Pairing-Friendly Curves
draft-yonezawa-pairing-friendly-curves-00
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| Authors | Shoko Yonezawa , Sakae Chikara , Tetsutaro Kobayashi , Tsunekazu Saito | ||
| Last updated | 2019-01-28 (Latest revision 2019-01-27) | ||
| Replaces | draft-kato-threat-pairing | ||
| Replaced by | draft-irtf-cfrg-pairing-friendly-curves | ||
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draft-yonezawa-pairing-friendly-curves-00
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
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Copyright Notice
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This document is subject to BCP 78 and the IETF Trust's Legal
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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
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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.
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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
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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
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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
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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
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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.
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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.
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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 | | |
| | | | | |
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| | 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.
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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.
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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>.
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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]