Network Working Group K. Kasamatsu
Internet-Draft S. Kanno
Intended status: Informational NTT Software Corporation
Expires: August 14, 2014 T. Kobayashi
Y. Kawahara
NTT
February 10, 2014

Barreto-Naehrig Curves
draft-kasamatsu-bncurves-01

Abstract

Elliptic curves with pairings are useful tools for constructing cryptographic primitives. In this memo, we specify domain parameters of Barreto-Naehrig curves (BN-curves) [5]. The BN-curve is an elliptic curve suitable for pairings and allows us to achieve high security and efficiency of cryptographic schemes. This memo specifies domain parameters of two 254-bit BN-curves [1] [2] which allow us to obtain efficient implementations and domain parameters of 224, 256, 384, and 512-bit BN-curves which are compliant with ISO/IEC 15946-5[3]. Furthermore, this memo organizes differences between types of elliptic curves specified in ISO document and often used in open source software, which are called M-type and D-type, respectively[21].

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Table of Contents

1. Introduction

Elliptic curves with a special map called a pairing or bilinear map allow cryptographic primitives to achieve functions or efficiency which cannot be realized by conventional mathematical tools. There are identity-based encryption (IBE), attribute-based encryption (ABE), ZSS signature, broadcast encryption (BE) as examples of such primitives. IBE realizes powerful management of public keys by allowing us to use a trusted identifier as a public key. ABE provides a rich decryption condition based on boolean functions and attributes corresponding to a secret key or a ciphertext. The ZSS signature gives a shorter size of signature than that of ECDSA. BE provides an efficient encryption procedure in a broadcast setting.

Some of these cryptographic schemes based on elliptic curves with pairings were proposed in the IETF (e.g. [6], [7], and [8]) and used in some protocols (e.g. [9], [10], [11], [12], and [13]). These cryptographic primitives will be used actively more in the IETF due to their functions or efficiency.

We need to choose an appropriate type of elliptic curve and parameters for the pairing-based cryptographic schemes, because the choice has great impact on security and efficiency of these schemes. However, an RFC on elliptic curves with pairings has not yet been provided in the IETF.

In this memo, we specify domain parameters of Barreto-Naehrig curve (BN-curve) [5]. The BN-curve allows us to achieve high security and efficiency with pairings due to an optimum embedding degree. This memo specifies domain parameters of two 254-bit BN-curves ([1] and [2]) because of these efficiencies. These BN-curves are known as efficient curves in academia and particularly provide efficient pairing computation which is generally slowest operation in pairing-based cryptography. There are optimized source codes of ([1] and [2]) as open source software ([19] and [20]), respectively. Furthermore, this memo specifies domain parameters of 224, 256, 384, and 512-bit curves which are compliant with ISO document [3] and organizes differences between types of elliptic curves specified in ISO document and used in open source software, which are called M-type and D-type respectively [21].

2. Requirements Terminology

The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT", "SHOULD", "SHOULD NOT", "RECOMMENDED", "MAY", and "OPTIONAL" in this memo are to be interpreted as described in [4].

3. Preliminaries

In this section, we introduce the definition of elliptic curve and bilinear map, notation used in this memo.

3.1. Elliptic Curve

Throughout this memo, 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 such that A, B are in F_p, 
	    4 * A^3 + 27 * B^2 != 0 mod p

Solutions (x,y) for an elliptic curve E, as well as the point at infinity, are called F_p-rational points. The additive group is constructed by a well-defined operation in the set of F_p-rational points. Typically, the cyclic additive group with prime order q and base point G in E(F_p) is used for the cryptographic applications. Furthermore, we define terminology used in this memo as follows.

3.2. Bilinear Map

Let G_1 be an additive group of prime order p and let G_2 and G_T be additive and multiplicative groups, respectively, of the same order. Let P, Q be generators of G_1, G_2 respectively. We say that (G_1, G_2, G_T) are asymmetric bilinear map groups if there exists a bilinear map e: (G_1, G_2) -> G_T satisfying the following properties:

  1. Bilinearity: for any S in G_1, for any T in G_2, for any a, b in Z_q, we have the relation e(aS, bT) = e(S,T)^{ab}.
  2. Non-degeneracy: for any S in G_1, e(S,T) = 1 for any T in G_2 only if S = O_E.
  3. Computability: for any S in G_1, for any T in G_2, the bilinear map is efficiently computable.
  4. There exists an efficient, publicly computable isomorphism I: G_2 -> G_1 such that I(Q) = P.

For BN-curves, G_1 is a q-order cyclic subgroup of E(F_p) and G_2 is a subgroup of E(F_{p^k}), where k is the embedding degree of the curve. The group G_T is the set of q-th roots of unity in the finite field F_{p^k}.

4. Domain Parameter Specification

In this section, this memo specifies the domain parameters for two 254-bit elliptic curves which allow us to efficiently compute the operation of a pairing at high levels of security and domain parameters for 224, 256, 384, and 512-bit elliptic curves which are compliant with the ISO document [3].

4.1. Notation for Domain Parameters and Types of Sextic Twists

Here, we define notations for specifying domain parameters and explain types of pairing friendly curves.

Domain parameters of the elliptic curve E(F_p) and E(F_{p^12}) are needed for computation of the pairing. Barreto and Naehrig proposed a method of point and pairing compression by using output of a map I from a sextic twist E'(F_{p^2}) to E(F_{p^12}) instead of E(F_{p^12}). Generally, this method is used with BN-curves. Hence, this memo follows the method. For the details of the method, refer to [5].

The pairing friendly curves are classified in two types, which are called D-type and M-type respectively [21]. The D-type sextic twist curve is defined by equation y'^2 = x'^3 + b/s when elliptic curve E(F_p) is set to be y^2 = x^3 + b and represent of F_{p^12} is set to be F_{p^2}[u]/(u^6 - s), where s is in F_{p^2}^*. Let z be a root of u^6 - s, where z is in F_{p^12}. The corresponding map I: E'(F_{p^2}) -> E(F_{p^12}) is (x', y') -> (z^2 * x', z^3 * y').

The M-type sextic twist curve is defined by equation y'^2 = x'^3 + b * s when elliptic curve E(F_p) is set to be y^2 = x^3 + b and represent of F_{p^12} is set to be F_{p^2}[u]/(u^6 - s), where s is in F_{p^2}^*. The corresponding map I: E'(F_{p^2}) -> E(F_{p^12}) is (x', y') -> (x' * s^{-1} * z^4, y' * s^{-1} * z^3), with z^6 = s.

These domain parameters are described in the following way.

For elliptic curve E

For twisted curve E'

4.2. Efficient Domain Parameters for 254-Bit-Curves

In this section, this memo specifies the domain parameters for two 254-bit elliptic curves which are more efficient than parameters of ISO document with D-type.

4.2.1. Domain Parameters by Beuchat et al.

Here, we describe the domain parameters for 254-bit elliptic curve[1] with D-type.

The domain parameters described in this subsection are defined by elliptic curve E(F_p) : y^2 = x^3 + 5 and sextic twist E'(F_{p^2}) : x'^3 + 5/s = x'^3 - u, where F_{p^2} = F_{p}[u]/(u^2 + 5), F_{p^6} = F_{p^2}[v]/(v^3 - u), F_{p^12} = F_{p^6}[w]/(w^2 - v), s = - 5/u. We describe domain parameters of elliptic curves E and E'. For the details of these parameters, refer to [1].

4.2.2. Domain Parameters by Aranha et al.

Here, we describe the domain parameters for 254-bit elliptic curve [2] with D-type.

The domain parameters described in this subsection are defined by elliptic curve E(F_p) : y^2 = x^3 + 2 and sextic twist E'(F_{p^2}) : x'^3 + 2/s = x'^3 + 1 - u, where ,F_{p^2} = F_p [u]/(u^2 + 1), F_{p^6} = F_{p^2} [v]/(v^3 - (1+u)), F_{p^12} = F_{p^6} [w]/(w^2 - v), 1/s = 1/(1 + u). We describes domain parameters of elliptic curves E and E'. For the details of these parameters, refer to [2].

4.3. Domain Parameters Based on ISO Document

Here, we describe the domain parameters for 224, 256, 384, and 512-bit elliptic curves which are compliant with the ISO document and are based on M-type. The domain parameters described in below subsections are defined by Elliptic curve E(F_p): y^2 = x^3 + 3 and sextic twist E'(F_{p^2}): y'^2 = x'^3 + 3 * s, where F_{p^2} = F_p[X]/(X^2 + 1), F_{p^12} = F_{p^2}[X]/(X^6 - s), s = 1 + X. We describe domain parameters of elliptic curves E. Detailed information on these domain parameters is given in [3].

4.3.1. Domain Parameters for 224-Bit Curves

4.3.2. Domain Parameters for 256-Bit Curves

4.3.3. Domain Parameters for 384-Bit Curves

4.3.4. Domain Parameters for 512-Bit Curves

4.3.5. Differences between D-Type and M-Type on ISO parameters

Although ISO document is based on M-type, open source software are often based on D-type. We need to be aware of the differences. Hence we also describe elliptic curves with D-type based on ISO document [3]. The elliptic curve E(F_p) is defined by equation y^2 = x^3 + 3 and the sextic twist E'(F_{p^2}) is defined by y'^2 = x'^3 + 3/s, where F_{p^2} = F_p[X]/(X^2 + 1), F_{p^12} = F_{p^2}[X]/(X^6 - s), 1/s = -8 + 8 * i, i = X^2 + 1. Detailed information on these domain parameters is given in [5].

5. Object Identifiers

We need to define the following object identifiers. Which organization is suitable for the allotment of these object identifiers?

6. Security Considerations

Elliptic curves which are specified in this memo have hardness of the problems below and enough security margin against the attacks below.

Pairing-based cryptographic primitives are often based on the hardness of the following problems, so when the elliptic curves from this document are used in such schemes, these problems would apply. (For details of problems, refer to section 2 of [14].)

Algorithms to efficiently solve the problems above, aside from special cases, are unknown. Mainly, there are Pollard-rho algorithm [16] against point of an elliptic curve and Number Field Sieve method [15] against output of pairing as generic attacks against ellitpic curve with pairing.

The Smart, Semaev, and Sato-Araki algorithm [17], and Cheon algorithm [14] are main algorithms which improve efficiency in specific cases. The Smart-Semaev algorithm and Sato-Araki algorithm are polynotmial time algorithms against the ECDLP in the case where #E(F_{p}) equals to p. These algorithms are independently proposed. Cheon algorithm [14] is against the ECDLP with auxiliary inputs. It is prevented by satisfy the following condition, where n is the order of the curve.

Table 1 shows the security level of elliptic curves described in this memo ([1], [2]). Schemes based on the elliptic curves must be combined with cryptographic primitives which have similar or greater security level than the scheme.

	    |  Curve-ID  | Security Level (bits) |
	    --------------------------------------
        |  Fp224BN   |          112          |
        |  Fp254BNa  |          128          |   
        |  Fp254BNb  |          128          |   
        |  Fp256BN   |          128          |   
        |  Fp384BN   |          128          |   
        |  Fp512BN   |          128          |   
	   

Table 1: security level of elliptic curve specified in this memo

7. Acknowledgements

This memo was inspired by the content and structure of [18].

8. Change log

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

9. References

9.1. Normative References

[1] Beuchat, J., Gonzalez-Diaz, J., Mitsunari, S., Okamoto, E., Rodriguez-Henriquez, F. and T. Teruya, "High-Speed Software Implementation of the Optimal Ate Pairing over Barreto-Naehrig Curves", Proceedings Lecture notes in computer sciences; Pairing-Based Cryptography --Pairing2010, 2010.
[2] Aranha, D., Karabina, K., Longa, P., Gebotys, C., Rodriguez-Henriquez, F. and J. Lopez, "Faster Explicit Formulas for Computing Pairings over Ordinary Curves", Proceedings Lecture notes in computer sciences; EUROCRYPT --EUROCRYPT2011, 2011.
[3] International Organization for Standardization, "Information Technology - Security Techniques -- Cryptographic techniques based on elliptic curves . Part 5: Elliptic curve generation", ISO/IEC 15946-5, 2009.
[4] Bradner, S., "Key words for use in RFCs to Indicate Requirement Levels", RFC 2119, March 1997.

9.2. Informative References

, "
[5] Barreto, P. and M. Naehrig, "Pairing-Friendly Elliptic Curves of Prime Order", Proceedings Lecture notes in computer sciences; 3897 in Selected Areas in Cryptgraphy -- SAC2005, 2006.
[6] Boyen, X. and L. Martin, "Identity-Based Cryptography Standard (IBCS) #1: Supersingular Curve Implementations of the BF and BB1 Cryptosystems", RFC 5091, December 2007.
[7] Groves, M., "Sakai-Kasahara Key Encryption (SAKKE)", RFC 6508, February 2012.
[8] Hitt, L., "ZSS Short Signature Scheme for Supersingular and BN Curves", Internet-Draft draft-irtf-cfrg-zss-02, 2013.
[9] Martin, L. and M. Schertler, "Using the Boneh-Franklin and Boneh-Boyen Identity-Based Encryption Algorithms with the Cryptographic Message Syntax (CMS)", RFC 5409, January 2009.
[10] Cakulev, V. and G. Sundaram, "MIKEY-IBAKE: Identity-Based Authenticated Key Exchange (IBAKE) Mode of Key Distribution in Multimedia Internet KEYing (MIKEY)", RFC 6267, June 2011.
[11] Groves, M., "Elliptic Curve-Based Certificateless Signatures for Identity-Based Encryption (ECCSI)", RFC 6507, February 2012.
[12] Groves, M., "MIKEY-SAKKE: Sakai-Kasahara Key Encryption in Multimedia Internet KEYing (MIKEY)", RFC 6509, February 2012.
[13] Cakulev, V., Sundaram, G. and I. Broustis, "IBAKE: Identity-Based Authenticated Key Exchange", RFC 6539, March 2012.
[14] Cheon, J., "Security Analysis of the Strong Diffie-Hellman Problem", Proceedings Lecture notes in computer sciences; 4004 in Advances in Cryptogoly -- Eurocrypt2006, 2006.
[15] Joux, A., Lercier, R., Smart, P. and F. Vercauteren, "The number field sieve in the medium prime case", Proceedings Lecture notes in computer sciences; 4117 in Comput. Sci. -- CRYPTO2006, 2006.
[16] Pollard, J., "Monte Carlo Methods for Index Computation ( mod p)", Proceedings Mathematics of Computation, Vol.32, 1978.
[17] Satoh, T. and K. Araki, "Fermat quotients and the polynomial time discrete log algorithm for anomalous elliptic curves", Proceedings Comm. Math. UnivSancti Pauli 47, 1998.
[18] Lochter, M. and J. Merkle, Elliptic Curve Cryptography (ECC) Brainpool Standard Curves and Curve Generation", RFC 5639, March 2010.
[19]University of Tsukuba Elliptic Curve and Pairing Library", 2013.
[20] Aranha, D. and C. Gouv, "RELIC is an Efficient LIbrary for Cryptography", 2013.
[21] Aranha, D., Barreto, P., Longa, P. and J. Rocardini, "The Realm of the Pairings", SAC 2013, to appear, 2013.

Authors' Addresses

Kohei Kasamatsu NTT Software Corporation EMail: kasamatsu.kohei-at-po.ntts.co.jp
Satoru Kanno NTT Software Corporation EMail: kanno.satoru-at-po.ntts.co.jp
Tetsutaro Kobayashi NTT EMail: kobayashi.tetsutaro-at-lab.ntt.co.jp
Yuto Kawahara NTT EMail: kawahara.yuto-at-lab.ntt.co.jp