b + c.
     
        4. The encoding of p is parsed as the integer r, from which p is
        given by p = 12 * r * q - 1.
     
        5. The point P is reconstructed from its encoding (x, y') using the
        point decompression technique of Algorithm 7.1.2 (DecodePoint).
     
        6. Each of P_1, P_2, and P_3 is reconstructed in a similar manner
        from its encoding using Algorithm 7.1.2 (DecodePoint).
     
        7. The extension field element v is reconstructed as v = Pairing(E,
        p, q, P_1, P_2) using Algorithm 4.5.1 (Pairing).
     
     10.3. Master secret blocks
     
        Algorithm 10.3.1 (BBencodeMaster): encodes a BB1 master secret in an
        exportable format.
     
        Input:
     
        o  A master secret triple of integers (alpha, beta, gamma) in (Z+_q)
           ^3.
     
        Output:
     
        o  A master secret block sigma, represented as a byte string.
     
        Method:
     
     
     
     
     
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        1. Encode each integer as an unsigned big-endian byte-string of fixed
        length Ceiling(8 * lg(q)), or, when q is a Solinas prime q = 2^a +/-
        2^b +/- 1, of length Ceiling((a + 1) / 8):
     
           (a) sigma_alpha to represent alpha.
     
           (b) sigma_beta to represent beta.
     
           (c) sigma_gamma to represent gamma.
     
        2. Sigma = sigma_alpha || sigma_beta || sigma_gamma is the
        concatenation of these strings.
     
        Algorithm 10.3.2 (BBdecodeMaster): decodes a BB1 master secret from a
        block in exportable format.
     
        Input:
     
        o  A master secret block sigma, represented as a byte string.
     
        Output:
     
        o  A master secret triple of integers (alpha, beta, gamma) in (Z+_q)
           ^3.
     
        Method:
     
        1. Parse sigma as sigma_alpha || sigma_beta || sigma_gamma, where
        each substring is a byte string of fixed length Ceiling(8 * lg(q)),
        or, when q is a Solinas prime q = 2^a +/- 2^b +/- 1, of length
        Ceiling((a + 1) / 8)).
     
        2. Decode each substring as an integer in unsigned big-endian byte-
        string representation:
     
        (a) Let alpha = Value(sigma_alpha).
     
        (b) Let beta = Value(sigma_beta).
     
        (c) Let gamma = Value(sigma_gamma).
     
     10.4. Private key blocks
     
        Algorithm 10.4.1 (BBencodePrivate): encodes a BB1 private key in an
        exportable format.
     
        Input:
     
     
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        o  A private key pair of points (D_0, D_1) in E(F_p) x E(F_p).
     
        Output:
     
        o  A private key block kappa, represented as a byte string.
     
        Method:
     
        1. Encode each point separately:
     
           (a) The first component of kappa, kappa_0 is obtained by applying
        Algorithm 7.1.1 (EncodePoint) to D_0.
     
           (b) The second componente of kappa, kappa_1 is obtained by
        applying Algorithm 7.1.1 (EncodePoint) to D_0.
     
        2. Let kappa = kappa_0 || kappa_1.
     
        Algorithm 10.4.2 (BBdecodePrivate): decodes a BB1 private key from an
        exportable format.
     
        Input:
     
        o  A private key block kappa, represented as a byte string.
     
        Output:
     
        o  A private key pair of point (D_0, D_1) in E(F_p) x E(F_p).
     
        Method:
     
        1. Decode each point separately:
     
           (a) The first prefix of kappa is parsed and decoded into a point
        D_0 in E(F_p) using Algorithm 7.1.2 (DecodePoint).
     
           (b) The remainder of kappa is parsed and decoded into a point D_1
        in E(F_p) using Algorithm 7.1.2 (DecodePoint).
     
     10.5. Ciphertext blocks
     
        Algorithm 10.5.1 (BBencodeCiphertext). Encodes a BB1 ciphertext tuple
        in an exportable format.
     
        Input:
     
     
     
     
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        o  A ciphertext tuple (u, C_0, C_1, y) in Z_q x E(F_p) x E(F_p) x {0,
           . . . , 255}*.
     
        Output:
     
        o  A ciphertext block omega, represented as a byte string.
     
        Method:
     
        1. Let chi_0 be the fixed-length encoding of C_0 = (x_0, y_0) using
        Algorithm 7.1.1 (EncodePoint).
     
        2. Let chi_1 be the fixed-length encoding of C_1 = (x_1, y_1) using
        Algorithm 7.1.1 (EncodePoint).
     
        3. Let nu be the encoding of u as an unsigned big-endian byte-string
        of fixed length Ceiling(8 * lg(q)), or, when q is a Solinas prime q =
        2^a +/- 2^b +/- 1, of length Ceiling((a + 1)/8).
     
        4. The value of omega = chi_0 || chi_1 || nu || y is the
        concatenation of these three strings and y.
     
        Algorithm 10.5.2 (BBdecodeCiphertext): decodes a BB1 ciphertext tuple
        from an exportable format.
     
        Input:
     
        o  A ciphertext block omega, represented as a byte string.
     
        Output:
     
        o  A ciphertext tuple (u, C_0, C_1, y) in Z_q x E(F_p) x E(F_p) x {0,
           . . . , 255}*.
     
        Method:
     
        1. The value of omega is parsed as a quadruple comprising a fixed-
        length encoding of C_0, a fixed-length encoding of C_1, a fixed-
        length encoding of u, and the arbitrary-length string y:
     
        (a) C_0 in E(F_p) is first recovered by applying Algorithm 7.1.2
        (DecodePoint) on the first parsed component of omega.
     
        (b) C_1 in E(F_p) is next recovered by applying Algorithm 7.1.2
        (DecodePoint) on the second parsed component of omega.
     
     
     
     
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        (c) The value of u in Z_q is then recovered from its unsigned big-
        endian byte-string representation in the third parsed component of
        omega, of length Ceiling(8 * lg(q)), or, when q is a Solinas prime q
        = 2^a +/- 2b +/- 1, of length Ceiling((a + 1)/8).
     
        (d) The value of y is finally taken as the remainder of omega.
     
     11. Test vectors
     
        The following data can be used to verify the correct operation of
        selected algorithms that are defined in this document.
     
     11.1. Algorithm 3.2.2 (PointMultiply)
     
        Input:
     
        q = 0xfffffffffffffffffffffffffffbffff
     
        p = 0xbffffffffffffffffffffffffffcffff3
     
        E/F_p: y^2 = x^3 + 1
     
        A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
        0x510c6972d795ec0c2b081b81de767f808)
     
        l = 0xb8bbbc0089098f2769b32373ade8f0daf
     
        Output:
     
        [l]A = (0x073734b32a882cc97956b9f7e54a2d326,
        0x9c4b891aab199741a44a5b6b632b949f7)
     
     11.2. Algorithm 4.1.1 (HashToRange)
     
        Input:
     
        s =
        54:68:69:73:20:41:53:43:49:49:20:73:74:72:69:6e:67:20:77:69:74:68:6f:
        75:74:20:6e:75:6c:6c:2d:74:65:72:6d:69:6e:61:74:6f:72 ("This ASCII
        string without null-terminator")
     
        n = 0xffffffffffffffffffffefffffffffffffffffff
     
        Output:
     
        v = 0x79317c1610c1fc018e9c53d89d59c108cd518608
     
     
     
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     11.3. Algorithm 4.5.1 (Pairing)
     
        q = 0xfffffffffffffffffffffffffffbffff
     
        p = 0xbffffffffffffffffffffffffffcffff3
     
        E/F_p: y^2 = x^3 + 1
     
        A = (0x489a03c58dcf7fcfc97e99ffef0bb4634,
        0x510c6972d795ec0c2b081b81de767f808)
     
        B = (0x40e98b9382e0b1fa6747dcb1655f54f75,
        0xb497a6a02e7611511d0db2ff133b32a3f)
     
        Output:
     
        e'(A, B) = (0x8b2cac13cbd422658f9e5757b85493818,
        0xbc6af59f54d0a5d83c8efd8f5214fad3c) in F_p^2 where F_p^2 is
        represented as F_p[x]/(x^2 + 1)
     
     11.4. Algorithm 5.2.1 (BFderivePubl)
     
        Input:
     
        id = 6f:42:62 ("Bob")
     
        t = 1
     
        p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
     
        q = 0xffffffffffffffffffffffeffffffffffff
     
        P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
        0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
     
        P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
        0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
     
        Output:
     
        Q_id = (0x22fa1207e0d19e1a4825009e0e88e35eb57ba79391498f59,
        0x982d29acf942127e0f01c881b5ec1b5fe23d05269f538836)
     
     11.5. Algorithm 5.3.1 (BFextractPriv)
     
        Input:
     
     
     
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        s = 0x749e52ddb807e0220054417e514742b05a0
     
        t = 1
     
        p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
     
        q = 0xffffffffffffffffffffffeffffffffffff
     
        P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
        0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
     
        P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
        0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
     
        Output:
     
        Q_id = (0x8212b74ea75c841a9d1accc914ca140f4032d191b5ce5501,
        0x950643d940aba68099bdcb40082532b6130c88d317958657)
     
     11.6. Algorithm 5.4.1 (BFencrypt)
     
        (Note that the following values can also be used to test Algorithm
        5.5.1 (BFdecrypt))
     
        Input:
     
        m = 48:69:20:74:68:65:72:65:21 ("Hi there!")
     
        id = 6f:42:62 ("Bob")
     
        t = 1
     
        p = 0xa6a0ffd016103ffffffffff595f002fe9ef195f002fe9efb
     
        q = 0xffffffffffffffffffffffeffffffffffff
     
        P = (0x6924c354256acf5a0ff7f61be4f0495b54540a5bf6395b3d,
        0x024fd8e2eb7c09104bca116f41c035219955237c0eac19ab)
     
        P_pub = (0xa68412ae960d1392701066664d20b2f4a76d6ee715621108,
        0x9e7644e75c9a131d075752e143e3f0435ff231b6745a486f)
     
        Output:
     
        Using the random value rho =
        0xed5397ff77b567ba5ecb644d7671d6b6f2082968, we get the following
        output:
     
     
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        U =
        (0x1b5f6c461497acdfcbb6d6613ad515430c8b3fa23b61c585e9a541b199e2a6cb,
        0x9bdfbed1ae664e51e3d4533359d733ac9a600b61048a7d899104e826a0ec4fa4)
     
        V = e0:1d:ad:81:32:6c:b1:73:af:c2:8d:72:2e:7a:32:1a:7b:29:8a:aa
     
        W = f9:04:ba:40:30:e9:ce:6e:ff
     
     11.7. Algorithm 8.3.1 (BBextractPriv)
     
        Inputs:
     
        alpha = 0xa60c395285ded4d70202c8283d894bad4f0
     
        beta = 0x48bf012da19f170b13124e5301561f45053
     
        gamma = 0x226fba82bc38e2ce4e28e56472ccf94a499
     
        t = 1
     
        p = 0x91bbe2be1c8950750784befffffffffffff6e441d41e12fb
     
        q = 0xfffffffffbfffffffffffffffffffffffff
     
        P = (0x13cc538fe950411218d7f5c17ae58a15e58f0877b29f2fe1,
        0x8cf7bab1a748d323cc601fabd8b479f54a60be11e28e18cf)
     
        P_1 = (0x0f809a992ed2467a138d72bc1d8931c6ccdd781bedc74627,
        0x11c933027beaaf73aa9022db366374b1c68d6bf7d7a888c2)
     
        P_2 = (0x0f8ac99a55e575bf595308cfea13edb8ec673983919121b0,
        0x3febb7c6369f5d5f18ee3ea6ca0181448a4f3c4f3385019c)
     
        P_3 = (0x2c10b43991052e78fac44fdce639c45824f5a3a2550b2a45,
        0x6d7c12d8a0681426a5bbc369c9ef54624356e2f6036a064f)
     
        v = (0x38f91032de6847a89fc3c83e663ed0c21c8f30ce65c0d7d3,
        0x44b9aa10849cc8d8987ef2421770a340056745da8b99fba2) in F_p^2 where
        F_p^2 is represented as F_p[x]/(x^2 + 1)
     
        id = 6f:42:62 ("Bob")
     
        Output:
     
        Using the random value r = 0x695024c25812112187162c08aa5f65c7a2c, we
        get the following output:
     
     
     
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        D_0 = (0x3264e13feeb7c506493888132964e79ad657a952334b9e53,
        0x3eeaefc14ba1277a1cd6fdea83c7c882fe6d85d957055c7b)
     
        D_1 = (0x8d7a72ad06909bb3bb29b67676d935018183a905e7e8cb18,
        0x2b346c6801c1db638f270af915a21054f16044ab67f6c40e)
     
     11.8. Algorithm 8.4.1 (BBencrypt)
     
        (Note that the following values can also be used to test Algorithm
        8.5.1 (BFdecrypt))
     
        Input:
     
        m = 48:69:20:74:68:65:72:65:21 ("Hi there!")
     
        id = 6f:42:62 ("Bob")
     
        t = 1
     
        k = 2
     
        E: y^2 = x^3 + 1
     
        p = 0x91bbe2be1c8950750784befffffffffffff6e441d41e12fb
     
        q = 0xfffffffffbfffffffffffffffffffffffff
     
        P = (0x13cc538fe950411218d7f5c17ae58a15e58f0877b29f2fe1,
        0x8cf7bab1a748d323cc601fabd8b479f54a60be11e28e18cf)
     
        P_1 = (0x0f809a992ed2467a138d72bc1d8931c6ccdd781bedc74627,
        0x11c933027beaaf73aa9022db366374b1c68d6bf7d7a888c2)
     
        P_2 = (0x0f8ac99a55e575bf595308cfea13edb8ec673983919121b0,
        0x3febb7c6369f5d5f18ee3ea6ca0181448a4f3c4f3385019c)
     
        P_3 = (0x2c10b43991052e78fac44fdce639c45824f5a3a2550b2a45,
        0x6d7c12d8a0681426a5bbc369c9ef54624356e2f6036a064f)
     
        v = (0x38f91032de6847a89fc3c83e663ed0c21c8f30ce65c0d7d3,
        0x44b9aa10849cc8d8987ef2421770a340056745da8b99fba2) in F_p^2 where
        F_p^2 is represented as F_p[x]/(x^2 + 1)
     
        Output:
     
        Using the random value s = 0x62759e95ce1af248040e220263fb41b965e, we
        get the following output:
     
     
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        u = 0xad1ebfa82edf0bcb5111e9dc08ff0737c68
     
        C_0 = (0x79f8f35904579f1aaf51897b1e8f1d84e1c927b8994e81f9,
        0x1cf77bb2516606681aba2e2dc14764aa1b55a45836014c62)
     
        C_1 = (0x410cfeb0bccf1fa4afc607316c8b12fe464097b20250d684,
        0x8bb76e7195a7b1980531b0a5852ce710cab5d288b2404e90)
     
        y = 82:a6:42:b9:bb:e9:82:c4:57
     
     12. ASN.1 module
     
        This section defines the ASN.1 module for the encodings discussed in
        sections 7 and 10.
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
     
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        IBCS { joint-iso-itu(2) country(16) us(840) organization(1)
           identicrypt(114334) ibcs(1) module(5) version(1) }
     
        DEFINITIONS IMPLICIT TAGS ::= BEGIN
     
        --
        -- Identity-based cryptography standards (IBCS): supersingular curve
        -- implementations of the BF and BB1 cryptosystems.
        --
        -- This version of the IBCS standard only supports IBE over
        -- type-1 curves. In the current version, the Curve type is
        -- always set to NULL, although future versions will use it.
        --
     
        IMPORTS Curve
           FROM X9-62-module
              { iso(1) member-body(2) us(840) ansi-x9-62(10045) module(5) 1
        };
     
        ibcs OBJECT IDENTIFIER ::= {
           joint-iso-itu(2) country(16) us(840) organization(1)
              identicrypt(114334) ibcs(1)
        }
     
        --
        -- IBCS1
        --
        -- IBCS1 defines the algorithms used to implement IBE
        --
     
        ibcs1 OBJECT IDENTIFIER ::= {
           ibcs ibcs1(1)
        }
     
        --
        -- Supporting types
        --
     
        --
        -- Encoding of a point on an elliptic curve E/Fp.
        --
     
        FpPoint ::= SEQUENCE {
           x  INTEGER,
           y  INTEGER
        }
     
     
     
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        --
        -- Encoding of a Solinas prime.
        --
        -- Encodes a Solinas prime of the form
        -- q = 2^a + s * 2^b +c with the integers a, b, c, and s.
        --
     
        SolinasPrime ::= SEQUENCE {
           a  INTEGER,
           b  INTEGER,
           c  INTEGER { positive(1), negative(-1) },
           s  INTEGER { positive(1), negative(-1) }
        }
     
        --
        -- Algorithms
        --
     
        ibe-algorithms OBJECT IDENTIFIER ::= {
           ibcs1 ibe-algorithms(2)
        }
     
        ---
        --- Boneh-Franklin IBE
        ---
     
        bf OBJECT IDENTIFIER ::= { ibe-algorithms bf(1) }
     
        --
        -- Encoding of a BF public parameters block.
        -- The only version currently supported is version 1.
        -- For type-1 curves, the curve is fixed, so Curve is set to NULL
        -- For the BF prime p and subprime q, we have q * r = p + 1,
        -- and we encode the values of r and q in the public parameters.
        -- The points P and P_pub are encoded as pointP and pointPpub
        respectively.
        --
     
        BFPublicParamaters ::= SEQUENCE {
           version     INTEGER { v1(1) },
           curve       Curve { NULL },
           r           INTEGER,
           q           SolinasPrime,
           pointP      FpPoint,
           pointPpub   FpPoint
        }
     
     
     
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        --
        -- A BF private key is a point on an elliptic curve,
        -- which is an FpPoint.
        --
     
        BFPrivateKeyBlock ::= FpPoint
     
        --
        -- A BF master secret is an integer.
        --
     
        BFMasterSecret ::= INTEGER
     
        --
        -- BF ciphertext block
        --
     
        BFCiphertextBlock ::= SEQUENCE {
           U  FpPoint,
           v  OCTET STRING,
           w  OCTET STRING
        }
     
        --
        -- Boneh-Boyen (BB1) IBE
        --
     
        bb1 OBJECT IDENTIFIER ::= { ibe-algorithms bb1(2) }
     
        --
        -- Encoding of a BB1 public parameters block.
        -- The version is currently fixed to 1.
        -- The embedding degree is currently fixed to 2.
        -- For type-1 curves, curve is set to NULL.
        -- For the BB1 prime p and subprime q, we have q * r = p + 1,
        -- and we encode the values of r and q in the public parameters.
        --
     
        BB1PublicParameters ::= SEQUENCE {
           Version              INTEGER { v1(1) },
           embedding-degree     INTEGER { degree-2(2) },
           curve                Curve { NULL },
           r                    INTEGER,
           q                    SolinasPrime,
           pointP               FpPoint,
           pointP1              FpPoint,
           pointP2              FpPoint,
     
     
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           pointP3              FpPoint
        }
     
        --
        -- BB1 master secret block
        --
     
        BB1MasterSecret ::= SEQUENCE {
           alpha INTEGER,
           beta  INTEGER,
           gamma INTEGER
        }
     
        --
        -- BB1 private Key block
        --
     
        BB1PrivateKeyBlock ::= SEQUENCE {
           pointD0  FpPoint,
           pointD1  FpPoint
        }
     
        --
        -- BB1 ciphertext block
        --
     
        BB1CiphertextBlock ::= SEQUENCE {
           pointChi0   FpPoint,
           pointChi1   FpPoint,
           nu          INTEGER,
           y           OCTET STRING
        }
     
        END
     
     
     13. Security considerations
     
        This document describes cryptographic algorithms, for which we assume
        that the security of the algorithm relies entirely on the secrecy of
        the relevant private key, so that an adversary will need to intercept
        encrypted messages and perform computationally-intensive
        cryptanalytic attacks against the ciphertext that he obtains in this
        way to recover either plaintext or a secret cryptographic key.
     
        We assume that users of the algorithms described in this document
        will require one of three levels of cryptographic strength: the
     
     
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        equivalent of 80 bits, the equivalent of 112 bits, or the equivalent
        of 128 bits. The 80-bit level is suitable for legacy applications and
        SHOULD not be used to protect information whose useful life extends
        past the year 2010. The 112-bit level is suitable for use in key
        transport of Triple-DES keys and should be adequate to protect
        information whose useful life extends up to the year 2030. The 128-
        bit level is suitable for use in key transport of 128-bit AES keys.
     
        The following table describes the security parameters for the BF and
        BB1 algorithms that will attain 80-bit, 112-bit and 128-bit levels of
        security. In this table, |p| represents the number of bits in a prime
        number p and |q| represents the number of bits in a subprime q. This
        table assumes that a Type-1 supersingular curve is used.
     
        Bits of Security   |p|    |q|
        80                 512    160
        112                1024   224
        128                1536   256
     
        Note that this document specifies the use of the SHA1 hashing
        algorithm to hash identities to either a point on an elliptic curve
        or an integer. Recent attacks on SHA1 have discovered ways to find
        collisions in SHA1 with much less work that the 80 bits of strength
        in resistance to collisions that we would expect for a hashing
        algorithm that creates a 160-bit digest. If we can find a collision
        in SHA1 we could use the colliding preimages to create two identities
        which have the same IBE private key. The practical use of such a SHA1
        collision seems extremely unlikely, particularly if IBE is used as
        described in [IBECMS], in which components of an identity are defined
        to be either a time or a URI. Any protocol using IBE SHOULD define
        part of an identity to avoid the possible use of collisions in SHA1
        in this way.
     
     
     14. IANA considerations
     
        All of the OIDs used in this document were assigned by the National
        Institute of Standards and Technology (NIST), so no further action by
        the IANA is necessary for this document.
     
     15. Acknowledgments
     
        This document is based on the IBCS #1 v2 document of Voltage
        Security, Inc. Any substantial use of material from this document
        should acknowledge Voltage Security, Inc. as the source of the
        information.
     
     
     
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     16. References
     
     16.1. Normative references
     
        [IBECMS] L. Martin, M. Schertler, "Using the Boneh-Franklin identity-
        based encryption algorithm with the Cryptographic Message Syntax
        (CMS)," draft-ietf-smime-bfibecms-01.txt, September 2006.
     
     16.2. Informative references
     
        [BB1] D. Boneh, X. Boyen, "Efficient selective-ID secure identity
        based encryption without random oracles," In Proc. of EUROCRYPT 04,
        LNCS 3027, pp. 223-238, 2004.
     
        [BF] D. Boneh, M. Franklin, "Identity-based encryption from the Weil
        pairing," In Proc. of CRYPTO 01, LNCS 2139, pp. 213-229, 2001.
     
        [ECC] I. Blake, G. Seroussi, N. Smart, Elliptic Curves in
        Cryptography, Cambridge University Press, 1999.
     
     Authors' Addresses
     
        Xavier Boyen
        Voltage Security
        1070 Arastradero Rd Suite 100
        Palo Alto, CA 94304
     
        Email: xavier@voltage.com
     
     
        Luther Martin
        Voltage Security
        1070 Arastradero Rd Suite 100
        Palo Alto, CA 94304
     
        Email: martin@voltage.com
     
     
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        made any independent effort to identify any such rights.  Information
     
     
     
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        Copyright (C) The Internet Society (2006).
     
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