M. Lochter
Bundesamt fuer Sicherheit in der Informationstechnik (BSI)
J. Merkle
secunet Security Networks AG
June 2007
ECC Brainpool Standard Curves and Curve Generation
draft-lochter-pkix-brainpool-ecc-00.txt
Status of this Memo
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Abstract
This RFC proposes several elliptic curve domain parameters over
finite prime fields for use in cryptographic applications. The domain
parameters are consistent with the relevant international standards,
and can be used in X.509 certificates and CRLs, IKE, TLS, XML
signatures, and all applications or protocols based on the
cryptographic message syntax (CMS).
1 Introduction
Although several standards for elliptic curves and domain parameters
exist (e.g. [ANSI1], [NIST] or [SEC2]), some major issues have still
not been addressed:
- The generation of the prime p and the seed from which the curve
parameters were derived is irreproducible, leaving out an
essential part of the security analysis.
- No proofs are provided that the proposed parameters do not belong
to those classes of parameters which are susceptible to
cryptanalytic attacks with sub-exponential complexity.
- Recent research results seem to indicate a potential for new
attacks on elliptic curve cryptosystems. At least for applications
with highest security demands or under circumstances which
complicate a change of parameters in response to new attacks,
the inclusion of a corresponding security requirement for domain
parameters (the class group condition, see section 2) is justified.
- Some of the proposed subgroups have a non-trivial cofactor, which
demands additional checks by cryptographic applications to prevent
small subgroup attacks (see [ANSI1] or [SEC1]).
- The domain parameters specified do not cover all bit lengths that
correspond to the commonly used key lengths for symmetric
cryptographic algorithms. In particular, there is no 512 bit curve
defined but only one with 521 bit length, which may be
disadvantageous for some implementations.
Furthermore, many of the parameters specified by the existing
standards are identical (see [SEC2] for a comparison). Thus, there is
still a need for additional elliptic curve domain parameters which
overcome the above limitations.
1.1 Scope and relation to other specifications
This RFC specifies elliptic curve domain parameters over prime
fields GF(p) with p having a length of 160, 192, 224, 256, 320, 384
and 512 bits. These parameters were generated in a pseudo-random yet
completely systematic and reproducible way and have been verified to
resist current cryptanalytic approaches. The parameters are compliant
with ANSI X9.62 ([ANSI1]) and X9.63 [ANSI2], ISO/IEC 14888 [ISO1] and
ISO/IEC 15946 [ISO2], ETSI TS 102176 ([ETSI]), as well as with the
specifications of NIST ([NIST]), SecG ([SEC1] and [SEC2]) and IEEE
([IEEE]).
Furthermore, this document identifies and explains the requirements
for the parameters that have led to the methods for the generation and
security validation of the parameters. Complementing information,
including the pseudo-random generation methods for the parameters and
the security proofs, are given in [EBP].
Finally, this RFC defines ASN.1 object identifiers for all elliptic
curve domain parameters specified herein, e.g. for use in X.509
certificates.
This document does neither address the cryptographic algorithms to be
used with the specified parameters nor their application in other
standards. However, it is consistent with the following RFCs and
internet drafts which specify the usage of elliptic curve cryptography
in protocols and applications:
- [RFC 3278] for the cryptographic message syntax (CMS)
- [RFC 3279] and [PKIX] for X.509 certificates and CRLs
- [RFC 4050] for XML signatures
- [RFC 4492] for TLS
- [IPSEC] for IKE
2 Requirements on the elliptic curve domain parameters
Throughout this memo let p > 3 be a prime and GF(p) a finite field
(sometimes also referred to as Galois Field or F_p) with p elements.
For given A and B with non-zero 4*A^3 + 27*B^2 mod p, the set of
solutions (x,y) for the equation E: y^2 = x^3 + A*x + B mod p over
GF(p) together with a neutral element O and well-defined laws for
addition and inversion define a group - the elliptic curve E(GF(p)).
Typically, for cryptographic applications, an element G of prime order
q is chosen.
A comprehensive introduction to elliptic curves and their
cryptographic applications can be found in [BSS].
Note 1: We choose {0,...,p-1} as a set of representatives for the
elements of GF(p). This choice induces a natural ordering on GF(p).
2.1 Security Requirements.
The following security requirements are either motivated by known
cryptographic analysis or aim to enhance trust in the recommended
curves.
1. Immunity to attacks using the Weil- or Tate-Pairing. These
attacks allow the embedding of the cyclic subgroup generated by G
into the group of units of a degree-l extension GF(p^l) of GF(p),
where sub-exponential attacks on the discrete logarithm problem
(DLP) exist. Here we have l = min{t | q divides p^t-1}, i.e. l is
the order of p mod q. By Fermat's little theorem, l divides q-1. We
require (q-1)/l < 100, which means that l is close to the maximum
possible value. This requirement is considerably stronger than those
of [SEC2] and [ANSI2] and also excludes supersingular curves, as
those are the curves of order p+1. Detailed information on this
requirement can be found in [BSS].
2. The trace is not equal to one. Trace one curves (or anomalous
curves) are curves with #E(GF(p)) = p. Satoh and Araki [SA], Semaev
[Sem] and Smart [Sma] independently proposed efficient solutions to
the elliptic curve discrete logarithm problem (ECDLP) on trace one
curves. Note that these curves are also excluded by requirement 5 of
section 2.2.
3. Large class number. The class number of the maximal order of the
endomorphism ring End(E) of E is larger than 10^7. Generally, E
cannot be "lifted" to a curve E' over an algebraic number field L
with End(E) = End(E’) unless the degree of L over the rationals is
larger than the class number of End(E). Although there are no
efficient attacks exploiting a small class number, recent work
([JMV] and [HR]) also may be seen as argument for the class number
condition. (See [EBP] for more details on class group computations.)
This condition excludes curves that are generated by the well-known
CM-method.
4. Prime group order. The group order #E(GF(p)) shall be a prime
number in order to counter small-subgroup attacks ([HMV]).
Therefore, all groups proposed in this RFC have cofactor 1. Note
that curves with prime order have no point of order 2 and
therefore no point with y-coordinate 0.
5. Verifiably pseudo-random. The elliptic curve domain parameters
shall be generated in a pseudo-random manner using seeds that are
generated in a systematic and comprehensive way. Our method of
construction is explained in [EBP].
6. Proof of security. For all curves a proof should be given that
all security requirements are met. These proofs are provided in
[EBP].
In [BG], attacks are described which apply to elliptic curve domain
parameters where q-1 has a factor u in the order of q^(1/3)). However,
the circumstances under which these attacks are applicable can be
avoided in most applications. Therefore, no corresponding security
requirement is stated here. However, it is highly recommended that
developers verify the security of their implementations against this
kind of attack.
2.2 Technical Requirements
Commercial demands and experience with existing implementations
lead to the following technical requirements for the elliptic curve
domain parameters.
1. For each of the bit lengths 160, 192, 224, 256, 320, 384 and 512
one curve shall be proposed. This requirement follows from the need
for curves providing different levels of security which are
appropriate for the underlying symmetric algorithms.
The existing standards specify a 521-bit curve instead of a 512-bit
curve.
2. The prime number p shall be congruent 3 mod 4. This requirement
allows efficient point compression: One method for the transmission
of curve points P=(x,y) is to transmit only x and the least
significant bit LSB(y) of y. Using the curve equation and p = 3 mod
4 we get (y^2)^(p+1)/4 = y*y^(p-1)/2 which is either y or -y by
Fermat's little theorem and hence, y can be computed very
efficiently.
This requirement is not always met by the parameters defined in
existing standards.
3. The curves shall be GF(p)-isomorphic to a "cryptographically good
curve" (i.e. a curve that meets all security requirements defined
in section 2.2) with A = -3 mod p. This property permits the use
of the arithmetical advantages of curves with A = -3 mod p as shown
by Brier and Joyce [BJ]. The requirement is fulfilled by a quadratic
twist E' of the given curve E with a square in GF(p): If -3 = A*Z^4
mod p is solvable, then E and E': y^2 = x^3 + Z^4*A*x + Z^6*B mod
p are GF(p)-isomorphic via the isomorphism F(x,y) := (x*Z^2, y*Z^3).
Especially, #E(GF(p)) = #E'(GF(p)) and, most importantly, E and E'
have the same algebraic structure and hence, offer the same level of
security. Approximately half of the isomorphism classes of elliptic
curves over GF(p) with p = 3 mod 4 contain a curve with A = -3 mod
p.
This constraint has also been used by [SEC2] and [NIST].
4. The prime p must not be of any special form; this requirement is
met by a verifiably pseudo-random generation of the parameters (see
requirement 5 in section 2.1).
Although parameters specified by existing standards do not meet this
requirement, the need for such curves over (pseudo-)randomly chosen
fields has already been foreseen by the Standards for Efficient
Cryptography Group (SECG), see [SEC2].
5. #E(GF(p)) < p. As a consequence of the Hasse-Weil-Theorem the
number of points #E(GF(p)) may be greater than the characteristic p
of the prime field GF(p). In some cases even the bit-length of
#E(GF(p)) can exceed the bit-length of p. To avoid overruns in
implementations we require that #E(GF(p)) < p. In order to thwart
attacks on digital signature schemes, some authors propose to use
q > p, but the attacks described e.g. in [BRS] appear infeasible in
a well-designed PKI.
6. B shall be a non-square mod p. Otherwise, the compressed
representations of the curve-points (0,0) and (0,X) with X being the
square root of B with a least significant bit of 0 would be
identical. As there are implementations of elliptic curves that
encode the point at infinity as (0,0) we try to avoid ambiguities.
Note that this condition is stable under quadratic twists as
described in condition 3 above. Condition 6 makes the attack
described in [G] impossible. It can therefore also be seen as a
security requirement.
This constraint has not been specified by existing standards.
3 Parameter specification
In this section the elliptic curve domain parameters proposed are
specified in the following way.
For all curves an ID is given by which it can be referenced.
p is the prime specifying the base field.
A and B are the coefficients of the equation y^2 = x^3 + A*x + B
mod p defining the elliptic curve E.
G is the base point, i.e. a point in E of prime order. x and y are
its x- and y-coordinates, respectively.
q is the prime order of the group generated by G.
h is the cofactor of G in E, i.e. #E(GF(p))/q.
For the twisted curve E' the coefficient Z that defines the
isomorphism F (see requirement 3 in section 2.2), the coefficients
A' = Z^4*A mod p and B' = Z^6*B mod p of the curve equation and the
base point G' are given.
The methods for the generation of the parameters and complete security
proofs regarding the security requirements specified in section 2.1
are given in [EBP].
3.1 Parameters for 160 bit curves
Curve-ID: brainpoolP160r1
p = E95E4A5F737059DC60DFC7AD95B3D8139515620F
A = 340E7BE2A280EB74E2BE61BADA745D97E8F7C300
B = 1E589A8595423412134FAA2DBDEC95C8D8675E58
x = BED5AF16EA3F6A4F62938C4631EB5AF7BDBCDBC3
y = 1667CB477A1A8EC338F94741669C976316DA6321
q = E95E4A5F737059DC60DF5991D45029409E60FC09
h = 1
#Twisted curve
Curve-ID: brainpoolP160t1
Z = 24DBFF5DEC9B986BBFE5295A29BFBAE45E0F5D0B
A' = E95E4A5F737059DC60DFC7AD95B3D8139515620C
B' = 7A556B6DAE535B7B51ED2C4D7DAA7A0B5C55F380
x = B199B13B9B34EFC1397E64BAEB05ACC265FF2378
y = ADD6718B7C7C1961F0991B842443772152C9E0AD
q = E95E4A5F737059DC60DF5991D45029409E60FC09
h = 1
3.2 Parameters for 192 bit curves
Curve-ID: brainpoolP192r1
p = C302F41D932A36CDA7A3463093D18DB78FCE476DE1A86297
A = 6A91174076B1E0E19C39C031FE8685C1CAE040E5C69A28EF
B = 469A28EF7C28CCA3DC721D044F4496BCCA7EF4146FBF25C9
x = C0A0647EAAB6A48753B033C56CB0F0900A2F5C4853375FD6
y = 14B690866ABD5BB88B5F4828C1490002E6773FA2FA299B8F
q = C302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1
h = 1
#Twisted curve
Curve-ID: brainpoolP192t1
Z = 1B6F5CC8DB4DC7AF19458A9CB80DC2295E5EB9C3732104CB
A' = C302F41D932A36CDA7A3463093D18DB78FCE476DE1A86294
B' = 13D56FFAEC78681E68F9DEB43B35BEC2FB68542E27897B79
x = 3AE9E58C82F63C30282E1FE7BBF43FA72C446AF6F4618129
y = 97E2C5667C2223A902AB5CA449D0084B7E5B3DE7CCC01C9
q = C302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1
h = 1
3.3 Parameters for 224 bit curves
Curve-ID: brainpoolP224r1
p = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FF
A = 68A5E62CA9CE6C1C299803A6C1530B514E182AD8B0042A59CAD29F43
B = 2580F63CCFE44138870713B1A92369E33E2135D266DBB372386C400B
x = D9029AD2C7E5CF4340823B2A87DC68C9E4CE3174C1E6EFDEE12C07D
y = 58AA56F772C0726F24C6B89E4ECDAC24354B9E99CAA3F6D3761402CD
q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F
h = 1
#Twisted curve
Curve-ID: brainpoolP224t1
Z = 2DF271E14427A346910CF7A2E6CFA7B3F484E5C2CCE1C8B730E28B3F
A' = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FC
B' = 4B337D934104CD7BEF271BF60CED1ED20DA14C08B3BB64F18A60888D
x = 6AB1E344CE25FF3896424E7FFE14762ECB49F8928AC0C76029B4D580
y = 374E9F5143E568CD23F3F4D7C0D4B1E41C8CC0D1C6ABD5F1A46DB4C
q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F
h = 1
3.4 Parameters for 256 bit curves
Curve-ID: brainpoolP256r1
p = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377
A = 7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9
B = 26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6
x = 8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262
y = 547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997
q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7
h = 1
#Twisted curve
Curve-ID: brainpoolP256t1
Z = 3E2D4BD9597B58639AE7AA669CAB9837CF5CF20A2C852D10F655668DFC150EF0
A' = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5374
B' = 662C61C430D84EA4FE66A7733D0B76B7BF93EBC4AF2F49256AE58101FEE92B04
x = A3E8EB3CC1CFE7B7732213B23A656149AFA142C47AAFBC2B79A191562E1305F4
y = 2D996C823439C56D7F7B22E14644417E69BCB6DE39D027001DABE8F35B25C9BE
q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7
h = 1
3.5 Parameters for 320 bit curves
Curve-ID: brainpoolP320r1
p =
D35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC28FCD412
B1F1B32E27
A =
3EE30B568FBAB0F883CCEBD46D3F3BB8A2A73513F5EB79DA66190EB085FFA9F492F375
A97D860EB4
B =
520883949DFDBC42D3AD198640688A6FE13F41349554B49ACC31DCCD884539816F5EB4
AC8FB1F1A6
x =
43BD7E9AFB53D8B85289BCC48EE5BFE6F20137D10A087EB6E7871E2A10A599C710AF8D
0D39E20611
y =
14FDD05545EC1CC8AB4093247F77275E0743FFED117182EAA9C77877AAAC6AC7D35245
D1692E8EE1
q =
D35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658E9869155
5B44C59311
h = 1
#Twisted curve
Curve-ID: brainpoolP320t1
Z =
15F75CAF668077F7E85B42EB01F0A81FF56ECD6191D55CB82B7D861458A18FEFC3E5AB
7496F3C7B1
A' =
D35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC28FCD412
B1F1B32E24
B' =
A7F561E038EB1ED560B3D147DB782013064C19F27ED27C6780AAF77FB8A547CEB5B4FE
F422340353
x =
925BE9FB01AFC6FB4D3E7D4990010F813408AB106C4F09CB7EE07868CC136FFF3357F6
24A21BED52
y =
63BA3A7A27483EBF6671DBEF7ABB30EBEE084E58A0B077AD42A5A0989D1EE71B1B9BC0
455FB0D2C3
q =
D35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658E9869155
5B44C59311
h = 1
3.6 Parameters for 384 bit curves
Curve-ID: brainpoolP384r1
p =
8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB71123ACD3A7
29901D1A71874700133107EC53
A =
7BC382C63D8C150C3C72080ACE05AFA0C2BEA28E4FB22787139165EFBA91F90F8AA581
4A503AD4EB04A8C7DD22CE2826
B =
4A8C7DD22CE28268B39B55416F0447C2FB77DE107DCD2A62E880EA53EEB62D57CB4390
295DBC9943AB78696FA504C11
x =
1D1C64F068CF45FFA2A63A81B7C13F6B8847A3E77EF14FE3DB7FCAFE0CBD10E8E826E0
3436D646AAEF87B2E247D4AF1E
y =
8ABE1D7520F9C2A45CB1EB8E95CFD55262B70B29FEEC5864E19C054FF99129280E4646
217791811142820341263C5315
q =
8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425A7CF3AB6
AF6B7FC3103B883202E9046565
h = 1
#Twisted curve
Curve-ID: brainpoolP384t1
Z =
41DFE8DD399331F7166A66076734A89CD0D2BCDB7D068E44E1F378F41ECBAE97D2D63D
BC87BCCDDCCC5DA39E8589291C
A' =
8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB71123ACD3A7
29901D1A71874700133107EC50
B' =
7F519EADA7BDA81BD826DBA647910F8C4B9346ED8CCDC64E4B1ABD11756DCE1D2074AA
263B88805CED70355A33B471EE
x =
18DE98B02DB9A306F2AFCD7235F72A819B80AB12EBD653172476FECD462AABFFC4FF19
1B946A5F54D8D0AA2F418808CC
y =
25AB056962D30651A114AFD2755AD336747F93475B7A1FCA3B88F2B6A208CCFE469408
584DC2B2912675BF5B9E582928
q =
8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425A7CF3AB6
AF6B7FC3103B883202E9046565
h = 1
3.7 Parameters for 512 bit curves
Curve-ID: brainpoolP512r1
p =
AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308717D4D9B
009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A48F3
A =
7830A3318B603B89E2327145AC234CC594CBDD8D3DF91610A83441CAEA9863BC2DED5D
5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7B9E7C1AC4D77FC94CA
B =
3DF91610A83441CAEA9863BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7
B9E7C1AC4D77FC94CADC083E67984050B75EBAE5DD2809BD638016F723
x =
81AEE4BDD82ED9645A21322E9C4C6A9385ED9F70B5D916C1B43B62EEF4D0098EFF3B1F
78E2D0D48D50D1687B93B97D5F7C6D5047406A5E688B352209BCB9F822
y =
7DDE385D566332ECC0EABFA9CF7822FDF209F70024A57B1AA000C55B881F8111B2DCDE
494A5F485E5BCA4BD88A2763AED1CA2B2FA8F0540678CD1E0F3AD80892
q =
AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA70330870553E5C
414CA92619418661197FAC10471DB1D381085DDADDB58796829CA90069
h = 1
#Twisted curve
Curve-ID: brainpoolP512t1
Z =
12EE58E6764838B69782136F0F2D3BA06E27695716054092E60A80BEDB212B64E585D9
0BCE13761F85C3F1D2A64E3BE8FEA2220F01EBA5EEB0F35DBD29D922AB
A' =
AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308717D4D9B
009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A48F0
B' =
7CBBBCF9441CFAB76E1890E46884EAE321F70C0BCB4981527897504BEC3E36A62BCDFA
2304976540F6450085F2DAE145C22553B465763689180EA2571867423E
x =
640ECE5C12788717B9C1BA06CBC2A6FEBA85842458C56DDE9DB1758D39C0313D82BA51
735CDB3EA499AA77A7D6943A64F7A3F25FE26F06B51BAA2696FA9035DA
y =
5B534BD595F5AF0FA2C892376C84ACE1BB4E3019B71634C01131159CAE03CEE9D99321
84BEEF216BD71DF2DADF86A627306ECFF96DBB8BACE198B61E00F8B332
q =
AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA70330870553E5C
414CA92619418661197FAC10471DB1D381085DDADDB58796829CA90069
h = 1
4 Object identifiers for the elliptic curve domain parameters
The root of the tree for the object identifier of the domain
parameters defined in this specification is given by
ecStdCurvesAndGeneration OBJECT IDENTIFIER::= {iso(1)
identifified-organization(3) teletrust(36) algorithm(3)
signature-algorithm(3) ecSign(2) 8}
The object identifier ellipticCurve represents the tree containing the
object identifiers for each set of domain parameters specified in this
RFC. It has the following value:
ellipticCurve OBJECT IDENTIFIER ::= {ecStdCurvesAndGeneration 1}
The tree for the domain parameters defined in this RFC is
versionOne OBJECT IDENTIFIER ::= {ellipticCurve 1}
The following object identifiers represent the domain parameters
defined in this RFC:
brainpoolP160r1 OBJECT IDENTIFIER ::= {versionOne 1}
brainpoolP160t1 OBJECT IDENTIFIER ::= {versionOne 2}
brainpoolP192r1 OBJECT IDENTIFIER ::= {versionOne 3}
brainpoolP192t1 OBJECT IDENTIFIER ::= {versionOne 4}
brainpoolP224r1 OBJECT IDENTIFIER ::= {versionOne 5}
brainpoolP224t1 OBJECT IDENTIFIER ::= {versionOne 6}
brainpoolP256r1 OBJECT IDENTIFIER ::= {versionOne 7}
brainpoolP256t1 OBJECT IDENTIFIER ::= {versionOne 8}
brainpoolP320r1 OBJECT IDENTIFIER ::= {versionOne 9}
brainpoolP320t1 OBJECT IDENTIFIER ::= {versionOne 10}
brainpoolP384r1 OBJECT IDENTIFIER ::= {versionOne 11}
brainpoolP384t1 OBJECT IDENTIFIER ::= {versionOne 12}
brainpoolP512r1 OBJECT IDENTIFIER ::= {versionOne 13}
brainpoolP512t1 OBJECT IDENTIFIER ::= {versionOne 14}
The ASN.1 syntax for elliptic curve domain parameters according to ANSI X9.62
[ANSI1] allows indicating whether a curve and base point have been generated
verifiably at random or not. The parameters specified in section 3 have all been
generated verifyably at random; however, the algorithms used for their generation
deviate from those specified in ANSI X9.62. Consequently, applications following
ANSI X9.62 will not be able to verify the randomness of the parameters. In order
to avoid rejection of the paramaters, the ASN.1 encoding SHOULD NOT specify that
the curve or base point has been generated verifiably at random. In particular,
CAs SHOULD encode SpecifiedECDomain in the following way:
- The field Version is set to ecdpVer1(1).
- The field curve.seed is absent.
- The field hash is absent.
5 Intellectual Property Rights
The authors have no knowledge about any intellectual property rights
which cover the usage of the domain parameters defined herein.
However, readers should be aware that implementations based on these
domain parameters may require use of inventions covered by patent
rights.
6 References
[ANSI1] ANSI X9.62-2005, Public Key Cryptography For The Financial
Services Industry: The Elliptic Curve Digital Signature
Algorithm (ECDSA). 2005.
[ANSI2] ANSI X9.63-2001, Public Key Cryptography For The Financial
Services Industry: Key Agreement and Key Transport Using
The Elliptic Curve Cryptography. 2001.
[BJ] E. Brier, M. Joyce, Fast multiplication on Elliptic Curves
through Isogenies. In: M. Fossorier, T. Hoholdt, and A.
Poli, eds., Applied Algebra, Algebraic Algorithms and
Error-Correcting Codes, LNCS 2643, Springer 2003.
[BG] J. Brown and R. P. Gallant, The static Diffie-Hellman
Problem, Technical Report CACR 2004-10, Centre for Applied
Cryptographic Research, University of Waterloo, 2005,
accessible via http://eprint.iacr.org/
[BRS] J. Bohli, S. Röhrich, R. Steinwandt. Key substitution
attacks revisited: taking into account malicious signers.
Preprint, 2004.
[BSS] I. Blake, G. Seroussi, N. Smart, Elliptic Curves in
Cryptography, Cambridge University Press, 1999.
[EBP] ECC Brainpool Standard Curves and Curve Generation, ECC
Brainpool, October 2005, available at
http://www.ecc-brainpool.org/download/
BP-Kurven-aktuell.pdf.
[ETSI] ETSI TS 102 176-1, Algorithms and Parameters for Secure
Electronic Signatures, Part 1: Hash functions and
asymmetric algorithms. Version 1.2.1, 2005
[NIST] National Institute of Standards and Technology. FIPS PUB
186-2: Digital Signature Standard (DSS). 2000.
[G] L. Goubin. A refined power-analysis-attack on Elliptic
Curve Cryptosystems. In: Public-Key-Cryptography - PKC2003,
Lecture Notes in Computer Science, 2567, Springer 2003.
[HMV] D. Hankerson, A. Menezes, S. Vanstone. Guide to Elliptic
Curve Cryptography. Springer 2004.
[HR] Ming-Deh Huang and Wayne Raskind. Global methods for
discrete logarithm problem III. Preprint 2005.
[IPSEC] IPSec Working Group. IKE and IKEv2 Authentication Using
ECDSA . Internet
draft. July 2006.
[ISO1] ISO/IEC 14888-3. Information technology — Security
techniques — Digital signatures with appendix — Part 3:
Discrete logarithm based mechanisms. Second Edition. 2006.
[ISO2] ISO/IEC 15946-2. Information technology — Security
techniques — Cryptographic techniques based on elliptic
curves — Part 2: Digital signatures. First Edition. 2002.
[JMV] D. Jao, S. D. Miller, R. Venkatesan, Ramanujan graphs and
the random reducibility of discrete log on isogenous
elliptic curves, IACR Cryptology ePrint Archive, 2004.
[PKIX] PKIX Working Group. Additional Algorithms and Identifiers
for use of Elliptic Curve Cryptography with PKIX
, October 2006.
[RFC2409] Harkins, D. and D. Carrel, "The Internet Key Exchange
(IKE)", RFC 2409, November 1998.
[RFC3278] Blake-Wilson, S., Brown, D., and P. Lambert, "Use of
Elliptic Curve Cryptography (ECC) Algorithms in
Cryptographic Message Syntax (CMS)", RFC 3278, April 2002.
[RFC3279] Bassham, L., Polk, W., and R. Housley, "Algorithms and
Identifiers for the Internet X.509 Public Key
Infrastructure Certificate and Certificate Revocation List
(CRL) Profile", RFC 3279, April 2002.
[RFC4050] Blake-Wilson, S., Karlinger, G., Kobayashi, T., and Y.
Wang, "Using the Elliptic Curve Signature Algorithm (ECDSA)
for XML Digital Signatures", RFC 4050, April 2005.
[RFC4492] Blake-Wilson, S., Bolyard, N., Gupta, V., Hawk, C., and B.
Moeller, "Elliptic Curve Cryptography (ECC) Cipher Suites
for Transport Layer Security (TLS)", RFC 4492, May 2006.
[SA] T. Satoh, K. Araki. Fermat quotients and the polynomial
time discrete log algorithm for anomalous elliptic curves.
Comm. Math. Univ. Sancti Pauli 47, 81-92, 1998.
[SEC1] Certicom Research. Standards for Efficient Cryptography -
SEC 1: Elliptic Curve Cryptography. Version 1.0, 2000.
[SEC2] Certicom Research. Standards for Efficient Cryptography -
SEC 2: Recomended Elliptic Curve Domain Parameters. Version
1.0, 2000.
[Sem] I. A. Semaev. Evaluation of discrete logarithms on some
elliptic curves. Math. Comp., 67, 353-356, 1998.
[Sma] N. P. Smart. The discrete logarithm problem on elliptic
curves of trace one. J. Cryptology 12, 193-196, 1999.
7 Authors' addresses
Dr. Manfred Lochter
Bundesamt fuer Sicherheit in der Informationstechnik
Postfach 200363
53133 Bonn
Germany
email: manfred.lochter@bsi.bund.de
Tel.: +49 228 9582 5643
Dr. Johannes Merkle
secunet Security Networks
Mergenthaler Allee 77
65760 Eschborn
Germany
email: johannes.merkle@secunet.com
Tel.: +49 6196 95888 55
Fax.: +49 6196 95888 88
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