CFRG Y. Sakemi, Ed.
Internet-Draft Lepidum
Intended status: Experimental T. Kobayashi
Expires: 28 September 2020 T. Saito
NTT
27 March 2020
Pairing-Friendly Curves
draft-irtf-cfrg-pairing-friendly-curves-03
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.
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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 . . . . . . . . . . . . . . . . . . 6
2.4. Barreto-Lynn-Scott Curve . . . . . . . . . . . . . . . . 6
2.5. Representation Convention for an Extension Field . . . . 7
3. Security of Pairing-Friendly Curves . . . . . . . . . . . . . 8
3.1. Evaluating the Security of Pairing-Friendly Curves . . . 8
3.2. Impact of the Recent Attack . . . . . . . . . . . . . . . 9
4. Selection of Pairing-Friendly Curves . . . . . . . . . . . . 9
4.1. Adoption Status of Pairing-friendly Curves . . . . . . . 9
4.1.1. International Standards . . . . . . . . . . . . . . . 13
4.1.2. Cryptographic Libraries . . . . . . . . . . . . . . . 14
4.1.3. Applications . . . . . . . . . . . . . . . . . . . . 15
4.2. For 100 Bits of Security . . . . . . . . . . . . . . . . 15
4.3. For 128 Bits of Security . . . . . . . . . . . . . . . . 15
4.3.1. BN Curves . . . . . . . . . . . . . . . . . . . . . . 16
4.4. For 192 Bits of Security . . . . . . . . . . . . . . . . 18
4.5. For 256 Bits of Security . . . . . . . . . . . . . . . . 18
5. Security Considerations . . . . . . . . . . . . . . . . . . . 22
6. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 22
7. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 22
8. References . . . . . . . . . . . . . . . . . . . . . . . . . 22
8.1. Normative References . . . . . . . . . . . . . . . . . . 22
8.2. Informative References . . . . . . . . . . . . . . . . . 23
Appendix A. Computing Optimal Ate Pairing . . . . . . . . . . . 29
A.1. Optimal Ate Pairings over Barreto-Naehrig Curves . . . . 30
A.2. Optimal Ate Pairings over Barreto-Lynn-Scott Curves . . . 30
Appendix B. Test Vectors of Optimal Ate Pairing . . . . . . . . 31
Appendix C. Parameters of the Barreto-Lynn-Scott Curve of
embedding degree 12 . . . . . . . . . . . . . . . . . . . 40
Authors' Addresses . . . . . . . . . . . . . . . . . . . . . . . 42
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 (Elliptic Curve Digital Signature
Algorithm), are widely used in many applications.
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Pairing-based cryptography, a variant of elliptic curve cryptography,
has attracted the attention for its flexible and applicable
functionality. Pairing is a special map defined over elliptic
curves. 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 are now in
practical use.
As the importance of pairing grows, elliptic curves where pairing is
efficiently computable are studied and the special curves called
pairing-friendly curves are proposed.
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 publishes RFCs for pairing-based cryptography such as Identity-
Based Cryptography [RFC5091], Sakai-Kasahara Key Encryption (SAKKE)
[RFC6508], and Identity-Based Authenticated Key Exchange (IBAKE)
[RFC6539]. SAKKE is applied to Multimedia Internet KEYing (MIKEY)
[RFC6509] and used in 3GPP [SAKKE].
Pairing-based key agreement protocols are standardized in ISO/IEC
[ISOIEC11770-3]. In [ISOIEC11770-3], a key agreement scheme by Joux
[Joux00], identity-based key agreement schemes by Smart-Chen-Cheng
[CCS07] and by Fujioka-Suzuki-Ustaoglu [FSU10] are specified.
MIRACL implements M-Pin, a multi-factor authentication protocol
[M-Pin]. 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) [TPM]. 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
[FIDO] and W3C [W3C] also published ECDAA algorithm similar to TCG.
Intel introduces Intel Enhanced Privacy ID (EPID) which enables
remote attestation of a hardware device while preserving the privacy
of the device as a functionality of Intel Software Guard Extensions
(SGX) [EPID]. They extend TPM ECDAA to realize such functionality.
A pairing-based EPID has been proposed [BL10] and distributed along
with Intel SGX applications.
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Zcash implements their own zero-knowledge proof algorithm named zk-
SNARKs (Zero-Knowledge Succinct Non-Interactive Argument of
Knowledge) [Zcash]. zk-SNARKs is used for protecting privacy of
transactions of Zcash. They use pairing for constructing zk-SNARKS.
Cloudflare introduces Geo Key Manager [Cloudflare] 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. In addition,
Cloudflare published a new cryptographic library CIRCL[CIRCL]
(Cloudflare Interoperable, Reusable Cryptographic Library) in 2019.
They plan for supporting secure pairing-friendly curves in CIRCL.
Recently, Boneh-Lynn-Shacham (BLS) signature schemes are being
standardized [I-D.boneh-bls-signature] and utilized in several
blockchain projects such as Ethereum [Ethereum], Algorand [Algorand],
Chia Network [Chia] and DFINITY [DFINITY]. The aggregation
functionality of BLS signatures is effective for their applications
of decentralization and scalability.
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-friendly 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, 192 and 256 bits of
security.
1.4. Requirements Terminology
The key words "MUST", "MUST NOT", "REQUIRED", "SHALL", "SHALL NOT",
"SHOULD", "SHOULD NOT", "RECOMMENDED", "NOT RECOMMENDED", "MAY", and
"OPTIONAL" in this document are to be interpreted as described in BCP
14 [RFC2119] [RFC8174] when, and only when, they appear in all
capitals, as shown here.
2. Preliminaries
2.1. Elliptic Curve
Let p > 3 be a prime and q = p^n for a natural number n. Let F_q 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,
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where x and y are in F_q, and A and B in F_q satisfy the discriminant
inequality 4 * A^3 + 27 * B^2 != 0 mod q. This is called Weierstrass
normal form of an elliptic curve.
Solutions (x, y) for an elliptic curve E, as well as the point at
infinity, O_E, are called F_q-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 passes
through P and Q. We can define P + O_E = P = O_E + P as well.
Similarly, we can define 2P = P + P and a scalar multiplication S =
[a]P for a positive integer a can be defined as an (a-1)-time
addition of P.
The additive group, denoted by E(F_q), is constructed by the set of
F_q-rational points and the addition law described above. We can
define the cyclic additive group with a prime order r by taking a
base point BP in E(F_q) as a generator. This group is used for the
elliptic curve cryptography.
We define terminology used in this memo as follows.
O_E: the point at infinity over an elliptic curve E.
E(F_q): a group constructed by F_q-rational points of E.
#E(F_q): the number of F_q-rational points of E.
h: a cofactor such that h = #E(F_q) / r.
2.2. Pairing
Pairing is a kind of the bilinear map defined over two elliptic
curves E and E'. Examples include Weil pairing, Tate pairing,
optimal Ate pairing [Ver09] 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 a prime field F_p and E' be
an elliptic curve defined over an extension field of F_p. Let k be a
minimum integer such that r is a divisor of p^k - 1, which is called
an embedding degree. Let G_1 be a cyclic subgroup on the elliptic
curve E with order r, and G_2 be a cyclic subgroup on the elliptic
curve E' with order r. Let G_T be an order r subgroup of a
multiplicative group (F_p^k)^*.
Pairing is defined as a bilinear map e: (G_1, G_2) -> G_T satisfying
the following properties:
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1. Bilinearity: for any S in G_1, T in G_2, and integers a and b,
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 [BN05] 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 defined by elliptic curves E and E' parameterized by a
well chosen integer t. E is defined over F_p, where p is a prime
more than or equal to 5, and E(F_p) has a subgroup of prime order r.
The characteristic p and the order r are parameterized by
p = 36 * t^4 + 36 * t^3 + 24 * t^2 + 6 * t + 1
r = 36 * t^4 + 36 * t^3 + 18 * t^2 + 6 * t + 1
for an integer t.
The elliptic curve E 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 m is an element which is
neither a square nor a cube in an extension field F_p^2, the twisted
curve E' of E is defined over an extension field F_p^2 by the
equation E': y^2 = x^3 + b' with b' = b / m or b' = b * m. BN curves
are called D-type if b' = b / m, and M-type if b' = b * m. The
embedded degree k is 12.
A pairing e is defined by taking G_1 as a subgroup of E(F_p) of order
r, G_2 as a subgroup of E'(F_p^2), and G_T as a subgroup of a
multiplicative group (F_p^12)^* of order r.
2.4. Barreto-Lynn-Scott Curve
A BLS curve [BLS02] 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 elliptic curves E and E' parameterized by a well
chosen integer t. E is defined over a finite field F_p by an
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equation of the form E: y^2 = x^3 + b, and its twisted curve, E': y^2
= x^3 + b', is defined in the same way as BN curves. In contrast to
BN curves, E(F_p) does not have a prime order. Instead, its order is
divisible by a large parameterized prime r and denoted by h * r with
cofactor h. The pairing will be defined on the r-torsions points.
In the same way as BN curves, BLS curves can be categorized into
D-type and M-type.
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 = (t - 1)^2 * (t^4 - t^2 + 1) / 3 + t
r = t^4 - t^2 + 1
BLS48:
p = (t - 1)^2 * (t^16 - t^8 + 1) / 3 + t
r = t^16 - t^8 + 1
for a well chosen integer t.
A pairing e is defined by taking G_1 as a subgroup of E(F_p) of order
r, G_2 as an order r subgroup of E'(F_p^2) for BLS12 and of E'(F_p^8)
for BLS48, and G_T as an order r subgroup of a multiplicative group
(F_p^12)^* for BLS12 and of a multiplicative group (F_p^48)^* for
BLS48.
2.5. Representation Convention for an Extension Field
Pairing-friendly curves use a tower of some extension fields. In
order to encode an element of an extension field, focusing on
interoperability, we adopt the representation convention shown in
Appendix J.4 of [I-D.ietf-lwig-curve-representations] as a standard
and effective method.
Let F_p be a finite field of characteristic p and F_p^d be an
extension field of F_p of degree d and an indeterminate i.
For an element s in F_p^d such that s = s_0 + s_1 * i + ... + s_{d -
1} * i^{d - 1} for s_0, s_1, ... , s_{d - 1} in a basefield F_p, s
is represented as octet string by oct(s) = s_0 || s_1 || ... || s_{d
- 1}.
Let F_p^d' be an extension field of F_p^d of degree d' / d and an
indeterminate j.
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For an element s' in F_p^d' such that s' = s'_0 + s'_1 * j + ... +
s'_{d' / d - 1} * j^{d' / d - 1} for s'_0, s'_1, ..., s'_{d' / d - 1}
in a basefield F_p^d, s' is represented as integer by oct(s') =
oct(s'_0) || oct(s'_1) || ... || oct(s'_{d' / d - 1}), where
oct(s'_0), ... , oct(s'_{d' / d - 1}) are octet strings encoded by
above convention.
In general, one can define encoding between integer and an element of
any finite field tower by inductively applying the above convention.
The parameters and test vectors of extension fields described in this
memo are encoded by this convention and represented in octet stream.
When applications communicate elements in an extension field, using
the compression method [MP04] may be more effective. In that case,
you need to use it with care for interoperability.
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.
* The elliptic curve discrete logarithm problem (ECDLP) in G_1 and
G_2
* The finite field discrete logarithm problem (FFDLP) in G_T
There are other hard problems over pairing-friendly curves used for
proving the security of pairing-based cryptography. Such problems
include computational bilinear Diffie-Hellman (CBDH) problem and
bilinear Diffie-Hellman (BDH) Problem, decision bilinear Diffie-
Hellman (DBDH) problem, gap DBDH problem, etc [ECRYPT]. 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 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 include Pollard's rho algorithm
[Pollard78], Index Calculus [HR83] and so on. In order to make index
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calculus algorithms more efficient, number field sieve (NFS)
algorithms are utilized.
3.2. Impact of the Recent Attack
In 2016, Kim and Barbulescu proposed a new variant of the NFS
algorithms, the extended tower number field sieve (exTNFS), which
drastically reduces the complexity of solving FFDLP [KB16]. Due to
exTNFS, the security level of pairing-friendly curves asymptotically
dropped down. For instance, Barbulescu and Duquesne estimated that
the security of the BN curves which had been believed to provide 128
bits of security (BN256, for example) dropped down to approximately
100 bits [BD18].
Some papers showed the minimum bit length 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,
Barbulescu and Duquesne estimated the minimum bit length of p of BN
curves after exTNFS as 461 bits, and that of BLS12 curves as 461 bits
[BD18]. For 256 bits of security, Kiyomura et al. estimated the
minimum bit length of p^k of BLS48 curves as 27,410 bits, which
implied 572 bits of p [KIK17].
4. Selection of Pairing-Friendly Curves
In this section, we introduce secure pairing-friendly curves that
consider the impact of exTNFS.
First, we show the adoption status of pairing-friendly curves in
standards, libraries and applications, and classify them according to
security level 128 bits, 192 bits, and 256 bits. Then, from the
viewpoint of "security" and "widely use", pairing-friendly curves
corresponding to each security level are selected and their
parameters are indicated.
In our selection policy, it is important that selected curves are
shown in peer-reviewed paper for security and that they are widely
used in cryptographic libraries. In addition, "efficiency" is one of
the important aspects but it is greatly depending on implementations,
so we consider that viewpoint of "security" and "widely use" are more
important than "efficiency" when considering interconnections and
interoperability on future Internet.
4.1. Adoption Status of Pairing-friendly Curves
We show the pairing-friendly curves selected by existing standards,
cryptographic libraries and applications.
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Table 1 summarizes the adoption status of pairing-friendly curves.
The details are described as following subsections. A BN curve with
a XXX-bit characteristic p is denoted as BNXXX and a BLS curve of
embedding degree k with a XXX-bit p denoted as BLSk_XXX. Due to
space limitations, Table 1 omits libraries that have not been
maintained since 2016 in which exTNFS was proposed and curves that
had security levels below 128 bits since before 2016 (ex. BN160).
The full version of Table1 is available at https://lepidum.co.jp/
blog/2020-03-27/ietf-draft-pfc/. In this table, security level for
each curve is evaluated according to [BD18],[GME19], [MAF19] and
[FK18]. Note that the curves marked as (*) indicate that the
evaluation of security level does not take into account the impact of
the exTNFS because [BD18] does not show the security level of these
curves.
+-------------+----------+------------+-----------------------------+
| Category | Name | Curve Type | Sec |
| | | | uri |
| | | | ty |
| | | | Lev |
| | | | els |
| | | | (bi |
| | | | t) |
| | | +---+-----+---+-----+---+-----+
| | | | ~ | Ard | ~ | Ard | ~ | Ard |
| | | | | 128 | | 192 | | 256 |
+=============+==========+============+===+=====+===+=====+===+=====+
| Standard | ISO/IEC | BN256I | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN384 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN512I | | | X | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | Freeman224 | | * | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | Freeman256 | | * | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | MNT256 | | * | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | TCG | BN256I | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN638 | | | X | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | FIDO/W3C | BN256I | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN256D | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN512I | | | X | | | |
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| | +------------+---+-----+---+-----+---+-----+
| | | BN638 | | | X | | | |
+-------------+----------+------------+---+-----+---+-----+---+-----+
| Library | mcl | BLS12_381 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN_SNARK1 | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN382M | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN462 | | X | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | TEPLA | BN254B | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | RELIC | BLS12_381 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_446 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_455 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_638 | | | X | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS24_477 | | | | X | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS48_575 | | | | | | X |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN256D | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN382R | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN446 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN638 | | | X | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | CP8_544 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | K54_569 | | | | | | X |
| | +------------+---+-----+---+-----+---+-----+
| | | KSS18_508 | | | X | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | OT8_511 | | X | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | AMCL | BLS12_381 | | X | | | | |
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| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_383 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_461 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS24_479 | | | | X | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS48_556 | | | | | | X |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254CX | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN256I | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN512I | | | X | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| |Intel IPP | BN256I | X | | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | Kyushu | BLS48_581 | | | | | | X |
| | Univ. | | | | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | MIRACL | BLS12_381 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_383 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS12_461 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS24_479 | | | | X | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS48_556 | | | | | | X |
| | +------------+---+-----+---+-----+---+-----+
| | | BLS48_581 | | | | | | X |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254CX | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN256I | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN462 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN512I | | | X | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | Adjoint | BLS12_381 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN_SNARK1 | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
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| | | BN254B | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254S1 | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254S2 | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN462 | | X | | | | |
+-------------+----------+------------+---+-----+---+-----+---+-----+
| Application | Zcash | BLS12_381 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN_SNARK1 | X | | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | Ethereum | BLS12_381 | | X | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | Chia | BLS12_381 | | X | | | | |
| | Network | | | | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | DFINITY | BLS12_381 | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN254N | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN_SNARK1 | X | | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN382M | | X | | | | |
| | +------------+---+-----+---+-----+---+-----+
| | | BN462 | | X | | | | |
| +----------+------------+---+-----+---+-----+---+-----+
| | Algorand | BLS12_381 | | X | | | | |
+-------------+----------+------------+---+-----+---+-----+---+-----+
Table 1: Adoption Status of Pairing-Friendly Curves
4.1.1. International Standards
ISO/IEC 15946 series specifies public-key cryptographic techniques
based on elliptic curves. ISO/IEC 15946-5 [ISOIEC15946-5] shows
numerical examples of MNT curves[MNT01] with 160-bit p and 256-bit p,
Freeman curves[Freeman06] with 224-bit p and 256-bit p, and BN curves
with 160-bit p, 192-bit p, 224-bit p, 256-bit p, 384-bit p and
512-bit p. These parameters do not take into account the effects of
the exTNFS. On the other hand, the parameters may be revised in the
future version since ISO/IEC 15946-5 is currently under development.
As described below, BN curves with 256-bit p and 512-bit p specified
in ISO/IEC 15946-5 used by other standards and libraries, these
curves are especially denoted as BN256I and BN512I.
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TCG adopts the BN256I and a BN curve with 638-bit p specified by
their own[TPM]. FIDO Alliance [FIDO] and W3C [W3C] adopt BN256I,
BN512I, the BN638 by TCG and the BN curve with 256-bit proposed by
Devegili et al.[DSD07] (named BN256D).
4.1.2. Cryptographic Libraries
There are a lot of cryptographic libraries that support pairing
calculations.
PBC is a library for pairing-based cryptography published by Stanford
University and it supports BN curves, MNT curves, Freeman curves, and
supersingular curves[PBC]. Users can generate pairing parameters by
PBC and use pairing operations with the generated parameters.
mcl[mcl] is a library for pairing-based cryptography which supports
four BN curves and BLS12_381. These BN curves include BN254 proposed
by Nogami et al. [NASKM08] (named BN254N), BN_SNARK1 suitable for
SNARK applications[libsnark], BN382M, and BN462. Kyushu university
publishes a library that supports the BLS48_581[BLS48]. University
of Tsukuba Elliptic Curve and Pairing Library (TEPLA)[TEPLA] supports
two BN curves, one is BN254N and the other is BN254 proposed by
Beuchat et al. [BGMORT10] (named BN254B). Intel publishes a
cryptographic library named Intel Integrated Performance
Primitives(Intel-IPP)[Intel-IPP] and the library supports BN256I.
RELIC[RELIC] uses various types of pairing-friendly curves that
include six BN curves (BN158, BN254R, BN256R, BN382R, BN446, and
BN638), where BN254R, BN256R and BN382R are RELIC specific parameters
and they are different from BN254N, BN254B, BN256I, BN256D and
BN382M. In addition, RELIC supports six BLS curves (BLS12_381,
BLS12_446, BLS12_445, BLS12_638, BLS24_477 and BLS48_575[MAF19]),
Cocks-Pinch curves of embedding degree 8 with 544-bit p[GME19],
pairing-friendly curves constructed by Scott et al.[SG19] based on
Kachisa-Scott-Schaefer curve with embedding degree 54 with 569-bit p
(named K54_569)[MAF19], a KSS curve[KSS08] of embedding degree 18
with 508-bit p (named KSS18_508)[AFKMR12], Optimal TNFS-secure curve
[FM19] of embedding degree 8 with 511-bit p(OT8_511), and a
supersingular curve[S86] with 1536-bit p (SS_1536).
Apache Milagro Crypto Library (AMCL)[AMCL] supports four BLS curves
(BLS12_381, BLS12_461, BLS24_479 and BLS48_556) and four BN curves
(BN254N, BN254CX which is proposed by CertiVox, BN256I and BN512I).
In addition to AMCL's supported curves, MIRACL[MIRACL] supports BN462
and BLS48_581.
Adjoint publishes a library that supports the BLS12_381 and six BN
curves (BN_SNARK1, BN254B, BN254N, BN254S1, BN254S2, and BN462)
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[AdjointLib], where BN254S1 and BN254S2 are BN curves adopted by old
version of AMCL [AMCLv2].
4.1.3. Applications
Several applications adopt pairing-friendly curves such as BN curves
and BLS curves.
Zcash implements a BN curve (named BN128) in their library libsnark
[libsnark]. After exTNFS, they propose a new parameter of BLS12 as
BLS12_381 [BLS12-381] and publish its experimental implementation
[zkcrypto].
Ethereum 2.0 adopts the BLS12_381 and uses implementation by
Meyer[pureGo-bls]. Chia Network publishes their implementation
[Chia] by integrating the RELIC toolkit [RELIC]. DFINITY uses mcl
and Algorand publishes their implementation which supports BLS12_381.
4.2. For 100 Bits of Security
Before exTNFS, BN curves with 256-bit size of underlying finite field
(so-called BN256) were considered to achieve 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.
Implementers who will newly develop the applications of pairing-based
cryptography SHOULD NOT use pairing-friendly curves with 100 bits of
security (i.e. BN256).
There exists applications which already implemented pairing-based
cryptography with 100-bit secure pairing-friendly curves. In such a
case, implementers MAY use 100 bits of security only if they need to
keep interoperability with the existing applications.
4.3. For 128 Bits of Security
Table 1 shows that a lot of pairing-friendly curves whose curve types
are BN curves and BLS curves are adopted as curves of 128 bits
security level. Among them, the one that best matches our selection
policy is BN462, so we introduce the parameters of BN462 in this
section.
On the other hand, from the viewpoint of "widely use", BLS12_381 is
an attractive curve because a lot of libraries and applications adopt
it. However, because it is not published as a curve of 128-bit
security level in peer-reviewed papers, it does not match our
selection policy. In addition, according to [BD18], the bit length
of p for BLS12 to achieve 128 bits of security is calculated as 461
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bits and more, which BLS12_381 does not satisfy. Since BLS12_381 has
a large influence from the viewpoint of interoperability, we
introduce parameters of BLS12_381 in Appendix C.
4.3.1. BN Curves
A BN curve with 128 bits of security is shown in [BD18], which we
call BN462. BN462 is defined by a parameter
t = 2^114 + 2^101 - 2^14 - 1
for the definition in Section 2.3.
For the finite field F_p, the towers of extension field F_p^2, F_p^6
and F_p^12 are defined by indeterminates u, v, w as follows:
F_p^2 = F_p[u] / (u^2 + 1)
F_p^6 = F_p^2[v] / (v^3 - u - 2)
F_p^12 = F_p^6[w] / (w^2 - v).
Defined by t, the elliptic curve E and its twisted curve E' are
represented by E: y^2 = x^3 + 5 and E': y^2 = x^3 - u + 2,
respectively. The size of p becomes 462-bit length. A pairing e is
defined by taking G_1 as a cyclic group of order r generated by a
base point BP = (x, y) in F_p, G_2 as a cyclic group of order r
generated by a based point BP' = (x', y') in F_p^2, and G_T as a
subgroup of a multiplicative group (F_p^12)^* of order r. BN462 is
D-type.
We give the following parameters for BN462.
* G_1 defined over E: y^2 = x^3 + b
- p : a characteristic
- r : an order
- BP = (x, y) : a base point
- h : a cofactor
- b : a coefficient of E
* G_2 defined over E': y^2 = x^3 + b'
- r' : an order
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- BP' = (x', y') : a base point (encoded with
[I-D.ietf-lwig-curve-representations])
o x' = x'_0 + x'_1 * u (x'_0, x'_1 in F_p)
o y' = y'_0 + y'_1 * u (y'_0, y'_1 in F_p)
- h' : a cofactor
- b' : a coefficient of E'
p:
0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908f41c802
0ffffffffff6ff66fc6ff687f640000000002401b00840138013
r:
0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908ee1c201
f7fffffffff6ff66fc7bf717f7c0000000002401b007e010800d
x:
0x21a6d67ef250191fadba34a0a30160b9ac9264b6f95f63b3edbec3cf4b2e689d
b1bbb4e69a416a0b1e79239c0372e5cd70113c98d91f36b6980d
y:
0x0118ea0460f7f7abb82b33676a7432a490eeda842cccfa7d788c659650426e6a
f77df11b8ae40eb80f475432c66600622ecaa8a5734d36fb03de
h: 1
b: 5
r':
0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908ee1c201
f7fffffffff6ff66fc7bf717f7c0000000002401b007e010800d
x'_0:
0x0257ccc85b58dda0dfb38e3a8cbdc5482e0337e7c1cd96ed61c913820408208f
9ad2699bad92e0032ae1f0aa6a8b48807695468e3d934ae1e4df
x'_1:
0x1d2e4343e8599102af8edca849566ba3c98e2a354730cbed9176884058b18134
dd86bae555b783718f50af8b59bf7e850e9b73108ba6aa8cd283
y'_0:
0x0a0650439da22c1979517427a20809eca035634706e23c3fa7a6bb42fe810f13
99a1f41c9ddae32e03695a140e7b11d7c3376e5b68df0db7154e
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y'_1:
0x073ef0cbd438cbe0172c8ae37306324d44d5e6b0c69ac57b393f1ab370fd725c
c647692444a04ef87387aa68d53743493b9eba14cc552ca2a93a
h':
0x240480360120023ffffffffff6ff0cf6b7d9bfca0000000000d812908fa1ce02
27fffffffff6ff66fc63f5f7f4c0000000002401b008a0168019
b': -u + 2
4.4. For 192 Bits of Security
As shown in Table 1, candidates of pairing-friendly curves for the
security level 192 bits are only two curves BLS24_477 and BLS24_479.
BLS24_477 has only one implementation and BLS24_479 is an
experimental parameter which is not shown in peer-reviewed paper.
Therefore, because none match our selection policy, we couldn't show
parameters for security level 192 bits here.
4.5. For 256 Bits of Security
As shown in Table 1, there are three candidats of pairing-friendly
curves for security level 256 bit. According to our selection
policy, we select BLS48_581 which is the most adopted by
cryptographic libraries.
The selected BLS48 curve is shown in [KIK17] and it is defined by a
parameter
t = -1 + 2^7 - 2^10 - 2^30 - 2^32.
For the finite field F_p, the towers of extension field F_p^2, F_p^4,
F_p^8, F_p^24 and F_p^48 are defined by indeterminates u, v, w, z, s
as follows:
F_p^2 = F_p[u] / (u^2 + 1)
F_p^4 = F_p^2[v] / (v^2 + u + 1)
F_p^8 = F_p^4[w] / (w^2 + v)
F_p^24 = F_p^8[z] / (z^3 + w)
F_p^48 = F_p^24[s] / (s^2 + z).
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. A pairing e is defined by
taking G_1 as a cyclic group of order r generated by a base point BP
= (x, y) in F_p, G_2 as a cyclic group of order r generated by a
based point BP' = (x', y') in F_p^8, and G_T as a subgroup of a
multiplicative group (F_p^48)^* of order r. The size of p becomes
581-bit length. BLS48-581 is D-type.
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We then give the parameters for BLS48-581 as follows.
* G_1 defined over E: y^2 = x^3 + b
- p : a characteristic
- r : a prime which divides an order of G_1
- BP = (x, y) : a base point
- h : a cofactor
- b : a coefficient of E
* G_2 defined over E': y^2 = x^3 + b'
- r' : an order
- BP' = (x', y') : a base point (encoded with
[I-D.ietf-lwig-curve-representations])
o x' = x'_0 + x'_1 * u + x'_2 * v + x'_3 * u * v + x'_4 * w +
x'_5 * u * w + x'_6 * v * w + x'_7 * u * v * w (x'_0, ...,
x'_7 in F_p)
o y' = y'_0 + y'_1 * u + y'_2 * v + y'_3 * u * v + y'_4 * w +
y'_5 * u * w + y'_6 * v * w + y'_7 * u * v * w (y'_0, ...,
y'_7 in F_p)
- h' : a cofactor
- b' : a coefficient of E'
p:
0x1280f73ff3476f313824e31d47012a0056e84f8d122131bb3be6c0f1f3975444
a48ae43af6e082acd9cd30394f4736daf68367a5513170ee0a578fdf721a4a48ac
3edc154e6565912b
r:
0x2386f8a925e2885e233a9ccc1615c0d6c635387a3f0b3cbe003fad6bc972c2e6
e741969d34c4c92016a85c7cd0562303c4ccbe599467c24da118a5fe6fcd671c01
x:
0x02af59b7ac340f2baf2b73df1e93f860de3f257e0e86868cf61abdbaedffb9f7
544550546a9df6f9645847665d859236ebdbc57db368b11786cb74da5d3a1e6d8c
3bce8732315af640
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y:
0x0cefda44f6531f91f86b3a2d1fb398a488a553c9efeb8a52e991279dd41b720e
f7bb7beffb98aee53e80f678584c3ef22f487f77c2876d1b2e35f37aef7b926b57
6dbb5de3e2587a70
x'_0:
0x05d615d9a7871e4a38237fa45a2775debabbefc70344dbccb7de64db3a2ef156
c46ff79baad1a8c42281a63ca0612f400503004d80491f510317b79766322154de
c34fd0b4ace8bfab
x'_1:
0x07c4973ece2258512069b0e86abc07e8b22bb6d980e1623e9526f6da12307f4e
1c3943a00abfedf16214a76affa62504f0c3c7630d979630ffd75556a01afa143f
1669b36676b47c57
x'_2:
0x01fccc70198f1334e1b2ea1853ad83bc73a8a6ca9ae237ca7a6d6957ccbab5ab
6860161c1dbd19242ffae766f0d2a6d55f028cbdfbb879d5fea8ef4cded6b3f0b4
6488156ca55a3e6a
x'_3:
0x0be2218c25ceb6185c78d8012954d4bfe8f5985ac62f3e5821b7b92a393f8be0
cc218a95f63e1c776e6ec143b1b279b9468c31c5257c200ca52310b8cb4e80bc3f
09a7033cbb7feafe
x'_4:
0x038b91c600b35913a3c598e4caa9dd63007c675d0b1642b5675ff0e7c5805386
699981f9e48199d5ac10b2ef492ae589274fad55fc1889aa80c65b5f746c9d4cbb
739c3a1c53f8cce5
x'_5:
0x0c96c7797eb0738603f1311e4ecda088f7b8f35dcef0977a3d1a58677bb03741
8181df63835d28997eb57b40b9c0b15dd7595a9f177612f097fc7960910fce3370
f2004d914a3c093a
x'_6:
0x0b9b7951c6061ee3f0197a498908aee660dea41b39d13852b6db908ba2c0b7a4
49cef11f293b13ced0fd0caa5efcf3432aad1cbe4324c22d63334b5b0e205c3354
e41607e60750e057
x'_7:
0x0827d5c22fb2bdec5282624c4f4aaa2b1e5d7a9defaf47b5211cf741719728a7
f9f8cfca93f29cff364a7190b7e2b0d4585479bd6aebf9fc44e56af2fc9e97c3f8
4e19da00fbc6ae34
y'_0:
0x00eb53356c375b5dfa497216452f3024b918b4238059a577e6f3b39ebfc435fa
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ab0906235afa27748d90f7336d8ae5163c1599abf77eea6d659045012ab12c0ff3
23edd3fe4d2d7971
y'_1:
0x0284dc75979e0ff144da6531815fcadc2b75a422ba325e6fba01d72964732fcb
f3afb096b243b1f192c5c3d1892ab24e1dd212fa097d760e2e588b423525ffc7b1
11471db936cd5665
y'_2:
0x0b36a201dd008523e421efb70367669ef2c2fc5030216d5b119d3a480d370514
475f7d5c99d0e90411515536ca3295e5e2f0c1d35d51a652269cbc7c46fc3b8fde
68332a526a2a8474
y'_3:
0x0aec25a4621edc0688223fbbd478762b1c2cded3360dcee23dd8b0e710e122d2
742c89b224333fa40dced2817742770ba10d67bda503ee5e578fb3d8b8a1e53373
16213da92841589d
y'_4:
0x0d209d5a223a9c46916503fa5a88325a2554dc541b43dd93b5a959805f112985
7ed85c77fa238cdce8a1e2ca4e512b64f59f430135945d137b08857fdddfcf7a43
f47831f982e50137
y'_5:
0x07d0d03745736b7a513d339d5ad537b90421ad66eb16722b589d82e2055ab750
4fa83420e8c270841f6824f47c180d139e3aafc198caa72b679da59ed8226cf3a5
94eedc58cf90bee4
y'_6:
0x0896767811be65ea25c2d05dfdd17af8a006f364fc0841b064155f14e4c819a6
df98f425ae3a2864f22c1fab8c74b2618b5bb40fa639f53dccc9e884017d9aa62b
3d41faeafeb23986
y'_7:
0x035e2524ff89029d393a5c07e84f981b5e068f1406be8e50c87549b6ef8eca9a
9533a3f8e69c31e97e1ad0333ec719205417300d8c4ab33f748e5ac66e84069c55
d667ffcb732718b6
h: 0x85555841aaaec4ac
b: 1
r':
0x2386f8a925e2885e233a9ccc1615c0d6c635387a3f0b3cbe003fad6bc972c2e6
e741969d34c4c92016a85c7cd0562303c4ccbe599467c24da118a5fe6fcd671c01
h':
0x170e915cb0a6b7406b8d94042317f811d6bc3fc6e211ada42e58ccfcb3ac076a
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7e4499d700a0c23dc4b0c078f92def8c87b7fe63e1eea270db353a4ef4d38b5998
ad8f0d042ea24c8f02be1c0c83992fe5d7725227bb27123a949e0876c0a8ce0a67
326db0e955dcb791b867f31d6bfa62fbdd5f44a00504df04e186fae033f1eb43c1
b1a08b6e086eff03c8fee9ebdd1e191a8a4b0466c90b389987de5637d5dd13dab3
3196bd2e5afa6cd19cf0fc3fc7db7ece1f3fac742626b1b02fcee04043b2ea9649
2f6afa51739597c54bb78aa6b0b99319fef9d09f768831018ee6564c68d054c62f
2e0b4549426fec24ab26957a669dba2a2b6945ce40c9aec6afdeda16c79e15546c
d7771fa544d5364236690ea06832679562a68731420ae52d0d35a90b8d10b688e3
1b6aee45f45b7a5083c71732105852decc888f64839a4de33b99521f0984a418d2
0fc7b0609530e454f0696fa2a8075ac01cc8ae3869e8d0fe1f3788ffac4c01aa27
20e431da333c83d9663bfb1fb7a1a7b90528482c6be7892299030bb51a51dc7e91
e9156874416bf4c26f1ea7ec578058563960ef92bbbb8632d3a1b695f954af10e9
a78e40acffc13b06540aae9da5287fc4429485d44e6289d8c0d6a3eb2ece350124
52751839fb48bc14b515478e2ff412d930ac20307561f3a5c998e6bcbfebd97eff
c6433033a2361bfcdc4fc74ad379a16c6dea49c209b1
b': -1 / w
5. 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, 192 and 256 bits of
security are introduced since the security level will different in
the requirements of the pairing-based applications. Implementers can
select these parameters according to their security requirements.
6. IANA Considerations
This document has no actions for IANA.
7. Acknowledgements
The authors would like to thank Akihiro Kato and Shoko Yonezawa for
their significant contribution to the early version of this memo.
The authors would also like to acknowledge Sakae Chikara, Kim
Taechan, Hoeteck Wee, Sergey Gorbunov and Michael Scott for their
valuable comments.
8. References
8.1. Normative References
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[I-D.ietf-lwig-curve-representations]
Struik, R., "Alternative Elliptic Curve Representations",
Work in Progress, Internet-Draft, draft-ietf-lwig-curve-
representations-08, 24 July 2019,
.
[Intel-IPP]
Intel Corporation, "Developer Reference for Intel
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techniques based on elliptic curves -- Part 5: Elliptic
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Brezing-Weng Pairing-Friendly Elliptic Curves Using
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2008, .
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.
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Optimal Ate Pairing over Elliptic Curves with Embedding
Degrees 54 and 48 at the 256-bit security level",
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2019, .
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cryptography library", 2016,
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Multimedia Internet KEYing (MIKEY)", RFC 6509,
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Identity-Based Authenticated Key Exchange", RFC 6539,
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Appendix A. Computing Optimal Ate Pairing
Before presenting the computation of optimal Ate pairing e(P, Q)
satisfying the properties shown in Section 2.2, we give subfunctions
used for pairing computation.
The following algorithm Line_Function shows the computation of the
line function. It takes A = (A[1], A[2]), B = (B[1], B[2]) in G_2
and P = ((P[1], P[2])) in G_1 as input and outputs an element of G_T.
if (A = B) then
l := (3 * A[1]^2) / (2 * A[2]);
else if (A = -B) then
return P[1] - A[1];
else
l := (B[2] - A[2]) / (B[1] - A[1]);
end if;
return (l * (P[1] -A[1]) + A[2] -P[2]);
When implementing the line function, implementers should consider the
isomorphism of E and its twisted curve E' so that one can reduce the
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computational cost of operations in G_2. We note that the function
Line_function does not consider such isomorphism.
Computation of optimal Ate pairing for BN curves uses Frobenius map.
Let a Frobenius map pi for a point Q = (x, y) over E' be pi(p, Q) =
(x^p, y^p).
A.1. Optimal Ate Pairings over Barreto-Naehrig Curves
Let c = 6 * t + 2 for a parameter t and c_0, c_1, ... , c_L in
{-1,0,1} such that the sum of c_i * 2^i (i = 0, 1, ..., L) equals to
c.
The following algorithm shows the computation of optimal Ate pairing
over Barreto-Naehrig curves. It takes P in G_1, Q in G_2, an integer
c, c_0, ...,c_L in {-1,0,1} such that the sum of c_i * 2^i (i = 0, 1,
..., L) equals to c, and an order r as input, and outputs e(P, Q).
f := 1; T := Q;
if (c_L = -1)
T := -T;
end if
for i = L-1 to 0
f := f^2 * Line_function(T, T, P); T := 2 * T;
if (c_i = 1 | c_i = -1)
f := f * Line_function(T, c_i * Q); T := T + c_i * Q;
end if
end for
Q_1 := pi(p, Q); Q_2 := pi(p, Q_1);
f := f * Line_function(T, Q_1, P); T := T + Q_1;
f := f * Line_function(T, -Q_2, P);
f := f^{(p^k - 1) / r}
return f;
A.2. Optimal Ate Pairings over Barreto-Lynn-Scott Curves
Let c = t for a parameter t and c_0, c_1, ... , c_L in {-1,0,1} such
that the sum of c_i * 2^i (i = 0, 1, ..., L) equals to c. The
following algorithm shows the computation of optimal Ate pairing over
Barreto-Lynn-Scott curves. It takes P in G_1, Q in G_2, a parameter
c, c_0, c_1, ..., c_L in {-1,0,1} such that the sum of c_i * 2^i (i =
0, 1, ..., L), and an order r as input, and outputs e(P, Q).
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f := 1; T := Q;
if (c_L = -1)
T := -T;
end if
for i = L-1 to 0
f := f^2 * Line_function(T, T, P); T := 2 * T;
if (c_i = 1 | c_i = -1)
f := f * Line_function(T, c_i * Q, P); T := T + c_i * Q;
end if
end for
f := f^{(p^k - 1) / r};
return f;
Appendix B. Test Vectors of Optimal Ate Pairing
We provide test vectors for Optimal Ate Pairing e(P, Q) given in
Appendix A for the curves BN462 and BLS48-581 given in Section 4.
Here, the inputs P = (x, y) and Q = (x', y') are the corresponding
base points BP and BP' given in Section 4.
For BN462, Q = (x', y') is given by
x' = x'_0 + x'_1 * u and
y' = y'_0 + y'_1 * u,
where u is a indeterminate and x'_0, x'_1, y'_0, y'_1 are elements of
F_p.
For BLS48-581, Q = (x', y') is given by
x' = x'_0 + x'_1 * u + x'_2 * v + x'_3 * u * v
+ x'_4 * w + x'_5 * u * w + x'_6 * v * w + x'_7 * u * v * w and
y' = y'_0 + y'_1 * u + y'_2 * v + y'_3 * u * v
+ y'_4 * w + y'_5 * u * w + y'_6 * v * w + y'_7 * u * v * w,
where u, v and w are indeterminates and x'_0, ..., x'_7 and y'_0,
..., y'_7 are elements of F_p. The representation of Q = (x', y')
given below is followed by [I-D.ietf-lwig-curve-representations].
BN462:
Input x value:
0x21a6d67ef250191fadba34a0a30160b9ac9264b6f95f63b3edbec3cf4b2e689d
b1bbb4e69a416a0b1e79239c0372e5cd70113c98d91f36b6980d
Input y value:
0x0118ea0460f7f7abb82b33676a7432a490eeda842cccfa7d788c659650426e6a
f77df11b8ae40eb80f475432c66600622ecaa8a5734d36fb03de
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Input x'_0 value:
0x0257ccc85b58dda0dfb38e3a8cbdc5482e0337e7c1cd96ed61c913820408208f
9ad2699bad92e0032ae1f0aa6a8b48807695468e3d934ae1e4df
Input x'_1 value:
0x1d2e4343e8599102af8edca849566ba3c98e2a354730cbed9176884058b18134
dd86bae555b783718f50af8b59bf7e850e9b73108ba6aa8cd283
Input y'_0 value:
0x0a0650439da22c1979517427a20809eca035634706e23c3fa7a6bb42fe810f13
99a1f41c9ddae32e03695a140e7b11d7c3376e5b68df0db7154e
Input y'_1 value:
0x073ef0cbd438cbe0172c8ae37306324d44d5e6b0c69ac57b393f1ab370fd725c
c647692444a04ef87387aa68d53743493b9eba14cc552ca2a93a
e_0:
0x0cf7f0f2e01610804272f4a7a24014ac085543d787c8f8bf07059f93f87ba7e2
a4ac77835d4ff10e78669be39cd23cc3a659c093dbe3b9647e8c
e_1:
0x00ef2c737515694ee5b85051e39970f24e27ca278847c7cfa709b0df408b830b
3763b1b001f1194445b62d6c093fb6f77e43e369edefb1200389
e_2:
0x04d685b29fd2b8faedacd36873f24a06158742bb2328740f93827934592d6f17
23e0772bb9ccd3025f88dc457fc4f77dfef76104ff43cd430bf7
e_3:
0x090067ef2892de0c48ee49cbe4ff1f835286c700c8d191574cb424019de11142
b3c722cc5083a71912411c4a1f61c00d1e8f14f545348eb7462c
e_4:
0x1437603b60dce235a090c43f5147d9c03bd63081c8bb1ffa7d8a2c31d6732308
60bb3dfe4ca85581f7459204ef755f63cba1fbd6a4436f10ba0e
e_5:
0x13191b1110d13650bf8e76b356fe776eb9d7a03fe33f82e3fe5732071f305d20
1843238cc96fd0e892bc61701e1844faa8e33446f87c6e29e75f
e_6:
0x07b1ce375c0191c786bb184cc9c08a6ae5a569dd7586f75d6d2de2b2f075787e
e5082d44ca4b8009b3285ecae5fa521e23be76e6a08f17fa5cc8
e_7:
0x05b64add5e49574b124a02d85f508c8d2d37993ae4c370a9cda89a100cdb5e1d
441b57768dbc68429ffae243c0c57fe5ab0a3ee4c6f2d9d34714
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e_8:
0x0fd9a3271854a2b4542b42c55916e1faf7a8b87a7d10907179ac7073f6a1de04
4906ffaf4760d11c8f92df3e50251e39ce92c700a12e77d0adf3
e_9:
0x17fa0c7fa60c9a6d4d8bb9897991efd087899edc776f33743db921a689720c82
257ee3c788e8160c112f18e841a3dd9a79a6f8782f771d542ee5
e_10:
0x0c901397a62bb185a8f9cf336e28cfb0f354e2313f99c538cdceedf8b8aa22c2
3b896201170fc915690f79f6ba75581f1b76055cd89b7182041c
e_11:
0x20f27fde93cee94ca4bf9ded1b1378c1b0d80439eeb1d0c8daef30db0037104a
5e32a2ccc94fa1860a95e39a93ba51187b45f4c2c50c16482322
BLS48-581:
Input x value:
0x02af59b7ac340f2baf2b73df1e93f860de3f257e0e86868cf61abdbaedffb9f7
544550546a9df6f9645847665d859236ebdbc57db368b11786cb74da5d3a1e6d8c
3bce8732315af640
Input y value:
0x0cefda44f6531f91f86b3a2d1fb398a488a553c9efeb8a52e991279dd41b720e
f7bb7beffb98aee53e80f678584c3ef22f487f77c2876d1b2e35f37aef7b926b57
6dbb5de3e2587a70
x'_0:
0x05d615d9a7871e4a38237fa45a2775debabbefc70344dbccb7de64db3a2ef156
c46ff79baad1a8c42281a63ca0612f400503004d80491f510317b79766322154de
c34fd0b4ace8bfab
x'_1:
0x07c4973ece2258512069b0e86abc07e8b22bb6d980e1623e9526f6da12307f4e
1c3943a00abfedf16214a76affa62504f0c3c7630d979630ffd75556a01afa143f
1669b36676b47c57
x'_2:
0x01fccc70198f1334e1b2ea1853ad83bc73a8a6ca9ae237ca7a6d6957ccbab5ab
6860161c1dbd19242ffae766f0d2a6d55f028cbdfbb879d5fea8ef4cded6b3f0b4
6488156ca55a3e6a
x'_3:
0x0be2218c25ceb6185c78d8012954d4bfe8f5985ac62f3e5821b7b92a393f8be0
cc218a95f63e1c776e6ec143b1b279b9468c31c5257c200ca52310b8cb4e80bc3f
09a7033cbb7feafe
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x'_4:
0x038b91c600b35913a3c598e4caa9dd63007c675d0b1642b5675ff0e7c5805386
699981f9e48199d5ac10b2ef492ae589274fad55fc1889aa80c65b5f746c9d4cbb
739c3a1c53f8cce5
x'_5:
0x0c96c7797eb0738603f1311e4ecda088f7b8f35dcef0977a3d1a58677bb03741
8181df63835d28997eb57b40b9c0b15dd7595a9f177612f097fc7960910fce3370
f2004d914a3c093a
x'_6:
0x0b9b7951c6061ee3f0197a498908aee660dea41b39d13852b6db908ba2c0b7a4
49cef11f293b13ced0fd0caa5efcf3432aad1cbe4324c22d63334b5b0e205c3354
e41607e60750e057
x'_7:
0x0827d5c22fb2bdec5282624c4f4aaa2b1e5d7a9defaf47b5211cf741719728a7
f9f8cfca93f29cff364a7190b7e2b0d4585479bd6aebf9fc44e56af2fc9e97c3f8
4e19da00fbc6ae34
y'_0:
0x00eb53356c375b5dfa497216452f3024b918b4238059a577e6f3b39ebfc435fa
ab0906235afa27748d90f7336d8ae5163c1599abf77eea6d659045012ab12c0ff3
23edd3fe4d2d7971
y'_1:
0x0284dc75979e0ff144da6531815fcadc2b75a422ba325e6fba01d72964732fcb
f3afb096b243b1f192c5c3d1892ab24e1dd212fa097d760e2e588b423525ffc7b1
11471db936cd5665
y'_2:
0x0b36a201dd008523e421efb70367669ef2c2fc5030216d5b119d3a480d370514
475f7d5c99d0e90411515536ca3295e5e2f0c1d35d51a652269cbc7c46fc3b8fde
68332a526a2a8474
y'_3:
0x0aec25a4621edc0688223fbbd478762b1c2cded3360dcee23dd8b0e710e122d2
742c89b224333fa40dced2817742770ba10d67bda503ee5e578fb3d8b8a1e53373
16213da92841589d
y'_4:
0x0d209d5a223a9c46916503fa5a88325a2554dc541b43dd93b5a959805f112985
7ed85c77fa238cdce8a1e2ca4e512b64f59f430135945d137b08857fdddfcf7a43
f47831f982e50137
y'_5:
0x07d0d03745736b7a513d339d5ad537b90421ad66eb16722b589d82e2055ab750
Sakemi, et al. Expires 28 September 2020 [Page 34]
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4fa83420e8c270841f6824f47c180d139e3aafc198caa72b679da59ed8226cf3a5
94eedc58cf90bee4
y'_6:
0x0896767811be65ea25c2d05dfdd17af8a006f364fc0841b064155f14e4c819a6
df98f425ae3a2864f22c1fab8c74b2618b5bb40fa639f53dccc9e884017d9aa62b
3d41faeafeb23986
y'_7:
0x035e2524ff89029d393a5c07e84f981b5e068f1406be8e50c87549b6ef8eca9a
9533a3f8e69c31e97e1ad0333ec719205417300d8c4ab33f748e5ac66e84069c55
d667ffcb732718b6
e_0:
0x0e26c3fcb8ef67417814098de5111ffcccc1d003d15b367bad07cef2291a93d3
1db03e3f03376f3beae2bd877bcfc22a25dc51016eda1ab56ee3033bc4b4fec596
2f02dffb3af5e38e
e_1:
0x069061b8047279aa5c2d25cdf676ddf34eddbc8ec2ec0f03614886fa828e1fc0
66b26d35744c0c38271843aa4fb617b57fa9eb4bd256d17367914159fc18b10a10
85cb626e5bedb145
e_2:
0x02b9bece645fbf9d8f97025a1545359f6fe3ffab3cd57094f862f7fb9ca01c88
705c26675bcc723878e943da6b56ce25d063381fcd2a292e0e7501fe572744184f
b4ab4ca071a04281
e_3:
0x0080d267bf036c1e61d7fc73905e8c630b97aa05ef3266c82e7a111072c0d205
6baa8137fba111c9650dfb18cb1f43363041e202e3192fced29d2b0501c882543f
b370a56bfdc2435b
e_4:
0x03c6b4c12f338f9401e6a493a405b33e64389338db8c5e592a8dd79eac7720dd
83dd6b0c189eeda20809160cd57cdf3e2edc82db15f553c1f6c953ea27114cb6bd
8a38e273f407dae0
e_5:
0x016e46224f28bfd8833f76ac29ee6e406a9da1bde55f5e82b3bd977897a9104f
18b9ee41ea9af7d4183d895102950a12ce9975669db07924e1b432d9680f5ce7e5
c67ed68f381eba45
e_6:
0x008ddce7a4a1b94be5df3ceea56bef0077dcdde86d579938a50933a47296d337
b7629934128e2457e24142b0eeaa978fd8e70986d7dd51fccbbeb8a1933434fec4
f5bc538de2646e90
Sakemi, et al. Expires 28 September 2020 [Page 35]
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e_7:
0x060ef6eae55728e40bd4628265218b24b38cdd434968c14bfefb87f0dcbfc76c
c473ae2dc0cac6e69dfdf90951175178dc75b9cc08320fcde187aa58ea047a2ee0
0b1968650eec2791
e_8:
0x0c3943636876fd4f9393414099a746f84b2633dfb7c36ba6512a0b48e66dcb2e
409f1b9e150e36b0b4311165810a3c721525f0d43a021f090e6a27577b42c7a57b
ed3327edb98ba8f8
e_9:
0x02d31eb8be0d923cac2a8eb6a07556c8951d849ec53c2848ee78c5eed40262eb
21822527a8555b071f1cd080e049e5e7ebfe2541d5b42c1e414341694d6f16d287
e4a8d28359c2d2f9
e_10:
0x07f19673c5580d6a10d09a032397c5d425c3a99ff1dd0abe5bec40a0d47a6b8d
aabb22edb6b06dd8691950b8f23faefcdd80c45aa3817a840018965941f4247f9f
97233a84f58b262e
e_11:
0x0d3fe01f0c114915c3bdf8089377780076c1685302279fd9ab12d07477aac03b
69291652e9f179baa0a99c38aa8851c1d25ffdb4ded2c8fe8b30338c14428607d6
d822610d41f51372
e_12:
0x0662eefd5fab9509aed968866b68cff3bc5d48ecc8ac6867c212a2d82cee5a68
9a3c9c67f1d611adac7268dc8b06471c0598f7016ca3d1c01649dda4b43531cffc
4eb41e691e27f2eb
e_13:
0x0aad8f4a8cfdca8de0985070304fe4f4d32f99b01d4ea50d9f7cd2abdc0aeea9
9311a36ec6ed18208642cef9e09b96795b27c42a5a744a7b01a617a91d9fb7623d
636640d61a6596ec
e_14:
0x0ffcf21d641fd9c6a641a749d80cab1bcad4b34ee97567d905ed9d5cfb74e9ae
f19674e2eb6ce3dfb706aa814d4a228db4fcd707e571259435393a27cac68b59a1
b690ae8cde7a94c3
e_15:
0x0cbe92a53151790cece4a86f91e9b31644a86fc4c954e5fa04e707beb69fc60a
858fed8ebd53e4cfd51546d5c0732331071c358d721ee601bfd3847e0e904101c6
2822dd2e4c7f8e5c
e_16:
0x0202db83b1ff33016679b6cfc8931deea6df1485c894dcd113bacf564411519a
Sakemi, et al. Expires 28 September 2020 [Page 36]
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42026b5fda4e16262674dcb3f089cd7d552f8089a1fec93e3db6bca43788cdb06f
c41baaa5c5098667
e_17:
0x070a617ed131b857f5b74b625c4ef70cc567f619defb5f2ab67534a1a8aa7297
5fc4248ac8551ce02b68801703971a2cf1cb934c9c354cadd5cfc4575cde8dbde6
122bd54826a9b3e9
e_18:
0x070e1ebce457c141417f88423127b7a7321424f64119d5089d883cb953283ee4
e1f2e01ffa7b903fe7a94af4bb1acb02ca6a36678e41506879069cee11c9dcf6a0
80b6a4a7c7f21dc9
e_19:
0x058a06be5a36c6148d8a1287ee7f0e725453fa1bb05cf77239f235b417127e37
0cfa4f88e61a23ea16df3c45d29c203d04d09782b39e9b4037c0c4ac8e8653e7c5
33ad752a640b233e
e_20:
0x0dfdfaaeb9349cf18d21b92ad68f8a7ecc509c35fcd4b8abeb93be7a204ac871
f2195180206a2c340fccb69dbc30b9410ed0b122308a8fc75141f673ae5ec82b6a
45fc2d664409c6b6
e_21:
0x0d06c8adfdd81275da2a0ce375b8df9199f3d359e8cf50064a3dc10a59241712
4a3b705b05a7ffe78e20f935a08868ecf3fc5aba0ace7ce4497bb59085ca277c16
b3d53dd7dae5c857
e_22:
0x0708effd28c4ae21b6969cb9bdd0c27f8a3e341798b6f6d4baf27be259b4a476
88b50cb68a69a917a4a1faf56cec93f69ac416512c32e9d5e69bd8836b6c2ba9c6
889d507ad571dbc4
e_23:
0x09da7c7aa48ce571f8ece74b98431b14ae6fb4a53ae979cd6b2e82320e8d25a0
ece1ca1563aa5aa6926e7d608358af8399534f6b00788e95e37ef1b549f43a58ad
250a71f0b2fdb2bf
e_24:
0x0a7150a14471994833d89f41daeaa999dfc24a9968d4e33d88ed9e9f07aa2432
c53e486ba6e3b6e4f4b8d9c989010a375935c06e4b8d6c31239fad6a61e2647b84
a0e3f76e57005ff7
e_25:
0x084696f31ff27889d4dccdc4967964a5387a5ae071ad391c5723c9034f16c255
7915ada07ec68f18672b5b2107f785c15ddf9697046dc633b5a23cc0e442d28ef6
eea9915d0638d4d8
Sakemi, et al. Expires 28 September 2020 [Page 37]
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e_26:
0x0398e76e3d2202f999ac0f73e0099fe4e0fe2de9d223e78fc65c56e209cdf48f
0d1ad8f6093e924ce5f0c93437c11212b7841de26f9067065b1898f48006bcc6f2
ab8fa8e0b93f4ba4
e_27:
0x06d683f556022368e7a633dc6fe319fd1d4fc0e07acff7c4d4177e83a911e733
13e0ed980cd9197bd17ac45942a65d90e6cb9209ede7f36c10e009c9d337ee97c4
068db40e34d0e361
e_28:
0x0d764075344b70818f91b13ee445fd8c1587d1c0664002180bbac9a396ad4a8d
c1e695b0c4267df4a09081c1e5c256c53fd49a73ffc817e65217a44fc0b20ef5ee
92b28d4bc3e38576
e_29:
0x0aa6a32fdc4423b1c6d43e5104159bcd8e03a676d055d4496f7b1bc8761164a2
908a3ff0e4c4d1f4362015c14824927011e2909531b8d87ee0acd676e7221a1ca1
c21a33e2cf87dc51
e_30:
0x1147719959ac8eeab3fc913539784f1f947df47066b6c0c1beafecdb5fa784c3
be9de5ab282a678a2a0cbef8714141a6c8aaa76500819a896b46af20509953495e
2a85eff58348b38d
e_31:
0x11a377bcebd3c12702bb34044f06f8870ca712fb5caa6d30c48ace96898fcbcd
dbcf31f331c9e524684c02c90db7f30b9fc470d6e651a7e8b1f684383f3705d7a4
7a1b4fe463d623c8
e_32:
0x0b8b4511f451ba2cc58dc28e56d5e1d0a8f557ecb242f4d994a627e07cf3fa44
e6d83cb907deacf303d2f761810b5d943b46c4383e1435ec23fec196a70e339461
73c78be3c75dfc83
e_33:
0x090962d632ee2a57ce4208052ce47a9f76ea0fdad724b7256bb07f3944e9639a
981d3431087241e30ae9bf5e2ea32af323ce7ed195d383b749cb25bc09f678d385
a49a0c09f6d9efca
e_34:
0x0931c7befc80acd185491c68af886fa8ee39c21ed3ebd743b9168ae3b298df48
5bfdc75b94f0b21aecd8dca941dfc6d1566cc70dc648e6ccc73e4cbf2a1ac83c82
94d447c66e74784d
e_35:
0x020ac007bf6c76ec827d53647058aca48896916269c6a2016b8c06f0130901c8
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975779f1672e581e2dfdbcf504e96ecf6801d0d39aad35cf79fbe7fe193c6c882c
15bce593223f0c7c
e_36:
0x0c0aed0d890c3b0b673bf4981398dcbf0d15d36af6347a39599f3a2258418482
8f78f91bbbbd08124a97672963ec313ff142c456ec1a2fc3909fd4429fd699d827
d48777d3b0e0e699
e_37:
0x0ef7799241a1ba6baaa8740d5667a1ace50fb8e63accc3bc30dc07b11d78dc54
5b68910c027489a0d842d1ba3ac406197881361a18b9fe337ff22d730fa44afabb
9f801f759086c8e4
e_38:
0x016663c940d062f4057257c8f4fb9b35e82541717a34582dd7d55b41ebadf40d
486ed74570043b2a3c4de29859fdeae9b6b456cb33bb401ecf38f9685646692300
517e9b035d6665fc
e_39:
0x1184a79510edf25e3bd2dc793a5082fa0fed0d559fa14a5ce9ffca4c61f17196
e1ffbb84326272e0d079368e9a735be1d05ec80c20dc6198b50a22a765defdc151
d437335f1309aced
e_40:
0x120e47a747d942a593d202707c936dafa6fed489967dd94e48f317fd3c881b10
41e3b6bbf9e8031d44e39c1ab5ae41e487eac9acd90e869129c38a8e6c97cf55d6
666d22299951f91a
e_41:
0x026b6e374108ecb2fe8d557087f40ab7bac8c5af0644a655271765d57ad71742
aa331326d871610a8c4c30ccf5d8adbeec23cdff20d9502a5005fce2593caf0682
c82e4873b89d6d71
e_42:
0x041be63a2fa643e5a66faeb099a3440105c18dca58d51f74b3bf281da4e689b1
3f365273a2ed397e7b1c26bdd4daade710c30350318b0ae9a9b16882c29fe31ca3
b884c92916d6d07a
e_43:
0x124018a12f0f0af881e6765e9e81071acc56ebcddadcd107750bd8697440cc16
f190a3595633bb8900e6829823866c5769f03a306f979a3e039e620d6d2f576793
d36d840b168eeedd
e_44:
0x0d422de4a83449c535b4b9ece586754c941548f15d50ada6740865be9c0b0667
88b6078727c7dee299acc15cbdcc7d51cdc5b17757c07d9a9146b01d2fdc7b8c56
2002da0f9084bde5
Sakemi, et al. Expires 28 September 2020 [Page 39]
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e_45:
0x1119f6c5468bce2ec2b450858dc073fea4fb05b6e83dd20c55c9cf694cbcc57f
c0effb1d33b9b5587852d0961c40ff114b7493361e4cfdff16e85fbce667869b6f
7e9eb804bcec46db
e_46:
0x061eaa8e9b0085364a61ea4f69c3516b6bf9f79f8c79d053e646ea637215cf65
90203b275290872e3d7b258102dd0c0a4a310af3958165f2078ff9dc3ac9e995ce
5413268d80974784
e_47:
0x0add8d58e9ec0c9393eb8c4bc0b08174a6b421e15040ef558da58d241e5f906a
d6ca2aa5de361421708a6b8ff6736efbac6b4688bf752259b4650595aa395c40d0
0f4417f180779985
Appendix C. Parameters of the Barreto-Lynn-Scott Curve of embedding
degree 12
In this part, we introduce parameters of the Barreto-Lynn-Scott curve
of embedding degree 12 with 381 bits p that adopted by a lot of
applications such as Zcash [Zcash], Ethereum [Ethereum] and so on.
BLS12_381 curve is shown in [BLS12-381] and it is defined by a
parameter
t = -2^63 - 2^62 - 2^60 - 2^57 - 2^48 - 2^16
where the size of p becomes 381-bit length.
For the finite field F_p, the towers of extension field F_p^2, F_p^6
and F_p^12 are defined by indeterminates u, v, w as follows:
F_p^2 = F_p[u] / (u^2 + 1)
F_p^6 = F_p^2[v] / (v^3 - u - 1)
F_p^12 = F_p^6[w] / (w^2 - v).
Defined by t, 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(u + 1).
A pairing e is defined by taking G_1 as a cyclic group of order r
generated by a base point BP = (x, y) in F_p, G_2 as a cyclic group
of order r generated by a based point BP' = (x', y') in F_p^2, and
G_T as a subgroup of a multiplicative group (F_p^12)^* of order r.
BLS12_381 is M-type.
We have to note that, according to [BD18], the bit length of p for
BLS12 to achieve 128 bits of security is calculated as 461 bits and
more, which BLS12_381 does not satisfy.
Sakemi, et al. Expires 28 September 2020 [Page 40]
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Parameters of BLS12_381 are given as follows.
* G_1 defined over E: y^2 = x^3 + b
- p : a characteristic
- r : an order
- BP = (x, y) : a base point
- h : a cofactor
- b : a coefficient of E
* G_2 defined over E': y^2 = x^3 + b'
- r' : an order
- BP' = (x', y') : a base point (encoded with
[I-D.ietf-lwig-curve-representations])
o x' = x'_0 + x'_1 * u (x'_0, x'_1 in F_p)
o y' = y'_0 + y'_1 * u (y'_0, y'_1 in F_p)
- h' : a cofactor
- b' : a coefficient of E'
p:
0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f624
1eabfffeb153ffffb9feffffffffaaab
r:
0x73eda753299d7d483339d80809a1d80553bda402fffe5bfeffffffff00000001
x:
0x17f1d3a73197d7942695638c4fa9ac0fc3688c4f9774b905a14e3a3f171bac58
6c55e83ff97a1aeffb3af00adb22c6bb
y:
0x08b3f481e3aaa0f1a09e30ed741d8ae4fcf5e095d5d00af600db18cb2c04b3ed
d03cc744a2888ae40caa232946c5e7e1
h: 0x396c8c005555e1568c00aaab0000aaab
b: 4
Sakemi, et al. Expires 28 September 2020 [Page 41]
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r':
0x1a0111ea397fe69a4b1ba7b6434bacd764774b84f38512bf6730d2a0f6b0f624
1eabfffeb153ffffb9feffffffffaaab
x'_0:
0x024aa2b2f08f0a91260805272dc51051c6e47ad4fa403b02b4510b647ae3d177
0bac0326a805bbefd48056c8c121bdb8
x'_1:
0x13e02b6052719f607dacd3a088274f65596bd0d09920b61ab5da61bbdc7f5049
334cf11213945d57e5ac7d055d042b7e
y'_0:
0x0ce5d527727d6e118cc9cdc6da2e351aadfd9baa8cbdd3a76d429a695160d12c
923ac9cc3baca289e193548608b82801
y'_1:
0x0606c4a02ea734cc32acd2b02bc28b99cb3e287e85a763af267492ab572e99ab
3f370d275cec1da1aaa9075ff05f79be
h':
0x5d543a95414e7f1091d50792876a202cd91de4547085abaa68a205b2e5a7ddfa
628f1cb4d9e82ef21537e293a6691ae1616ec6e786f0c70cf1c38e31c7238e5
b': 4 * (u + 1)
Authors' Addresses
Yumi Sakemi (editor)
Lepidum
Email: yumi.sakemi@lepidum.co.jp
Tetsutaro Kobayashi
NTT
Email: tetsutaro.kobayashi.dr@hco.ntt.co.jp
Tsunekazu Saito
NTT
Email: tsunekazu.saito.hg@hco.ntt.co.jp
Sakemi, et al. Expires 28 September 2020 [Page 42]