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Network Working Group                                          D. McGrew
Internet-Draft                                             Cisco Systems
Intended status: Informational                                   K. Igoe
Expires: August 23, 2010                        National Security Agency
                                                       February 19, 2010


           Fundamental Elliptic Curve Cryptography Algorithms
                  draft-mcgrew-fundamental-ecc-02.txt

Abstract

   This note describes the fundamental algorithms of Elliptic Curve
   Cryptography (ECC) as they are defined in some early references.
   These descriptions may be useful for implementing the fundamental
   algorithms without using any of the specialized methods that were
   developed in following years.  Only elliptic curves defined over
   fields of characteristic greater than three are in scope; these
   curves are those used in Suite B.

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Copyright Notice

   Copyright (c) 2010 IETF Trust and the persons identified as the
   document authors.  All rights reserved.




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   This document is subject to BCP 78 and the IETF Trust's Legal
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   described in the BSD License.


Table of Contents

   1.  Introduction . . . . . . . . . . . . . . . . . . . . . . . . .  4
     1.1.  Conventions Used In This Document  . . . . . . . . . . . .  4
   2.  Mathematical Background  . . . . . . . . . . . . . . . . . . .  5
     2.1.  Modular Arithmetic . . . . . . . . . . . . . . . . . . . .  5
     2.2.  Group Operations . . . . . . . . . . . . . . . . . . . . .  5
     2.3.  Finite Fields  . . . . . . . . . . . . . . . . . . . . . .  7
   3.  Elliptic Curve Groups  . . . . . . . . . . . . . . . . . . . .  7
     3.1.  Homogeneous Coordinates  . . . . . . . . . . . . . . . . .  8
     3.2.  Other Coordinates  . . . . . . . . . . . . . . . . . . . .  9
     3.3.  ECC Parameters . . . . . . . . . . . . . . . . . . . . . .  9
       3.3.1.  Security . . . . . . . . . . . . . . . . . . . . . . . 10
   4.  Elliptic Curve Diffie-Hellman (ECDH) . . . . . . . . . . . . . 10
     4.1.  Data Types . . . . . . . . . . . . . . . . . . . . . . . . 11
     4.2.  Compact Representation . . . . . . . . . . . . . . . . . . 11
   5.  Elliptic Curve ElGamal Signatures  . . . . . . . . . . . . . . 12
     5.1.  Background . . . . . . . . . . . . . . . . . . . . . . . . 12
     5.2.  Hash Functions . . . . . . . . . . . . . . . . . . . . . . 12
     5.3.  KT-IV Signatures . . . . . . . . . . . . . . . . . . . . . 13
       5.3.1.  Keypair Generation . . . . . . . . . . . . . . . . . . 13
       5.3.2.  Signature Creation . . . . . . . . . . . . . . . . . . 13
       5.3.3.  Signature Verification . . . . . . . . . . . . . . . . 13
     5.4.  KT-I Signatures  . . . . . . . . . . . . . . . . . . . . . 14
       5.4.1.  Keypair Generation . . . . . . . . . . . . . . . . . . 14
       5.4.2.  Signature Creation . . . . . . . . . . . . . . . . . . 14
       5.4.3.  Signature Verification . . . . . . . . . . . . . . . . 15
     5.5.  Converting KT-IV signatures to KT-I signatures . . . . . . 15
     5.6.  Rationale  . . . . . . . . . . . . . . . . . . . . . . . . 15
   6.  Interoperability . . . . . . . . . . . . . . . . . . . . . . . 17
     6.1.  ECDH . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
     6.2.  KT-I and ECDSA . . . . . . . . . . . . . . . . . . . . . . 17
   7.  Validating an Implementation . . . . . . . . . . . . . . . . . 18
     7.1.  ECDH . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
     7.2.  KT-I . . . . . . . . . . . . . . . . . . . . . . . . . . . 19
   8.  Intellectual Property  . . . . . . . . . . . . . . . . . . . . 19
     8.1.  Disclaimer . . . . . . . . . . . . . . . . . . . . . . . . 19



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   9.  Security Considerations  . . . . . . . . . . . . . . . . . . . 20
     9.1.  Subgroups  . . . . . . . . . . . . . . . . . . . . . . . . 20
     9.2.  Diffie-Hellman . . . . . . . . . . . . . . . . . . . . . . 21
     9.3.  Group Representation and Security  . . . . . . . . . . . . 21
     9.4.  Signatures . . . . . . . . . . . . . . . . . . . . . . . . 22
   10. IANA Considerations  . . . . . . . . . . . . . . . . . . . . . 22
   11. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . 22
   12. References . . . . . . . . . . . . . . . . . . . . . . . . . . 23
     12.1. Normative References . . . . . . . . . . . . . . . . . . . 23
     12.2. Informative References . . . . . . . . . . . . . . . . . . 24
   Appendix A.  Key Words . . . . . . . . . . . . . . . . . . . . . . 27
   Appendix B.  Random Integer Generation . . . . . . . . . . . . . . 27
   Appendix C.  Why Compact Representation Works  . . . . . . . . . . 28
   Appendix D.  Example ECC Parameter Set . . . . . . . . . . . . . . 29
   Appendix E.  Additive and multiplicative notation  . . . . . . . . 29
   Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . . 30



































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1.  Introduction

   ECC is a public-key technology that offers performance advantages at
   higher security levels.  It includes an Elliptic Curve version of
   Diffie-Hellman key exchange protocol [DH1976] and Elliptic Curve
   versions of the ElGamal Signature Algorithm [E1985].  The adoption of
   ECC has been slower than had been anticipated, perhaps due to the
   lack of freely available normative documents and uncertainty over
   intellectual property rights.

   This note contains a description of the fundamental algorithms of ECC
   over fields with characteristic greater than three, based directly on
   original references.  Its intent is to provide the Internet community
   with a summary of the basic algorithms that predate any specialized
   or optimized algorithms, which can be used as a normative
   specification.  The original descriptions and notations were followed
   as closely as possible.

   There are several standards that specify or incorporate ECC
   algorithms, including the Internet Key Exchange (IKE), ANSI X9.62,
   and IEEE P1363.  The algorithms in this note can interoperate with
   some of the algorithms in these standards, with a suitable choice of
   parameters and options.  The specifics are itemized in Section 6.

   The rest of the note is organized as follows.  Section 2.1,
   Section 2.2, and Section 2.3 furnish the necessary terminology and
   notation from modular arithmetic, group theory and the theory of
   finite fields, respectively.  Section 3 defines the groups based on
   elliptic curves over finite fields of characteristic greater than
   three.  Section 4 presents the fundamental ECDH algorithm.  Section 5
   presents elliptic curve versions of the ElGamal signature method.
   Sections 2 through 5, inclusive, contain all of the normative text
   (the text that defines the norm for implementations conforming to
   this specification), and all of the following sections are purely
   informative.  Interoperability is discussed in Section 6.  Validation
   testing is described in Section 7.  Section 8 reviews intellectual
   property issues.  Section 9 summarizes security considerations.
   Appendix B describes random number generation, and other appendices
   provide clarifying details.

1.1.  Conventions Used In This Document

   The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
   "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this
   document are to be interpreted as described in Appendix A.






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2.  Mathematical Background

   This section reviews mathematical preliminaries and establishes
   terminology and notation that is used below.

2.1.  Modular Arithmetic

   This section reviews modular arithmetic.  Two integers x and y are
   said to be congruent modulo n if x - y is an integer multiple of n.

   Two integers x and y are coprime when their greatest common divisor
   is 1; in this case, there is no third number z > 1 such that z
   divides x and z divides y.

   The set Zq = { 0, 1, 2, ..., q-1 } is closed under the operations of
   modular addition, modular subtraction, modular multiplication, and
   modular inverse.  These operations are as follows.

      For each pair of integers a and b in Zq, a + b mod q is equal to
      a + b if a + b < q, and is equal to a + b - q otherwise.

      For each pair of integers a and b in Zq, a - b mod q is equal to
      a - b if a - b >= 0, and is equal to a - b + q otherwise.

      For each pair of integers a and b in Zq, a * b mod q is equal to
      the remainder of a * b divided by q.

      For each integer x in Zq that is coprime with q, the inverse of x
      modulo q is denoted as 1 / x mod q, and can be computed using the
      extended euclidean algorithm (see Section 4.5.2 of [K1981v2], for
      example).

   Algorithms for these operations are well known; for instance, see
   Chapter 4 of [K1981v2].

2.2.  Group Operations

   This section establishes some terminology and notation for
   mathematical groups, which is needed later on.  Background references
   abound; see [D1966], for example.

   A group is a set of elements G together with an operation that
   combines any two elements in G and returns a third element in G. The
   operation is denoted as * and its application is denoted as a * b,
   for any two elements a and b in G. The operation is associative, that
   is, for all a, b and c in G, a * (b * c) is identical to (a * b) * c.
   Repeated application of the group operation N-1 times to the element
   a is denoted as a^N, for any element a in G and any positive integer



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   N. That is, a^2, = a * a, a^3 = a * a * a, and so on.  The
   associativity of the group operation ensures that the computation of
   a^n is unambiguous; any grouping of the terms gives the same result.

   The above definition of a group operation uses multiplicative
   notation.  Sometimes an alternative called additive notation is used,
   in which a * b is denoted as a + b, and a^N is denoted as N * a.  In
   multiplicative notation, g^N is called exponentiation, while the
   equivalent operation in additive notation is called scalar
   multiplication.  In this document, multiplicative notation is used
   throughout for consistency.  Appendix E elucidates the correspondence
   between the two notations.

   Every group has a special element called the identity element, which
   we denote as e.  For each element a in G, e * a = a * e = a.  By
   convention, a^0 is equal to the identity element for any a in G.

   Every group element a has a unique inverse element b such that
   a * b = b * a = e.  The inverse of a is denoted as a^-1 in
   multiplicative notation.  (In additive notation, the inverse of a is
   denoted as -a.)

   For any positive integer X, a^(-X) is defined to be (a^-1)^(X).
   Using this convention, exponentiation behaves as one would expect,
   namely for any integers X and Y:

      a^(X+Y) = (a^X)*(a^Y)

      (a^X)^Y = a^(XY) = (a^Y)^X.

   A group element a is said to have finite order if a^X = e for some
   positive integer X, and the order of a is the smallest such X. If no
   such X exists, a is said to have infinite order.  In cryptographic
   applications one typically deals with finite groups (groups with a
   finite number of elements), and all elements of finite groups have
   finite order.  If a group element a has order R, then for any
   integers X and Y,

      a^X = a^(X mod R),

      a^X = a^Y if and only if X is congruent to Y mod R,

      the set H = { a, a^2, a^3, ... , a^R=e } forms a subgroup of G,
      called the cyclic subgroup generated by a, and a is said to be a
      generator of H.

   Typically there are several group elements that generate H. Any group
   element of the form a^M, with M relatively prime to R, also generates



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   H. Note that a^M is equal to g^(M modulo R) for any non-negative
   integer M.

   Given the element a of order R, and an integer i between 1 and R-1,
   inclusive, the element a^i can be computed by the "square and
   multiply" method outlined in Section 2.1 of [M1983] (see also Knuth,
   Vol. 2, Section 4.6.3.), or other methods.

2.3.  Finite Fields

   This section establishes terminology and notation for finite fields
   with prime characteristic.

   When p is a prime number, then the set Zp, with the addition,
   subtraction, multiplication and division operations, is a finite
   field with characteristic p.  Each nonzero element x in Zp has an
   inverse 1/x.  There is a one-to-one correspondence between the
   integers between zero and p-1, inclusive, and the elements of the
   field.  The field Zp is sometimes denoted as Fp.

   Equations involving field elements do not explicitly denote the "mod
   p" operation, but it is understood to be implicit.  For example, the
   statement that x, y, and z are in Fp and

      z = x + y

   is equivalent to the statement that x, y, and z are in the set
   { 0, 1, ..., p-1 } and

      z = x + y mod p.


3.  Elliptic Curve Groups

   This note only covers elliptic curves over fields with characteristic
   greater than three; these are the curves used in Suite B [SuiteB].
   For other fields, the definition of the elliptic curve group would be
   different.

   An elliptic curve over a field Fp is defined by the curve equation

      y^2 = x^3 + a*x + b,

   where x, y, a, and b are elements of the field Fp [M1985].  A point
   on an elliptic curve is a pair (x,y) of values in Fp that satisfy the
   curve equation, such that x and y are both in Fp, or it is a special
   point (@,@) that represents the identity element (which is called the
   "point at infinity").  The order of an elliptic curve group is the



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   number of distinct points.

   Two elliptic curve points (x1,y1) and (x2,y2) are equal whenever
   x1=x2 and y1=y2, or when both points are the point at infinity.  The
   inverse of the point (x1,y1) is the point (x1,-y1).  The point at
   infinity is its own inverse.

   The group operation associated with the elliptic curve group is as
   follows [BC1989].  To an arbitrary pair of points P and Q specified
   by their coordinates (x1,y1) and (x2,y2) respectively, the group
   operation assigns a third point P*Q with the coordinates (x3,y3).
   These coordinates are computed as follows

      (x3,y3) = (@,@) when P is not equal to Q and x1 is equal to x2.

      x3 = ((y2-y1)/(x2-x1))^2 - x1 - x2 and
      y3 = (x1-x3)*(y2-y1)/(x2-x1) - y1 when P is not equal to Q and
      x1 is not equal to x2.

      (x3,y3) = (@,@) when P is equal to Q and y1 is equal to 0,

      x3 = ((3*x1^2 + a)/(2*y1))^2 - 2*x1 and
      y3 = (x1-x3)*(3*x1^2 + a)/(2*y1) - y1 if P is equal to Q and y1 is
      not equal to 0.

   In the above equations, a, x1, x2, x3, y1, y2, and y3 are elements of
   the field Fp; thus, computation of x3 and y3 in practice must reduce
   the right-hand-side modulo p.

   The representation of elliptic curve points as a pair of integers in
   Zp is known as the affine coordinate representation.  This
   representation is suitable as an external data representation for
   communicating or storing group elements, though the point at infinity
   must be treated as a special case.

   Some pairs of integers are not valid elliptic curve points.  A valid
   pair will satisfy the curve equation, while an invalid pair will not.

3.1.  Homogeneous Coordinates

   An alternative way to implement the group operation is to use
   homogeneous coordinates [K1987] (see also [KMOV1991]).  This method
   is typically more efficient because it does not require a modular
   inversion operation.

   An elliptic curve point (x,y) (other than the point at infinity
   (@,@)) is equivalent to a point (X,Y,Z) in homogeneous coordinates
   whenever x=X/Z mod p and y=Y/Z mod p.



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   Let P1=(X1,Y1,Z1) and P2=(X2,Y2,Z2) be points on an elliptic curve
   and suppose that the points P1, P2 are not equal to (@,@), P1 is not
   equal to P2, and P1 is not equal to P2^-1.  Then the product
   P3=(X3,Y3,Z3) = P1 * P2 is given by

      X3 = v * (Z2 * (Z1 * u^2 - 2 * X1 * v^2) - v^3) mod p

      Y3 = Z2 * (3 * X1 * u * v^2 - Y1 * v^3 - Z1 * u^3) + u * v^3 mod p

      Z3 = v^3 * Z1 * Z2 mod p

   where u = Y2 * Z1 - Y1 * Z2 mod p and v = X2 * Z1 - X1 * Z2 mod p.

   When the points P1 and P2 are equal, then (X1/Z1, Y1/Z1) is equal to
   (X2/Z2, Y2/Z2), which is true if and only if u and v are both equal
   to zero.

   The product P3=(X3,Y3,Z3) = P1 * P1 is given by

      X3 = 2 * Y1 * Z1 * (w^2 - 8 * X1 * Y1^2 * Z1) mod p

      Y3 = 4 * Y1^2 * Z1 * (3 * w * X1 - 2 * Y1^2 * Z1) - w^3 mod p

      Z3 = 8 * (Y1 * Z1)^3 mod p

   where w = 3 * X1^2 + a * Z1^2 mod p.  In the above equations, a, u,
   v, w, X1, X2, X3, Y1, Y2, Y3, Z1, Z2, and Z3 are integers in the set
   Fp.

   When converting from affine coordinates to homogeneous coordinates,
   it is convenient to set Z to 1.  When converting from homogeneous
   coordinates to affine coordinates, it is necessary to perform a
   modular inverse to find 1/Z mod p.

3.2.  Other Coordinates

   Some other coordinate systems have been described; several are
   documented in [CC1986], including Jacobi coordinates.

3.3.  ECC Parameters

   In cryptographic contexts, an elliptic curve parameter set consists
   of a cyclic subgroup of an elliptic curve together with a preferred
   generator of that subgroup.  When working over a prime order finite
   field with characteristic greater than three, an elliptic curve group
   is completely specified by the following parameters:





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      The prime number p that indicates the order of the field Fp.

      The value a used in the curve equation.

      The value b used in the curve equation.

      The generator g of the subgroup.

      The order n of the subgroup generated by g.

   An example of an ECC parameter set is provided in Appendix D.

   Each elliptic curve point is associated with a particular parameter
   set.  The elliptic curve group operation is only defined between two
   points in the same group.  It is an error to apply the group
   operation to two elements that are from different groups, or to apply
   the group operation to a pair of coordinates that is not a valid
   point.  (A pair (x,y) of coordinates in Fp is a valid point only when
   they satisfy the curve equation.)  See Section 9.3 for further
   information.

   For each elliptic curve group, the discriminant - 16*(4*a^3 + 27*b^2)
   must be nonzero modulo p.

   Parameter generation is out of scope for this note.

3.3.1.  Security

   Security is highly dependent on the choice of these parameters.  This
   section gives normative guidance on acceptable choices.  See also
   Section 9 for informative guidance.

   The order of the group generated by g MUST be divisible by a large
   prime, in order to preclude easy solution of the discrete logarithm
   problem [K1987]

   With some parameter choices, the discrete log problem is
   significantly easier to solve.  This includes parameter sets in which
   b = 0 and p = 3 (mod 4), and parameter sets in which a = 0 and
   p = 2 (mod 3) [MOV1993].  These parameter choices are inferior for
   cryptographic purposes and SHOULD NOT be used.


4.  Elliptic Curve Diffie-Hellman (ECDH)

   The Diffie-Hellman (DH) key exchange protocol [DH1976] allows two
   parties communicating over an insecure channel to agree on a secret
   key.  It was originally defined in terms of operations in the



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   multiplicative group of a field with a large prime characteristic.
   Massey [M1983] observed that it can be easily generalized so that it
   is defined in terms of an arbitrary mathematical group.  Miller
   [M1985] and Koblitz [K1987] analyzed the DH protocol over an elliptic
   curve group.  We describe DH following the former reference.

   Let G be a group, and g be a generator for that group, and let t
   denote the order of G. The DH protocol runs as follows.  Party A
   chooses an exponent j between 1 and t-1, inclusive, uniformly at
   random, computes g^j and sends that element to B. Party B chooses an
   exponent k between 1 and t-1, inclusive, uniformly at random,
   computes g^k and sends that element to A. Each party can compute
   g^(j*k); party A computes (g^k)^j, and party B computes (g^j)^k.

   See Appendix B regarding generation of random integers.

4.1.  Data Types

   An ECDH private key z is an integer in Zt.

   The corresponding ECDH public key Y is the group element, where Y =
   g^z.  Each public key is associated with a particular particular
   parameter set as per Section 3.3.

   The shared secret computed by both parties is a group element.

   Each run of the ECDH protocol is associated with a particular
   parameter set, and both of the public keys and the shared secret are
   elements of that group.

4.2.  Compact Representation

   As described in the final paragraph of [M1985], the x-coordinate of
   the shared secret value g^(j*k) is a suitable representative for the
   entire point whenever exponentiation is used as a one-way function.
   In the ECDH key exchange protocol, after the element g^(j*k) has been
   computed, the x-coordinate of that value can be used as the shared
   secret.  We call this compact output.

   Following [M1985] again, when compact output is used in ECDH, only
   the x-coordinate of an elliptic curve point needs to be transmitted,
   instead of both coordinates as in the typical affine coordinate
   representation.  We call this the compact representation.  Its
   mathematical background is explained in Appendix C.

   ECDH can be used with or without compact output.  Both parties in a
   particular run of the ECDH protocol MUST use the same method.  ECDH
   can be used with or without compact representation.  If compact



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   representation is used in a particular run of the ECDH protocol, then
   compact output MUST be used as well.


5.  Elliptic Curve ElGamal Signatures

5.1.  Background

   The ElGamal signature algorithm was introduced in 1984 [E1984a]
   [E1984b] [E1985].  It is based on the discrete logarithm problem, and
   was originally defined for the multiplicative group of the integers
   modulo a large prime number.  It is straightforward to extend it to
   use other finite groups, such as the multiplicative group of the
   field GF(2^w) [AMV1990] or an elliptic curve group [A1992].

   An ElGamal signature consists of a pair of components.  There are
   many possible generalizations of ElGamal signature methods that have
   been obtained by different rearrangements of the equation for the
   second component; see [HMP1994], [HP1994], [NR1994], [A1992], and
   [AMV1990].  These generalizations are independent of the mathematical
   group used, and have been described for the multiplicative group
   modulo a prime number, the multiplicative group of GF(2^w), and
   elliptic curve groups [HMP1994][NR1994] [AMV1990][A1992].

   The Digital Signature Algorithm (DSA) [FIPS186] is an important
   ElGamal signature variant.

5.2.  Hash Functions

   ElGamal signatures must use a collision-resistant hash function, so
   that it can sign messages of arbitrary length and can avoid
   existential forgery attacks; see Section 9.4.  (This is true for all
   ElGamal variants [HMP1994].)  We denote the hash function as h().
   Its input is a bit string of arbitrary length, and its output is a
   non-negative integer.

   Let H() denote a hash function whose output is a fixed-length bit
   string.  To use H in an ElGamal signature method, we define the
   mapping between that output and the non-negative integers; this
   realizes the function h() described above.  Given a bit string m, the
   function h(m) is computed as follows:

   1.  H(m) is evaluated; the result is a fixed-length bit string.

   2.  Convert the resulting bit string to an integer i by treating its
       leftmost (initial) bit as the most significant bit of i, and
       treating its rightmost (final) bit as the least significant bit
       of i.



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5.3.  KT-IV Signatures

   Koyama and Tsuruoka described a signature method based on Elliptic
   Curve ElGamal, in which the first signature component is the
   x-coordinate of an elliptic curve point reduced modulo q [KT1994].
   In this section we recall that method, which we refer to as KT-IV.

   The algorithm uses an elliptic curve group, as described in
   Section 3.3, with prime field order p, and curve equation parameters
   a and b.  We denote the generator as alpha, and the order of the
   generator as q.  We follow [FIPS186] in checking for exceptional
   cases.

5.3.1.  Keypair Generation

   The private key z is an integer between 1 and q - 1, inclusive,
   generated uniformly at random.  (See Appendix B regarding random
   integers.)  The public key is the group element
   Y = alpha^z.  Each public key is associated with a particular
   particular parameter set as per Section 3.3.

5.3.2.  Signature Creation

   To compute a KT-IV signature for a message m using the private key z:

   1.  Choose an integer k uniformly at random from the set of all
       integers between 1 and q-1, inclusive.  (See Appendix B regarding
       random integers.)

   2.  Calculate R = (r_x, r_y) = alpha^k.

   3.  Calculate s1 = r_x mod q.

   4.  Calculate s2 = k / (h(m) + z * s1) mod q.

   5.  As an option, one MAY check if s1 = 0 or s2 = 0.  If either
       s1 = 0 or s2 = 0, a new value of k SHOULD be generated and the
       signature SHOULD be recalculated.  (It is extremely unlikely that
       s1 = 0 or s2 = 0 if signatures are generated properly.)

   The signature is the ordered pair (s1, s2).  Both signature
   components are non-negative integers.

5.3.3.  Signature Verification

   Given the message m, the public key Y, and the signature (s1, s2)
   verification is as follows:




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   1.  Check to see that 0 < s1 < q and 0 < s2 < q; if either condition
       is violated, the signature SHALL be rejected.

   2.  Compute the non-negative integers u1 and u2, where

          u1 = h(m) * s2 mod q, and

          u2 = s1 * s2 mod q.

   3.  Compute the elliptic curve point R' = alpha^u1 * Y^u2.

   4.  If the x-coordinate of R' mod q is equal to s1, then the
       signature and message pass the verification; otherwise, they
       fail.

5.4.  KT-I Signatures

   Horster, Michels, and Petersen categorized many different ElGamal
   signature methods, demonstrated their equivalence, and showed how to
   convert signatures of one type to another type [HMP1994].  In their
   terminology, the signature method of Section 5.3 and [KT1994] is a
   Type IV method, which is why it is denoted as KT-IV.

   A Type I KT signature method has a second component that is computed
   in the same manner as that of the Digital Signature Algorithm.  In
   this section, we describe this method, which we refer to as KT-I.

5.4.1.  Keypair Generation

   Keypairs and keypair generation are exactly as in Section 5.3.1.

5.4.2.  Signature Creation

   To compute a KT-I signature for a message m using the private key z:

   1.  Choose an integer k uniformly at random from the set of all
       integers between 1 and q-1, inclusive.  (See Appendix B regarding
       random integers.)

   2.  Calculate R = (r_x, r_y) = alpha^k.

   3.  Calculate s1 = r_x mod q.

   4.  Calculate s2 = (h(m) + z * s1) / k mod q.

   5.  As an option, one MAY check if s1 = 0 or s2 = 0.  If either
       s1 = 0 or s2 = 0, a new value of k SHOULD be generated and the
       signature SHOULD be recalculated.  (It is extremely unlikely that



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       s1 = 0 or s2 = 0 if signatures are generated properly.)

   The signature is the ordered pair (s1, s2).  Both signature
   components are non-negative integers.

5.4.3.  Signature Verification

   Given the message m, the public key Y, and the signature (s1, s2)
   verification is as follows:

   1.  Check to see that 0 < s1 < q and 0 < s2 < q; if either condition
       is violated the signature SHALL be rejected.

   2.  Compute s2_inv = 1 / s2 mod q.

   3.  Compute the non-negative integers u1 and u2, where

          u1 = h(m) * s2_inv mod q, and

          u2 = s1 * s2_inv mod q.

   4.  Compute the elliptic curve point R' = alpha^u1 * Y^u2.

   5.  If the x-coordinate of R' mod q is equal to s1, then the
       signature and message pass the verification; otherwise, they
       fail.

5.5.  Converting KT-IV signatures to KT-I signatures

   A KT-IV signature for a message m and a public key Y can easily be
   converted into a KT-I signature for the same message and public key.
   If (s1, s2) is a KT-IV signature for a message m, then
   (s1, 1/s2 mod q) is a KT-I signature for the same message [HMP1994].

   The conversion operation uses only public information, and it can be
   performed by the creator of the pre-conversion KT-IV signature, the
   verifier of the post-conversion KT-I signature, or by any other
   entity.

   An implementation MAY use this method to compute KT-I signatures.

5.6.  Rationale

   This subsection is not normative for this specification and is
   provided only as background information.

   [HMP1994] presents many generalizations of ElGamal signatures.
   Equation (5) of that reference shows the general signature equation



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      A = x_A * B + k * C (mod q)

   where x_A is the private key, k is the secret value, and A, B, and C
   are determined by the Type of the equation, as shown in Table 1 of
   [HMP1994].  DSA [FIPS186] is an EG-I.1 signature method (as is KT-I),
   with A = m, B = -r, and C = s.  (Here we use the notation of
   [HMP1994] in which the first signature component is r and the second
   signature component is s; in KT-I and KT-IV these components are
   denoted as s1 and s2 respectively.  The private key x_A corresponds
   to the private key z.)  Its signature equation is

      m = - r * z + s * k (mod q).

   The signature method of [KT1994] and Section 5.3 is an EG-IV.1
   method, with A = m * s, B = - r * s, C = 1.  Its signature equation
   is

      m * s = - r * s * z + k (mod q)

   The functions f and g mentioned in Table 1 are merely multiplication,
   as described under the heading "Fifth generalization".

   No hash function is shown in the above equations, but as described in
   Section 9.4, a hash function should be applied to the message prior
   to signing in order to prevent existential forgery attacks.

   Nyberg and Rueppel [NR1994] studied many different ElGamal signature
   methods, and defined "strong equivalence" as follows:

      Two signature methods are called strongly equivalent if the
      signature of the first scheme can be transformed efficiently into
      signatures of the second scheme and vice versa, without knowledge
      of the private key.

   KT-I and KT-IV signatures are obviously strongly equivalent.

   A valid signature with s2=0 leaks the secret key, since in that case
   z = - h(m) / s1 mod q.  We follow [FIPS186] in checking for this
   exceptional case and the case that s1=0.  The s2=0 check was
   suggested by Rivest [R1992] and is discussed in [BS1992].

   [KT1994] uses "a positive integer q' that does not exceed q" when
   computing the signature component s1 from the x-coordinate r_x of the
   elliptic curve point R = (r_x, r_y).  The value q' is also used
   during signature validation when comparing the x-coordinate of a
   computed elliptic curve point to the value to s1.  In this note, we
   use the simplifying convention that q' = q.




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6.  Interoperability

   The algorithms in this note can be used to interoperate with some
   other ECC specifications.  This section provides details for each
   algorithm.

6.1.  ECDH

   Section 4 can be used with the Internet Key Exchange (IKE) versions
   one [RFC2409] or two [RFC4306].  These algorithms are compatible with
   the ECP groups defined by [RFC4753], [RFC2409], and [RFC2412].  The
   group definition used in this protocol uses an affine coordinate
   representation of the public key and uses neither the compact output
   nor the compact representation of Section 4.2.  Note that some groups
   use a negative curve parameter "a" and express this fact in the curve
   equation rather than in the parameter.  The test cases in Section 8
   of [RFC4753] can be used to test an implementation; these cases use
   the multiplicative notation, as does this note.  The KEi and KEr
   payloads are equal to g^j and g^k, respectively, with 64 bits of
   encoding data prepended to them.

   The algorithms in Section 4 can be used to interoperate with the IEEE
   [P1363] and ANSI [X9.62] standards for ECDH based on fields of
   characteristic greater than three.  IEEE P1363 ECDH can be used in a
   manner that will interoperate with this note, with the following
   options and parameter choices from that specification:

      prime curves with a cofactor of 1,

      the ECSVDP-DH primitive, and

      the Key Derivation Function must be the "identity" function
      (equivalently, the KDF step should be omitted and the shared
      secret value should be output directly).

6.2.  KT-I and ECDSA

   The Digital Signature Algorithm (DSA) is based on the discrete
   logarithm problem over the multiplicative subgroup of the finite
   field with large prime order [DSA1991][FIPS186].  The Elliptic Curve
   Digital Signature Algorithm (ECDSA) [P1363] [X9.62] is an elliptic
   curve version of DSA.

   KT-I is mathematically and functionally equivalent to ECDSA, and can
   interoperate with the IEEE [P1363] and ANSI [X9.62] standards for
   Elliptic Curve DSA (ECDSA) based on fields of characteristic greater
   than three.  KT-I signatures can be verified using the ECDSA
   verification algorithm, and ECDSA signatures can be verified using



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   the KT-I verification algorithm.


7.  Validating an Implementation

   It is essential to validate the implementation of a cryptographic
   algorithm.  This section outlines tests that should be performed on
   the algorithms defined in this note.

   A known answer test, or KAT, uses a fixed set of inputs to test an
   algorithm; the output of the algorithm is compared with the expected
   output, which is also a fixed value.  KATs for ECDH and KT-I are set
   out in the following subsections.

   A consistency test generates inputs for one algorithm being tested
   using a second algorithm that is also being tested, then checks the
   output of the first algorithm.  A signature creation algorithm can be
   tested for consistency against a signature verification algorithm.
   Implementations of KT-I should be tested in this way.  Their
   signature generation processes are non-deterministic, and thus cannot
   be tested using a KAT.  Signature verification algorithms, on the
   other hand, are deterministic and should be tested via a KAT.  This
   combination of tests provides coverage for all of the operations,
   including keypair generation.  Consistency testing should also be
   applied to ECDH.

7.1.  ECDH

   An ECDH implementation can be validated using the known answer test
   cases from [RFC4753].  The correspondence between the notation in
   that RFC and the notation in this note are summarized in the
   following table.  (Refer to Section 3.3 and Section 4), and express
   the generator g in the affine coordinate representation as (gx, gy).

     +----------------------+---------------------------------------+
     | ECDH                 | RFC 4753                              |
     +----------------------+---------------------------------------+
     | order p of field Fp  | p                                     |
     | curve coefficient a  | -3                                    |
     | curve coefficient b  | b                                     |
     | generator g          | g=(gx, gy)                            |
     | private keys j and k | i and r                               |
     | public keys g^j, g^k | g^i = (gix, giy) and g^r = (grx, gry) |
     +----------------------+---------------------------------------+







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7.2.  KT-I

   A KT-I implementation can be validated using the known answer test
   cases from [RFC4754].  The correspondence between the notation in
   that RFC and the notation in this note are summarized in the
   following table.

                +---------------------+------------------+
                | KT-I                | RFC 4754         |
                +---------------------+------------------+
                | order p of field Fp | p                |
                | curve coefficient a | -3               |
                | curve coefficient b | b                |
                | generator alpha     | g                |
                | group order q       | q                |
                | private key z       | w                |
                | public key Y        | g^w = (gwx,gwy)  |
                | random k            | ephem priv k     |
                | s1                  | r                |
                | s2                  | s                |
                | s2_inv              | sinv             |
                | u1                  | u = h*sinv mod q |
                | u2                  | v = r*sinv mod q |
                +---------------------+------------------+


8.  Intellectual Property

   Concerns about intellectual property have slowed the adoption of ECC,
   because a number of optimizations and specialized algorithms have
   been patented in recent years.

   All of the normative references for ECDH (as defined in Section 4)
   were published during or before 1989, and those for KT-I were
   published during or before May, 1994.  All of the normative text for
   these algorithms is based solely on their respective references.

8.1.  Disclaimer

   This document is not intended as legal advice.  Readers are advised
   to consult their own legal advisers if they would like a legal
   interpretation of their rights.

   The IETF policies and processes regarding intellectual property and
   patents are outlined in [RFC3979] and [RFC4879] and at
   https://datatracker.ietf.org/ipr/about/.





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9.  Security Considerations

   The security level of an elliptic curve cryptosystem is determined by
   the cryptanalytic algorithm that is the least expensive for an
   attacker to implement.  There are several algorithms to consider.

   The Pohlig-Hellman method is a divide-and-conquer technique [PH1978].
   If the group order n can be factored as

      n = q1 * q2 * ... * qz,

   then the discrete log problem over the group can be solved by
   independently solving a discrete log problem in groups of order q1,
   q2, ..., qz, then combining the results using the Chinese remainder
   theorem.  The overall computational cost is dominated by that of the
   discrete log problem in the subgroup with the largest order.

   Shanks algorithm [K1981v3] computes a discrete logarithm in a group
   of order n using O(sqrt(n)) operations and O(sqrt(n)) storage.  The
   Pollard rho algorithm [P1978] computes a discrete logarithm in a
   group of order n using O(sqrt(n)) operations, with a negligible
   amount of storage, and can be efficiently parallelized [VW1994].

   The Pollard lambda algorithm [P1978] can solve the discrete logarithm
   problem using O(sqrt(w)) operations and O(log(w)) storage, when the
   exponent is known to lie in an interval of width w.

   The algorithms described above work in any group.  There are
   specialized algorithms that specifically target elliptic curve
   groups.  There are no known subexponential algorithms against general
   elliptic curve groups, though there are methods that target certain
   special elliptic curve groups; see [MOV1993] and [FR1994].

9.1.  Subgroups

   A group consisting of a nonempty set of elements S with associated
   group operation * is a subgroup of the group with the set of elements
   G, if the latter group uses the same group operation and S is a
   subset of G. For each elliptic curve equation, there is an elliptic
   curve group whose group order is equal to the order of the elliptic
   curve; that is, there is a group that contains every point on the
   curve.

   The order m of the elliptic curve is divisible by the order n of the
   group associated with the generator; that is, for each elliptic curve
   group, m = n * c for some number c.  The number c is called the
   "cofactor" [P1363].  Each ECC parameter set as in Section 3.3 is
   associated with a particular cofactor.



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   It is possible and desirable to use a cofactor equal to 1.

9.2.  Diffie-Hellman

   Note that the key exchange protocol as defined in Section 4 does not
   protect against active attacks; Party A must use some method to
   ensure that (g^k) originated with the intended communicant B, rather
   than an attacker, and Party B must do the same with (g^j).

   It is not sufficient to authenticate the shared secret g^(j*k), since
   this leaves the protocol open to attacks that manipulate the public
   keys.  Instead, the values of the public keys g^x and g^y that are
   exchanged should be directly authenticated.  This is the strategy
   used by protocols that build on Diffie-Hellman and which use end-
   entity authentication to protect against active attacks, such as
   OAKLEY [RFC2412] and the Internet Key Exchange [RFC2409][RFC4306].

   When the cofactor of a group is not equal to 1, there are a number of
   attacks that are possible against ECDH.  See [VW1996], [AV1996], and
   [LL1997].

9.3.  Group Representation and Security

   The elliptic curve group operation does not explicitly incorporate
   the parameter b from the curve equation.  This opens the possibility
   that a malicious attacker could learn information about an ECDH
   private key by submitting a bogus public key [BMM2000].  An attacker
   can craft an elliptic curve group G' that has identical parameters to
   a group G that is being used in an ECDH protocol, except that b is
   different.  An attacker can submit a point on G' into a run of the
   ECDH protocol that is using group G, and gain information from the
   fact that the group operations using the private key of the device
   under attack are effectively taking place in G' instead of G.

   This attack can gain useful information about an ECDH private key
   that is associated with a static public key, that is, a public key
   that is used in more than one run of the protocol.  However, it does
   not gain any useful information against ephemeral keys.

   This sort of attack is thwarted if an ECDH implementation does not
   assume that each pair of coordinates in Zp is actually a point on the
   appropriate elliptic curve.

   These considerations also apply when ECDH is used with compact
   representation (see Appendix C).






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9.4.  Signatures

   Elliptic curve parameters should only be used if they come from a
   trusted source; otherwise, some attacks are possible [AV1996],
   [V1996].

   If no hash function is used in an ElGamal signature system, then the
   system is vulnerable to existential forgeries, in which an attacker
   who does not know a private key can generate valid signatures for the
   associated public key, but cannot generate a signature for a message
   of its own choosing.  (See [E1985] for instance.)  The use of a
   collision-resistant hash function eliminates this vulnerability.

   In principle, any collision-resistant hash function is suitable for
   use in KT signatures.  To facilitate interoperability, we recognize
   the following hashes as suitable for use as the function H defined in
   Section 5.2:

      SHA-256, which has a 256-bit output.

      SHA-384, which has a 384-bit output.

      SHA-512, which has a 512-bit output.

   All of these hash functions are defined in [FIPS180-2].

   The number of bits in the output of the hash used in KT signatures
   should be equal or close to the number of bits needed to represent
   the group order.


10.  IANA Considerations

   This note has no actions for IANA.  This section should be removed by
   the RFC editor before publication as an RFC.


11.  Acknowledgements

   The author expresses his thanks to the originators of elliptic curve
   cryptography, whose work made this note possible, and all of the
   reviewers, who provided valuable constructive feedback.  Thanks are
   especially due to Howard Pinder and Andrey Jivsov.


12.  References





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12.1.  Normative References

   [AMV1990]  Agnew, G., Mullin, R., and S. Vanstone, "Improved Digital
              Signature Scheme based on Discrete Exponentiation",
              Electronics Letters Vol. 26, No. 14, July, 1990.

   [BC1989]   Bender, A. and G. Castagnoli, "On the Implementation of
              Elliptic Curve Cryptosystems", Advances in Cryptology -
              CRYPTO '89 Proceedings Spinger Lecture Notes in Computer
              Science (LNCS) volume 435, 1989.

   [CC1986]   Chudnovsky, D. and G. Chudnovsky, "Sequences of numbers
              generated by addition in formal groups and new primality
              and factorization tests", Advances in Applied
              Mathematics Volume 7, Issue 4, December 1986.

   [D1966]    Deskins, W., "Abstract Algebra", MacMillan Company New
              York, 1966.

   [DH1976]   Diffie, W. and M. Hellman, "New Directions in
              Cryptography", IEEE Transactions in Information
              Theory IT-22, pp 644-654, 1976.

   [FR1994]   Frey, G. and H. Ruck, "A remark concerning m-divisibility
              and the discrete logarithm in the divisor class group of
              curves.", Mathematics of Computation Vol. 62, No. 206, pp.
              865-874, 1994.

   [HMP1994]  Horster, P., Michels, M., and H. Petersen, "Meta-ElGamal
              signature schemes", University of Technology Chemnitz-
              Zwickau Department of Computer Science Technical
              Report TR-94-5, May 1994.

   [K1981v2]  Knuth, D., "The Art of Computer Programming, Vol. 2:
              Seminumerical Algorithms", Addison Wesley , 1981.

   [K1987]    Koblitz, N., "Elliptic Curve Cryptosystems", Mathematics
              of Computation Vol. 48, 1987, 203-209, 1987.

   [KT1994]   Koyama, K. and Y. Tsuruoka, "Digital signature system
              based on elliptic curve and signer device and verifier
              device for said system", Japanese Unexamined Patent
              Application Publication H6-43809, February 18, 1994.

   [M1983]    Massey, J., "Logarithms in finite cyclic groups -
              cryptographic issues", Proceedings of the 4th Symposium on
              Information Theory , 1983.




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   [M1985]    Miller, V., "Use of elliptic curves in cryptography",
              Advances in Cryptology - CRYPTO '85 Proceedings Springer
              Lecture Notes in Computer Science (LNCS) volume 218, 1985.

   [MOV1993]  Menezes, A., Vanstone, S., and T. Okamoto, "Reducing
              Elliptic Curve Logarithms to Logarithms in a Finite
              Field", IEEE Transactions on Information Theory Vol 39,
              No. 5, pp. 1639-1646, September, 1993.

12.2.  Informative References

   [A1992]    Anderson, J., "Response to the proposed DSS",
              Communications of the ACM v.35 n.7 p.50-52, July 1992.

   [AV1996]   Anderson, R. and S. Vaudenay, "Minding Your P's and Q's",
              Advances in Cryptology - ASIACRYPT '96 Proceedings Spinger
              Lecture Notes in Computer Science (LNCS) volume 1163,
              1996.

   [BMM2000]  Biehl, I., Meyer, B., and V. Muller, "Differential fault
              analysis on elliptic curve cryptosystems", Advances in
              Cryptology - CRYPTO 2000 Proceedings Spinger Lecture Notes
              in Computer Science (LNCS) volume 1880, 2000.

   [BS1992]   Branstad, D. and M. Smid, "Response to Comments on the
              NIST Proposed Digital Signature Standard", Advances in
              Cryptology - CRYPTO '92 Proceedings Springer Lecture Notes
              in Computer Science (LNCS) volume 740, August 1992.

   [DSA1991]  "DIGITAL SIGNATURE STANDARD", Federal Register Vol. 56,
              August 1991.

   [E1984a]   ElGamal, T., "Cryptography and logarithms over finite
              fields", Stanford University UMI Order No. DA 8420519,
              1984.

   [E1984b]   ElGamal, T., "Cryptography and logarithms over finite
              fields", Advances in Cryptology - CRYPTO '84
              Proceedings Springer Lecture Notes in Computer Science
              (LNCS) volume 196, 1984.

   [E1985]    ElGamal, T., "A public key cryptosystem and a signature
              scheme based on discrete logarithms", IEEE Transactions on
              Information Theory Vol 30, No. 4, pp. 469-472, 1985.

   [FIPS180-2]
              "SECURE HASH STANDARD", Federal Information Processing
              Standard (FIPS) 180-2, August 2002.



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   [FIPS186]  "DIGITAL SIGNATURE STANDARD", Federal Information
              Processing Standard FIPS-186, May 1994.

   [HP1994]   Horster, P. and H. Petersen, "Verallgemeinerte ElGamal-
              Signaturen", Proceedings der Fachtagung SIS '94, Verlag
              der Fachvereine Zurich, 1994.

   [K1981v3]  Knuth, D., "The Art of Computer Programming, Vol. 3:
              Sorting and Searching", Addison Wesley , 1981.

   [KMOV1991]
              Koyama, K., Maurer, U., Vanstone, S., and T. Okamoto, "New
              Public-Key Schemes Based on Elliptic Curves over the Ring
              Zn", Advances in Cryptology - CRYPTO '91
              Proceedings Spinger Lecture Notes in Computer Science
              (LNCS) volume 576, 1991.

   [L1969]    "Computer technology applied to the theory of numbers",
              M.A.A. Studies in Mathematics 180-2, 1969.

   [LL1997]   Lim, C. and P. Lee, "A Key Recovery Attack on Discrete
              Log-based Schemes Using a Prime Order Subgroup", Advances
              in Cryptology - CRYPTO '97 Proceedings Spinger Lecture
              Notes in Computer Science (LNCS) volume 1294, 1997.

   [NR1994]   Nyberg, K. and R. Rueppel, "Message Recovery for Signature
              Schemes Based on the Discrete Logarithm Problem", Advances
              in Cryptology - EUROCRYPT '94 Proceedings Springer Lecture
              Notes in Computer Science (LNCS) volume 950, May 1994.

   [P1363]    "Standard Specifications for Public Key Cryptography",
              Institute of Electric and Electronic Engineers
              (IEEE) P1363, 2000.

   [P1978]    Pollard, J., "Monte Carlo methods for index computation
              mod p", Mathematics of Computation Vol. 32, 1978.

   [PH1978]   Pohlig, S. and M. Hellman, "An Improved Algorithm for
              Computing Logarithms over GF(p) and its Cryptographic
              Significance", IEEE Transactions on Information Theory Vol
              24, pp. 106-110, 1978.

   [R1988]    Rose, H., "A Course in Number Theory", Oxford University
              Press , 1988.

   [R1992]    Rivest, R., "Response to the proposed DSS", Communications
              of the ACM v.35 n.7 p.41-47., July 1992.




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   [RFC2119]  Bradner, S., "Key words for use in RFCs to Indicate
              Requirement Levels", BCP 14, RFC 2119, March 1997.

   [RFC2409]  Harkins, D. and D. Carrel, "The Internet Key Exchange
              (IKE)", RFC 2409, November 1998.

   [RFC2412]  Orman, H., "The OAKLEY Key Determination Protocol",
              RFC 2412, November 1998.

   [RFC3979]  Bradner, S., "Intellectual Property Rights in IETF
              Technology", BCP 79, RFC 3979, March 2005.

   [RFC4086]  Eastlake, D., Schiller, J., and S. Crocker, "Randomness
              Requirements for Security", BCP 106, RFC 4086, June 2005.

   [RFC4306]  Kaufman, C., "Internet Key Exchange (IKEv2) Protocol",
              RFC 4306, December 2005.

   [RFC4753]  Fu, D. and J. Solinas, "ECP Groups For IKE and IKEv2",
              RFC 4753, January 2007.

   [RFC4754]  Fu, D. and J. Solinas, "IKE and IKEv2 Authentication Using
              the Elliptic Curve Digital Signature Algorithm (ECDSA)",
              RFC 4754, January 2007.

   [RFC4879]  Narten, T., "Clarification of the Third Party Disclosure
              Procedure in RFC 3979", BCP 79, RFC 4879, April 2007.

   [SuiteB]   "NSA Suite B Cryptography", Web Page http://www.nsa.gov/
              ia/programs/suiteb_cryptography/index.shtml.

   [V1996]    Vaudenay, S., "Hidden Collisions on DSS", Advances in
              Cryptology - CRYPTO '96 Proceedings Spinger Lecture Notes
              in Computer Science (LNCS) volume 1109, 1996.

   [VW1994]   van Oorschot, P. and M. Wiener, "Parallel Collision Search
              with Application to Hash Functions and Discrete
              Logarithms", Proceedings of the 2nd ACM Conference on
              Computer and communications security  pp. 210-218, 1994.

   [VW1996]   van Oorschot, P. and M. Wiener, "On Diffie-Hellman key
              agreement with short exponents", Advances in Cryptology -
              EUROCRYPT '96 Proceedings Spinger Lecture Notes in
              Computer Science (LNCS) volume 1070, 1996.

   [X9.62]    "Public Key Cryptography for the Financial Services
              Industry: The Elliptic Curve Digital Signature Algorithm
              (ECDSA)", American National Standards Institute (ANSI)



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              X9.62.


Appendix A.  Key Words

   The definitions of these key words are quoted from [RFC2119] and are
   commonly used in Internet standards.  They are reproduced in this
   note in order to avoid a normative reference from after 1994.

   1.  MUST - This word, or the terms "REQUIRED" or "SHALL", mean that
       the definition is an absolute requirement of the specification.

   2.  MUST NOT - This phrase, or the phrase "SHALL NOT", mean that the
       definition is an absolute prohibition of the specification.

   3.  SHOULD - This word, or the adjective "RECOMMENDED", mean that
       there may exist valid reasons in particular circumstances to
       ignore a particular item, but the full implications must be
       understood and carefully weighed before choosing a different
       course.

   4.  SHOULD NOT - This phrase, or the phrase "NOT RECOMMENDED" mean
       that there may exist valid reasons in particular circumstances
       when the particular behavior is acceptable or even useful, but
       the full implications should be understood and the case carefully
       weighed before implementing any behavior described with this
       label.

   5.  MAY - This word, or the adjective "OPTIONAL", mean that an item
       is truly optional.  One vendor may choose to include the item
       because a particular marketplace requires it or because the
       vendor feels that it enhances the product while another vendor
       may omit the same item.  An implementation which does not include
       a particular option MUST be prepared to interoperate with another
       implementation which does include the option, though perhaps with
       reduced functionality.  In the same vein an implementation which
       does include a particular option MUST be prepared to interoperate
       with another implementation which does not include the option
       (except, of course, for the feature the option provides.)


Appendix B.  Random Integer Generation

   It is easy to generate an integer uniformly at random between zero
   and 2^t -1, inclusive, for some positive integer t.  Generate a
   random bit string that contains exactly t bits, and then convert the
   bit string to a non-negative integer by treating the bits as the
   coefficients in a base-two expansion of an integer.



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   It is sometimes necessary to generate an integer r uniformly at
   random so that r satisfies a certain property P, for example, lying
   within a certain interval.  A simple way to do this is with the
   rejection method:

   1.  Generate a candidate number c uniformly at random from a set that
       includes all numbers that satisfy property P (plus some other
       numbers, preferably not too many)

   2.  If c satisfies property P, then return c.  Otherwise, return to
       Step 1.

   For example, to generate a number between 1 and n-1, inclusive,
   repeatedly generate integers between zero and 2^t - 1, inclusive,
   stopping at the first integer that falls within that interval.

   Recommendations on how to generate random bit strings are provided in
   [RFC4086].


Appendix C.  Why Compact Representation Works

   In the affine representation, the x-coordinate of the point P^i does
   not depend on the y-coordinate of the point P, for any non-negative
   exponent i and any point P. This fact can be seen as follows.  When
   given only the x-coordinate of a point P, it is not possible to
   determine exactly what the y-coordinate is, but the y value will be a
   solution to the curve equation

      y^2 = x^3 + a*x + b (mod p).

   There are at most two distinct solutions y = w and y = -w mod p, and
   the point P must be either Q=(x,w) or Q^-1=(x,-w).  Thus P^n is equal
   to either Q^n or (Q^-1)^n = (Q^n)^-1.  These values have the same
   x-coordinate.  Thus, the x-coordinate of a point P^i can be computed
   from the x-coordinate of a point P by computing one of the possible
   values of the y coordinate of P, then computing the ith power of P,
   and then ignoring the y-coordinate of that result.

   In general, it is possible to compute a square root modulo p by using
   Shanks's method [K1987]; simple methods exist for some values of p.
   When p = 3 (mod 4), the square roots of z mod p are w and -w mod p,
   where

      w = z ^ ((p+1)/4) (mod p);

   this observation is due to Lehmer [L1969].  When p satisfies this
   property, y can be computed from the curve equation, and either y = w



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   or y = -w mod p, where

      w = (x^3 + a*x + b)^((p+1)/4) (mod p).

   Square roots modulo p only exist for a quadratic residue modulo p,
   [R1988]; if z is not a quadratic residue, then there is no number w
   such that w^2 = z (mod p).  A simple way to verify that z is a
   quadratic residue after computing w is to verify that
   w * w = z (mod p).  If this relation does not hold for the above
   equation, then the value x is not a valid x-coordinate for a valid
   elliptic curve point.  This is an important consideration when ECDH
   is used with compact output; see Section 9.3.

   The primes used in the P-256, P-384, and P-521 curves described in
   [RFC4753] all have the property that p = 3 (mod 4).


Appendix D.  Example ECC Parameter Set

   For concreteness, we recall an elliptic curve defined by Solinas and
   Fu in [RFC4753] and referred to as P-256, which is believed to
   provide a 128-bit security level.  We use the notation of
   Section 3.3, and express the generator in the affine coordinate
   representation g=(gx,gy), where the values gx and gy are in Fp.

   p: FFFFFFFF00000001000000000000000000000000FFFFFFFFFFFFFFFFFFFFFFFF

   a: - 3

   b: 5AC635D8AA3A93E7B3EBBD55769886BC651D06B0CC53B0F63BCE3C3E27D2604B

   n: FFFFFFFF00000000FFFFFFFFFFFFFFFFBCE6FAADA7179E84F3B9CAC2FC632551

   gx: 6B17D1F2E12C4247F8BCE6E563A440F277037D812DEB33A0F4A13945D898C296

   gy: 4FE342E2FE1A7F9B8EE7EB4A7C0F9E162BCE33576B315ECECBB6406837BF51F5

   Note that p can also be expressed as

      p = 2^(256)-2^(224)+2^(192)+2^(96)-1.


Appendix E.  Additive and multiplicative notation

   The early publications on elliptic curve cryptography used
   multiplicative notation, but most modern publications use additive
   notation.  This section includes a table mapping between those two
   conventions.  In this section, a and b are elements of an elliptic



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   curve group, and N is an integer.

            +-------------------------+-----------------------+
            | Multiplicative Notation | Additive Notation     |
            +-------------------------+-----------------------+
            | multiplication          | addition              |
            | a * b                   | a + b                 |
            | squaring                | doubling              |
            | a * a = a^2             | a + a = 2a            |
            | exponentiation          | scalar multiplication |
            | a^N = a * a * ... * a   | Na = a + a + ... + a  |
            | inverse                 | inverse               |
            | a^-1                    | -a                    |
            +-------------------------+-----------------------+


Authors' Addresses

   David A. McGrew
   Cisco Systems
   510 McCarthy Blvd.
   Milpitas, CA  95035
   USA

   Phone: (408) 525 8651
   Email: mcgrew@cisco.com
   URI:   http://www.mindspring.com/~dmcgrew/dam.htm


   Kevin M. Igoe
   National Security Agency
   Commercial Solutions Center
   United States of America

   Email: kmigoe@nsa.gov
















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