[Docs] [txt|pdf|xml|html] [Tracker] [WG] [Email] [Diff1] [Diff2] [Nits]

Versions: (draft-kato-threat-pairing) 00 01 02

Network Working Group                                        S. Yonezawa
Internet-Draft                                                   Lepidum
Intended status: Experimental                                 S. Chikara
Expires: August 1, 2019                                  NTT TechnoCross
                                                            T. Kobayashi
                                                                T. Saito
                                                                     NTT
                                                        January 28, 2019


                        Pairing-Friendly Curves
               draft-yonezawa-pairing-friendly-curves-00

Abstract

   This memo introduces pairing-friendly curves used for constructing
   pairing-based cryptography.  It describes recommended parameters for
   each security level and recent implementations of pairing-friendly
   curves.

Status of This Memo

   This Internet-Draft is submitted in full conformance with the
   provisions of BCP 78 and BCP 79.

   Internet-Drafts are working documents of the Internet Engineering
   Task Force (IETF).  Note that other groups may also distribute
   working documents as Internet-Drafts.  The list of current Internet-
   Drafts is at https://datatracker.ietf.org/drafts/current/.

   Internet-Drafts are draft documents valid for a maximum of six months
   and may be updated, replaced, or obsoleted by other documents at any
   time.  It is inappropriate to use Internet-Drafts as reference
   material or to cite them other than as "work in progress."

   This Internet-Draft will expire on August 1, 2019.

Copyright Notice

   Copyright (c) 2019 IETF Trust and the persons identified as the
   document authors.  All rights reserved.

   This document is subject to BCP 78 and the IETF Trust's Legal
   Provisions Relating to IETF Documents
   (https://trustee.ietf.org/license-info) in effect on the date of
   publication of this document.  Please review these documents
   carefully, as they describe your rights and restrictions with respect
   to this document.  Code Components extracted from this document must



Yonezawa, et al.         Expires August 1, 2019                 [Page 1]


Internet-Draft           Pairing-Friendly Curves            January 2019


   include Simplified BSD License text as described in Section 4.e of
   the Trust Legal Provisions and are provided without warranty as
   described in the Simplified BSD License.

Table of Contents

   1.  Introduction  . . . . . . . . . . . . . . . . . . . . . . . .   2
     1.1.  Pairing-Based Cryptography  . . . . . . . . . . . . . . .   2
     1.2.  Applications of Pairing-Based Cryptography  . . . . . . .   3
     1.3.  Goal  . . . . . . . . . . . . . . . . . . . . . . . . . .   4
     1.4.  Requirements Terminology  . . . . . . . . . . . . . . . .   4
   2.  Preliminaries . . . . . . . . . . . . . . . . . . . . . . . .   4
     2.1.  Elliptic Curve  . . . . . . . . . . . . . . . . . . . . .   4
     2.2.  Pairing . . . . . . . . . . . . . . . . . . . . . . . . .   5
     2.3.  Barreto-Naehrig Curve . . . . . . . . . . . . . . . . . .   5
     2.4.  Barreto-Lynn-Scott Curve  . . . . . . . . . . . . . . . .   6
   3.  Security of Pairing-Friendly Curves . . . . . . . . . . . . .   7
     3.1.  Evaluating the Security of Pairing-Friendly Curves  . . .   7
     3.2.  Impact of the Recent Attack . . . . . . . . . . . . . . .   7
   4.  Security Evaluation of Pairing-Friendly Curves  . . . . . . .   8
     4.1.  For 100 Bits of Security  . . . . . . . . . . . . . . . .   8
     4.2.  For 128 Bits of Security  . . . . . . . . . . . . . . . .   9
     4.3.  For 256 Bits of Security  . . . . . . . . . . . . . . . .   9
   5.  Implementations of Pairing-Friendly Curves  . . . . . . . . .   9
   6.  Security Considerations . . . . . . . . . . . . . . . . . . .  11
   7.  IANA Considerations . . . . . . . . . . . . . . . . . . . . .  11
   8.  Acknowledgements  . . . . . . . . . . . . . . . . . . . . . .  12
   9.  Change log  . . . . . . . . . . . . . . . . . . . . . . . . .  12
   10. References  . . . . . . . . . . . . . . . . . . . . . . . . .  12
     10.1.  Normative References . . . . . . . . . . . . . . . . . .  12
     10.2.  Informative References . . . . . . . . . . . . . . . . .  13
   Appendix A.  Test Vectors of Optimal Ate Pairing  . . . . . . . .  17
   Authors' Addresses  . . . . . . . . . . . . . . . . . . . . . . .  17

1.  Introduction

1.1.  Pairing-Based Cryptography

   Elliptic curve cryptography is one of the important areas in recent
   cryptography.  The cryptographic algorithms based on elliptic curve
   cryptography, such as ECDSA, is widely used in many applications.

   Pairing-based cryptography, a variant of elliptic curve cryptography,
   is attracted the attention for its flexible and applicable
   functionality.  Pairing is a special map defined over elliptic
   curves.  Generally, elliptic curves is defined so that pairing is not
   efficiently computable since elliptic curve cryptography is broken if
   pairing is efficiently computable.  As the importance of pairing



Yonezawa, et al.         Expires August 1, 2019                 [Page 2]


Internet-Draft           Pairing-Friendly Curves            January 2019


   grows, elliptic curves where pairing is efficiently computable are
   studied and the special curves called pairing-friendly curves are
   proposed.

   Thanks to the characteristics of pairing, it can be applied to
   construct several cryptographic algorithms and protocols such as
   identity-based encryption (IBE), attribute-based encryption (ABE),
   authenticated key exchange (AKE), short signatures and so on.
   Several applications of pairing-based cryptography is now in
   practical use.

1.2.  Applications of Pairing-Based Cryptography

   Several applications using pairing-based cryptography are
   standardized and implemented.  We show example applications available
   in the real world.

   IETF issues RFCs for pairing-based cryptography such as identity-
   based cryptography [9], certificateless signatures [10], Sakai-
   Kasahara Key Encryption (SAKKE) [11], and Identity-Based
   Authenticated Key Exchange (IBAKE) [12].  SAKKE is applied to
   Multimedia Internet KEYing (MIKEY) [13] and used in 3GPP [14].

   Pairing-based key agreement protocols are standardized in ISO/IEC
   [15].  In [15], a key agreement scheme by Joux [16], identity-based
   key agreement schemes by Smart-Chen-Cheng [17] and by Fujioka-Suzuki-
   Ustaoglu [18] are specified.

   MIRACL implements M-Pin, a multi-factor authentication protocol [19].
   M-Pin protocol includes a kind of zero-knowledge proof, where pairing
   is used for its construction.

   Trusted Computing Group (TCG) specifies ECDAA (Elliptic Curve Direct
   Anonymous Attestation) in the specification of Trusted Platform
   Module (TPM) [20].  ECDAA is a protocol for proving the attestation
   held by a TPM to a verifier without revealing the attestation held by
   that TPM.  Pairing is used for constructing ECDAA.  FIDO Alliance
   [21] and W3C [22] also published ECDAA algorithm similar to TCG.

   Zcash implements their own zero-knowledge proof algorithm named zk-
   SNARKs (Zero-Knowledge Succinct Non-Interactive Argument of
   Knowledge) [23]. zk-SNARKs is used for protecting privacy of
   transactions of Zcash.  They use pairing for constructing zk-SNARKS.

   Cloudflare introduced Geo Key Manager [24] to restrict distribution
   of customers' private keys to the subset of their data centers.  To
   achieve this functionality, attribute-based encryption is used and
   pairing takes a role as a building block.



Yonezawa, et al.         Expires August 1, 2019                 [Page 3]


Internet-Draft           Pairing-Friendly Curves            January 2019


   DFINITY utilized threshold signature scheme to generate the
   decentralized random beacons [25].  They constructed a BLS signature-
   based scheme, which is based on pairings.

   In Ethereum 2.0, project Prysm applies signature aggregation for
   scalability benefits by leveraging DFINITY's random-beacon chain
   playground [26].  Their codes are published on GitHub.

1.3.  Goal

   The goal of this memo is to consider the security of pairing-friendly
   curves used in pairing-based cryptography and introduce secure
   parameters of pairing-frindly curves.  Specifically, we explain the
   recent attack against pairing-friendly curves and how much the
   security of the curves is reduced.  We show how to evaluate the
   security of pairing-friendly curves and give the parameters for 100
   bits of security, which is no longer secure, 128 and 256 bits of
   security.

1.4.  Requirements Terminology

   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 [1].

2.  Preliminaries

2.1.  Elliptic Curve

   Let p > 3 be a prime and F_p be a finite field.  The curve defined by
   the following equation E is called an elliptic curve.

      E : y^2 = x^3 + A * x + B,

   where A, B are in F_p and satisfies 4 * A^3 + 27 * B^2 != 0 mod p.

   Solutions (x, y) for an elliptic curve E, as well as the point at
   infinity, O_E, are called F_p-rational points.  If P and Q are two
   points on the curve E, we can define R = P + Q as the opposite point
   of the intersection between the curve E and the line that intersects
   P and Q.  We can define P + O_E = P = O_E + P as well.  The additive
   group is constructed by the well-defined operation in the set of F_p-
   rational points.  Similarly, a scalar multiplication S = [a]P for a
   positive integer a can be defined as an a-time addition of P.

   Typically, the cyclic additive group with a prime order r and the
   base point G in its group is used for the elliptic curve




Yonezawa, et al.         Expires August 1, 2019                 [Page 4]


Internet-Draft           Pairing-Friendly Curves            January 2019


   cryptography.  Furthermore, we define terminology used in this memo
   as follows.

      O_E: the point at infinity over an elliptic curve E.

      #E(F_p): number of points on an elliptic curve E over F_p.

      h: a cofactor such that h = #E(F_p)/r.

      k: an embedding degree, a minimum integer such that r is a divisor
      of p^k - 1.

2.2.  Pairing

   Pairing is a kind of the bilinear map defined over an elliptic curve.
   Examples include Weil pairing, Tate pairing, optimal Ate pairing [2]
   and so on.  Especially, optimal Ate pairing is considered to be
   efficient to compute and mainly used for practical implementation.

   Let E be an elliptic curve defined over the prime field F_p.  Let G_1
   be a cyclic subgroup generated by a rational point on E with order r,
   and G_2 be a cyclic subgroup generated by a twisted curve E' of E
   with order r.  Let G_T be an order r subgroup of a field F_p^k, where
   k is an embedded degree.  Pairing is defined as a bilinear map e:
   (G_1, G_2) -> G_T satisfying the following properties:

   (1)  Bilinearity: for any S in G_1, T in G_2, a, b in Z_r, we have
        the relation e([a]S, [b]T) = e(S, T)^{a * b}.

   (2)  Non-degeneracy: for any T in G_2, e(S, T) = 1 if and only if S =
        O_E.  Similarly, for any S in G_1, e(S, T) = 1 if and only if T
        = O_E.

   (3)  Computability: for any S in G_1 and T in G_2, the bilinear map
        is efficiently computable.

2.3.  Barreto-Naehrig Curve

   A BN curve [3] is one of the instantiations of pairing-friendly
   curves proposed in 2005.  A pairing over BN curves constructs optimal
   Ate pairings.

   A BN curve is an elliptic curve E defined over a finite field F_p,
   where p is more than or equal to 5, such that p and its order r are
   prime numbers parameterized by

      p = 36u^4 + 36u^3 + 24u^2 + 6u + 1




Yonezawa, et al.         Expires August 1, 2019                 [Page 5]


Internet-Draft           Pairing-Friendly Curves            January 2019


      r = 36u^4 + 36u^3 + 18u^2 + 6u + 1

   for some well chosen integer u.  The elliptic curve has an equation
   of the form E: y^2 = x^3 + b, where b is an element of multiplicative
   group of order p.

   BN curves always have order 6 twists.  If w is an element which is
   neither a square nor a cube in a finite field F_p^2, the twisted
   curve E' of E is defined over a finite field F_p^2 by the equation
   E': y^2 = x^3 + b' with b' = b/w or b' = bw.

   A pairing e is defined by taking G_1 as a cyclic group composed by
   rational points on the elliptic curve E, G_2 as a cyclic group
   composed by rational points on the elliptic curve E', and G_T as a
   multiplicative group of order p^12.

2.4.  Barreto-Lynn-Scott Curve

   A BLS curve [4] is another instantiations of pairings proposed in
   2002.  Similar to BN curves, a pairing over BLS curves constructs
   optimal Ate pairings.

   A BLS curve is an elliptic curve E defined over a finite field F_p by
   an equation of the form E: y^2 = x^3 + b and has a twist of order 6
   defined in the same way as BN curves.  In contrast to BN curves, a
   BLS curve does not have a prime order but its order is divisible by a
   large parameterized prime r and the pairing will be defined on the
   r-torsions points.

   BLS curves vary according to different embedding degrees.  In this
   memo, we deal with BLS12 and BLS48 families with embedding degrees 12
   and 48 with respect to r, respectively.

   In BLS curves, parameterized p and r are given by the following
   equations:

      BLS12:

         p = (u - 1)^2 (u^4 - u^2 + 1)/3 + u

         r = u^4 - u^2 + 1

      BLS48:

         p = (u - 1)^2 (u^16 - u^8 + 1)/3 + u

         r = u^16 - u^8 + 1




Yonezawa, et al.         Expires August 1, 2019                 [Page 6]


Internet-Draft           Pairing-Friendly Curves            January 2019


   for some well chosen integer u.

3.  Security of Pairing-Friendly Curves

3.1.  Evaluating the Security of Pairing-Friendly Curves

   The security of pairing-friendly curves is evaluated by the hardness
   of the following discrete logarithm problems.

   o  The elliptic curve discrete logarithm problem (ECDLP) in G_1 and
      G_2

   o  The finite field discrete logarithm problem (FFDLP) in G_T

   There are other hard problems over pairing-friendly curves, which are
   used for proving the security of pairing-based cryptography.  Such
   problems include bilinear computational Diffie-Hellman (BCDH)
   problem, bilinear decisional Diffie-Hellman (BDDH) problem, gap BDDH
   problem, etc [27].  Almost all of these variants are reduced to the
   hardness of discrete logarithm problems described above and believed
   to be easier than the discrete logarithm problems.

   There would be the case where the attacker solves these reduced
   problems to break the pairing-based cryptography.  Since such attacks
   have not been discovered yet, we discuss the hardness of the discrete
   logarithm problems in this memo.

   The security level of pairing-friendly curves is estimated by the
   computational cost of the most efficient algorithm to solve the above
   discrete logarithm problems.  The well-known algorithms for solving
   the discrete logarithm problems includes Pollard's rho algorithm
   [28], Index Calculus [29] and so on.  In order to make index calculus
   algorithms more efficient, number field sieve (NFS) algorithms are
   utilized.

   In addition, the special case where the cofactors of G_1, G_2 and G_T
   are small should be taken care [30].  In such case, the discrete
   logarithm problem can be efficiently solved.  One has to choose
   parameters so that the cofactors of G_1, G_2 and G_T contain no prime
   factors smaller than |G_1|, |G_2| and |G_T|.

3.2.  Impact of the Recent Attack

   In 2016, Kim and Barbulescu proposed a new variant of the NFS
   algorithms, the extended number field sieve (exTNFS), which
   drastically reduces the complexity of solving FFDLP [5].  Due to
   exTNFS, the security level of pairing-friendly curves asymptotically
   dropped down.  For instance, Barbulescu and Duquesne estimates that



Yonezawa, et al.         Expires August 1, 2019                 [Page 7]


Internet-Draft           Pairing-Friendly Curves            January 2019


   the security of the BN curves which was believed to provide 128 bits
   of security (BN256, for example) dropped down to approximately 100
   bits [6].

   Some papers show the minimum bitlength of the parameters of pairing-
   friendly curves for each security level when applying exTNFS as an
   attacking method for FFDLP.  For 128 bits of security, Menezes,
   Sarkar and Singh estimated the minimum bitlength of p of BN curves
   after exTNFS as 383 bits, and that of BLS12 curves as 384 bits [7].
   For 256 bits of security, Kiyomura et al. estimated the minimum
   bitlength of p^k of BLS48 curves as 27,410 bits, which implied 572
   bits of p [8].

4.  Security Evaluation of Pairing-Friendly Curves

   We give security evaluation for pairing-friendly curves based on the
   evaluating method presented in Section 3.  We also introduce secure
   parameters of pairing-friendly curves for each security level.  The
   parameters introduced here are chosen with the consideration of
   security, efficiency and global acceptance.

   For security, we introduce 100 bits, 128 bits and 256 bits of
   security.  We note that 100 bits of security is no longer secure and
   recommend 128 bits and 256 bits of security for secure applications.
   We follow TLS 1.3 which specifies the cipher suites with 128 bits and
   256 bits of security as mandatory-to-implement for the choice of the
   security level.

   Implementors of the applications have to choose the parameters with
   appropriate security level according to the security requirements of
   the applications.  For efficiency, we refer to the benchmark by mcl
   [31] for 128 bits of security, and by Kiyomura et al. [8] for 256
   bits of security and choose sufficiently efficient parameters.  For
   global acceptance, we give the implementations of pairing-friendly
   curves in section Section 5.

4.1.  For 100 Bits of Security

   Before exTNFS, BN curves with 256-bit size of underlying finite field
   (so-called BN256) were considered to have 128 bits of security.
   After exTNFS, however, the security level of BN curves with 256-bit
   size of underlying finite field fell into 100 bits.

   Implementors who will newly develop the applications of pairing-based
   cryptography SHOULD NOT use BN256 as a pairing-friendly curve when
   their applications require 128 bits of security.  In case an
   application does not require higher security level and is sufficient
   to have 100 bits of security (i.e.  IoT), implementors MAY use BN256.



Yonezawa, et al.         Expires August 1, 2019                 [Page 8]


Internet-Draft           Pairing-Friendly Curves            January 2019


4.2.  For 128 Bits of Security

   A BN curve with 128 bits of security is shown in [6], which we call
   BN462.  BN462 is defined by a parameter u = 2^114 + 2^101 - 2^14 - 1
   for the definition in Section 2.3.  Defined by u, the elliptic curve
   E and its twisted curve E' are represented by E: y^2 = x^3 - 4 and
   E': y^2 = x^3 - 4 * (1 + i), where i is an element of an extension
   field F_p^2, respectively.  The size of p becomes 462-bit length.

   A BLS12 curve with 128 bits of security shown in [6] is parameterized
   by u = -2^77 - 2^71 - 2^64 + 2^37 + 2^35 + 2^22 - 2^5, which we call
   BLS12-461.  Defined by u, the elliptic curve E and its twisted curve
   E' are represented by E: y^2 = x^3 - 2 and E': y^2 = x^3 - 2 / (1 +
   i), respectively.  The size of p becomes 461-bit length.  The curve
   BLS12-461 is subgroup-secure.

   There is another BLS12 curve stating 128 bits of security, BLS12-381
   [32].  It is defined by a parameter u = -0xd201000000010000.  Defined
   by u, the elliptic curve E and its twisted curve E' are represented
   by E: y^2 = x^3 + 4 and E': y^2 = x^3 + 4(i + 1), respectively.

   We have to note that, according to [7], the bit length of p for BLS12
   to achieve 128 bits of security is calculated as 384 bits and more,
   which BLS12-381 does not satisfy.  Although the computational time is
   conservatively estimated by 2^110 when exTNFS is applied with index
   calculus, there is no currently published efficient method for such
   computational time.  They state that BLS12-381 achieves 127-bit
   security level evaluated by the computational cost of Pollard's rho.

4.3.  For 256 Bits of Security

   As shown in Section 3.2, it is unrealistic to achieve 256 bits of
   security by BN curves since the minimum size of p becomes too large
   to implement.  Hence, we consider BLS48 for 256 bits of security.

   A BLS48 curve with 256 bits of security is shown in [8], which we
   call BLS48-581.  It is defined by a parameter u = -1 + 2^7 - 2^10 -
   2^30 - 2^32 and the elliptic curve E and its twisted curve E' are
   represented by E: y^2 = x^3 + 1 and E': y^2 = x^3 - 1/w, where w is
   an element of an extension field F_p^8.  The size of p becomes
   581-bit length.

5.  Implementations of Pairing-Friendly Curves

   We show the pairing-friendly curves selected by existing standards,
   applications and cryptographic libraries.





Yonezawa, et al.         Expires August 1, 2019                 [Page 9]


Internet-Draft           Pairing-Friendly Curves            January 2019


   ISO/IEC 15946-5 [33] shows examples of BN curves with the size of
   160, 192, 224, 256, 384 and 512 bits of p.  There is no action so far
   after the proposal of exTNFS.

   TCG adopts an BN curve of 256 bits specified in ISO/IEC 15946-5
   (TPM_ECC_BN_P256) and of 638 bits specified by their own
   (TPM_ECC_BN_P638).  FIDO Alliance [21] and W3C [22] adopt the BN
   curves specified in TCG, a 512-bit BN curve shown in ISO/IEC 15946-5
   and another 256-bit BN curve.

   MIRACL [34] implements BN curves and BLS12 curves.

   Zcash implemented a BN curve (named BN128) in their library libsnark
   [35].  After exTNFS, they propose a new parameter of BLS12 as
   BLS12-381 [32] and publish its experimental implementation [36].

   Cloudflare implements a 256-bit BN curve (bn256) [37].  There is no
   action so far after exTNFS.

   Ethereum 2.0 adopts BLS12-381 (BLS12_381), BN curves with 254 bits of
   p (CurveFp254BNb) and 382 bits of p (CurveFp382_1 and CurveFp382_2)
   [38].  Their implementation calls mcl [31] for pairing computation.

   Cryptographic libraries which implement pairings include PBC [39],
   mcl [31], RELIC [40], TEPLA [41], AMCL [42], Intel IPP [43] and a
   library by Kyushu University [44].

   Table 1 shows the adoption of pairing-friendly curves in existing
   standards, applications and libraries.

   +--------------+------------+--------------+----------------+-------+
   | Category     | Name       | 100 bit      | 128 bit        | 256   |
   |              |            |              |                | bit   |
   +--------------+------------+--------------+----------------+-------+
   | standards    | ISO/IEC    | BN256        | BN384          |       |
   |              | [33]       |              |                |       |
   |              |            |              |                |       |
   |              | TCG        | BN256        |                |       |
   |              |            |              |                |       |
   |              | FIDO/W3C   | BN256        |                |       |
   |              |            |              |                |       |
   | applications | MIRACL     | BN254        | BLS12          |       |
   |              |            |              |                |       |
   |              | Zcash      | BN128        | BLS12-381      |       |
   |              |            | (CurveSNARK) |                |       |
   |              |            |              |                |       |
   |              | Cloudflare | BN256        |                |       |
   |              |            |              |                |       |



Yonezawa, et al.         Expires August 1, 2019                [Page 10]


Internet-Draft           Pairing-Friendly Curves            January 2019


   |              | Ethereum   | BN254        | BN382 (*) /    |       |
   |              |            |              | BLS12-381 (*)  |       |
   |              |            |              |                |       |
   | libraries    | PBC        | BN254 /      | BN381_1 (*) /  |       |
   |              |            | BN_SNARK1    | BN462 /        |       |
   |              |            |              | BLS12-381      |       |
   |              |            |              |                |       |
   |              | mcl        | BN254 /      | BN381_1 (*) /  |       |
   |              |            | BN_SNARK1    | BN462 /        |       |
   |              |            |              | BLS12-381      |       |
   |              |            |              |                |       |
   |              | RELIC [40] | BN254 /      | BLS12-381 /    |       |
   |              |            | BN256        | BLS12-455      |       |
   |              |            |              |                |       |
   |              | TEPLA      | BN254        |                |       |
   |              |            |              |                |       |
   |              | AMCL       | BN254 /      | BLS12-381 (*)  | BLS48 |
   |              |            | BN256        | / BLS12-383    |       |
   |              |            |              | (*) /          |       |
   |              |            |              | BLS12-461      |       |
   |              |            |              |                |       |
   |              | Intel IPP  | BN256        |                |       |
   |              |            |              |                |       |
   |              | Kyushu     |              |                | BLS48 |
   |              | Univ.      |              |                |       |
   +--------------+------------+--------------+----------------+-------+

    (*) There is no research result on the security evaluation, but the
        implementers states that they satisfy 128 bits of security.

               Table 1: Adoption of Pairing-Friendly Curves

6.  Security Considerations

   This memo entirely describes the security of pairing-friendly curves,
   and introduces secure parameters of pairing-friendly curves.  We give
   these parameters in terms of security, efficiency and global
   acceptance.  The parameters for 100, 128 and 256 bits of security are
   introduced since the security level will different in the
   requirements of the pairing-based applications.

7.  IANA Considerations

   This document has no actions for IANA.







Yonezawa, et al.         Expires August 1, 2019                [Page 11]


Internet-Draft           Pairing-Friendly Curves            January 2019


8.  Acknowledgements

   The authors would like to thank Akihiro Kato for his significant
   contribution to the early version of this memo.

9.  Change log

   NOTE TO RFC EDITOR: Please remove this section in before final RFC
   publication.

10.  References

10.1.  Normative References

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

   [2]        Vercauteren, F., "Optimal pairings", Proceedings IEEE
              Transactions on Information Theory 56(1): 455-461 (2010),
              2010.

   [3]        Barreto, P. and M. Naehrig, "Pairing-Friendly Elliptic
              Curves of Prime Order", Selected Areas in Cryptography-SAC
              2005.  volume 3897 of Lecture Notes in Computer Science,
              pages 319-331, 2006.

   [4]        Barreto, P., Lynn, B., and M. Scott, "Constructing
              Elliptic Curves with Prescribed Embedding Degrees",
              Security in Communication Networks - SCN 2002 LNCS 2576,
              pp. 257--167, Springer, 2002.

   [5]        Kim, T. and R. Barbulescu, "Extended tower number field
              sieve: a new complexity for the medium prime case.",
              CRYPTO 2016 LNCS, vol. 9814, pp. 543.571, 2016.

   [6]        Barbulescu, R. and S. Duquesne, "Updating Key Size
              Estimations for Pairings", Journal of Cryptology 2018,
              January 2018.

   [7]        Menezes, A., Sarkar, P., and S. Singh, "Challenges with
              Assessing the Impact of NFS Advances on the Security of
              Pairing-Based Cryptography", Paradigms in Cryptology -
              Mycrypt 2016 LNCS 10311, pp. 83--108, Springer, 2017.

   [8]        Kiyomura, Y., Inoue, A., Kawahara, Y., Yasuda, M., Takagi,
              T., and T. Kobayashi, "Secure and Efficient Pairing at
              256-Bit Security Level", ACNS 2017 LNCS, vol. 10355, pp.
              59.79, 2017, 2017.



Yonezawa, et al.         Expires August 1, 2019                [Page 12]


Internet-Draft           Pairing-Friendly Curves            January 2019


10.2.  Informative References

   [9]        Boyen, X. and L. Martin, "Identity-Based Cryptography
              Standard (IBCS) #1: Supersingular Curve Implementations of
              the BF and BB1 Cryptosystems", RFC 5091,
              DOI 10.17487/RFC5091, December 2007,
              <https://www.rfc-editor.org/info/rfc5091>.

   [10]       Groves, M., "Elliptic Curve-Based Certificateless
              Signatures for Identity-Based Encryption (ECCSI)",
              RFC 6507, DOI 10.17487/RFC6507, February 2012,
              <https://www.rfc-editor.org/info/rfc6507>.

   [11]       Groves, M., "Sakai-Kasahara Key Encryption (SAKKE)",
              RFC 6508, DOI 10.17487/RFC6508, February 2012,
              <https://www.rfc-editor.org/info/rfc6508>.

   [12]       Cakulev, V., Sundaram, G., and I. Broustis, "IBAKE:
              Identity-Based Authenticated Key Exchange", RFC 6267,
              DOI 10.17487/RFC6267, March 2012,
              <https://www.rfc-editor.org/info/rfc6539>.

   [13]       Groves, M., "MIKEY-SAKKE: Sakai-Kasahara Key Encryption in
              Multimedia Internet KEYing (MIKEY)", RFC 6509,
              DOI 10.17487/RFC6509, February 2012,
              <https://www.rfc-editor.org/info/rfc6509>.

   [14]       3GPP, "Security of the mission critical service (Release
              15)", 3GPP TS 33.180 15.3.0, September 2018.

   [15]       ISO/IEC, "ISO/IEC 11770-3:2015", ISO/IEC  Information
              technology -- Security techniques -- Key management --
              Part 3: Mechanisms using asymmetric techniques, 2015.

   [16]       Joux, A., "A One Round Protocol for Tripartite Diffie-
              Hellman", ANTS-IV LNCS 1838, pp. 385--393, Springer-
              Verlag, 2000.

   [17]       Chen, L., Cheng, Z., and N. Smart, "Indentity-based Key
              Agreement Protocols From Pairings", International Journal
              of Information Security Volume 6 Issue 4, pages 213--241,
              Springer-Verlag, June 2007.

   [18]       Fujioka, A., Suzuki, K., and B. Ustaoglu, "Ephemeral Key
              Leakage Resilient and Efficient ID-AKEs That Can Share
              Identities, Private and Master Keys", Pairing-Based
              Cryptography - Pairing 2010 LNCS 6487, pp. 187--205,
              Springer, 2010.



Yonezawa, et al.         Expires August 1, 2019                [Page 13]


Internet-Draft           Pairing-Friendly Curves            January 2019


   [19]       Scott, M., "M-Pin: A Multi-Factor Zero Knowledge
              Authentication Protocol", <https://www.miracl.com/miracl-
              labs/m-pin-a-multi-factor-zero-knowledge-authentication-
              protocol>.

   [20]       Trusted Computing Group (TCG), "TPM 2.0 Library
              Specification", September 2016,
              <https://trustedcomputinggroup.org/resource/
              tpm-library-specification/>.

   [21]       Lindemann, R., "FIDO ECDAA Algorithm - FIDO Alliance
              Review Draft 02", July 2018,
              <https://fidoalliance.org/specs/fido-v2.0-rd-20180702/
              fido-ecdaa-algorithm-v2.0-rd-20180702.html>.

   [22]       Balfanz, D., Czeskis, A., Hodges, J., Jones, J., Jones,
              M., Kumar, A., Liao, A., Lindemann, R., and E. Lundberg,
              "Web Authentication: An API for accessing Public Key
              Credentials Level 1 - W3C Candidate Recommendation", July
              2018, <https://www.w3.org/TR/webauthn/>.

   [23]       Lindemann, R., "What are zk-SNARKs?", July 2018,
              <https://z.cash/technology/zksnarks.html>.

   [24]       Sullivan, N., "Geo Key Manager: How It Works", September
              2017, <https://blog.cloudflare.com/
              geo-key-manager-how-it-works/>.

   [25]       Hanke, T., Movahedi, M., and D. Williams, "DFINITY
              Technology Overview Series Consensus System Rev. 1",
              <https://dfinity.org/pdf-viewer/library/
              dfinity-consensus.pdf>.

   [26]       Jordan, R., "Ethereum 2.0 Development Update #17 -
              Prysmatic Labs", November 2018, <https://medium.com/
              prysmatic-labs/ethereum-2-0-development-update-17-
              prysmatic-labs-ed5bcf82ec00>.

   [27]       ECRYPT, "Final Report on Main Computational Assumptions in
              Cryptography", January 2013.

   [28]       Pollard, J., "Monte Carlo Methods for Index Computation
              (mod p)", Proceedings Mathematics of Computation, Vol.32,
              1978.

   [29]       Hellman, M. and J. Reyneri, "Fast computation of discrete
              logarithms in GF(q)", Advances in Cryptology: Proceedings
              of CRYPTO '82 pp. 3-13, 1983.



Yonezawa, et al.         Expires August 1, 2019                [Page 14]


Internet-Draft           Pairing-Friendly Curves            January 2019


   [30]       Barreto, P., Costello, C., Misoczki, R., Naehrig, M.,
              Pereira, G., and G. Zanon, "Subgroup security in pairing-
              based cryptography", Cryptology ePrint
              Archive http://eprint.iacr.org/2015/247.pdf, 2015.

   [31]       Mitsunari, S., "mcl - A portable and fast pairing-based
              cryptography library", 2016,
              <https://github.com/herumi/mcl>.

   [32]       Bowe, S., "BLS12-381: New zk-SNARK Elliptic Curve
              Construction", March 2017,
              <https://blog.z.cash/new-snark-curve/>.

   [33]       ISO/IEC, "ISO/IEC 15946-5:2017", ISO/IEC Information
              technology -- Security techniques -- Cryptographic
              techniques based on elliptic curves -- Part 5: Elliptic
              curve generation, 2017.

   [34]       MIRACL Ltd., "MIRACL Cryptographic SDK", 2018,
              <https://github.com/miracl/MIRACL>.

   [35]       SCIPR Lab, "libsnark: a C++ library for zkSNARK proofs",
              2012, <https://github.com/zcash/libsnark>.

   [36]       zkcrypto, "zkcrypto - Pairing-friendly elliptic curve
              library", 2017, <https://github.com/zkcrypto/pairing>.

   [37]       Cloudflare, "package bn256",
              <https://godoc.org/github.com/cloudflare/bn256>.

   [38]       Prysmatic Labs, "go-bls - Go wrapper for a BLS12-381
              Signature Aggregation implementation in C++", 2018,
              <https://godoc.org/github.com/prysmaticlabs/go-bls>.

   [39]       Lynn, B., "PBC Library - The Pairing-Based Cryptography
              Library", 2006, <https://crypto.stanford.edu/pbc/>.

   [40]       Aranha, D. and C. Gouv, "RELIC is an Efficient LIbrary for
              Cryptography", 2013,
              <https://code.google.com/p/relic-toolkit/>.

   [41]       University of Tsukuba, "TEPLA: University of Tsukuba
              Elliptic Curve and Pairing Library", 2013,
              <http://www.cipher.risk.tsukuba.ac.jp/tepla/index_e.html>.

   [42]       The Apache Software Foundation, "The Apache Milagro
              Cryptographic Library (AMCL)", 2016,
              <https://github.com/apache/incubator-milagro-crypto>.



Yonezawa, et al.         Expires August 1, 2019                [Page 15]


Internet-Draft           Pairing-Friendly Curves            January 2019


   [43]       Intel Corporation, "Developer Reference for Intel
              Integrated Performance Primitives Cryptography 2019",
              2018, <https://software.intel.com/en-us/ipp-crypto-
              reference-arithmetic-of-the-group-of-elliptic-curve-
              points>.

   [44]       Kyushu University, "bls48 - C++ library for Optimal Ate
              Pairing on BLS48", 2017,
              <https://github.com/mk-math-kyushu/bls48>.










































Yonezawa, et al.         Expires August 1, 2019                [Page 16]


Internet-Draft           Pairing-Friendly Curves            January 2019


Appendix A.  Test Vectors of Optimal Ate Pairing

   (TBD)

Authors' Addresses

   Shoko Yonezawa
   Lepidum

   EMail: yonezawa@lepidum.co.jp


   Sakae Chikara
   NTT TechnoCross

   EMail: chikara.sakae@po.ntt-tx.co.jp


   Tetsutaro Kobayashi
   NTT

   EMail: kobayashi.tetsutaro@lab.ntt.co.jp


   Tsunekazu Saito
   NTT

   EMail: saito.tsunekazu@lab.ntt.co.jp























Yonezawa, et al.         Expires August 1, 2019                [Page 17]


Html markup produced by rfcmarkup 1.129d, available from https://tools.ietf.org/tools/rfcmarkup/