Brainpool Elliptic Curves for the IKE Group Description Registry
draft-harkins-brainpool-ike-groups-00
The information below is for an old version of the document.
| Document | Type |
This is an older version of an Internet-Draft that was ultimately published as RFC 6932.
|
|
|---|---|---|---|
| Author | Dan Harkins | ||
| Last updated | 2012-08-08 | ||
| RFC stream | (None) | ||
| Formats | |||
| Reviews | |||
| Stream | Stream state | (No stream defined) | |
| Consensus boilerplate | Unknown | ||
| RFC Editor Note | (None) | ||
| IESG | IESG state | Became RFC 6932 (Informational) | |
| Telechat date | (None) | ||
| Responsible AD | (None) | ||
| Send notices to | (None) |
draft-harkins-brainpool-ike-groups-00
Internet Engineering Task Force D. Harkins, Ed.
Internet-Draft Aruba Networks
Intended status: Informational August 8, 2012
Expires: February 9, 2013
Brainpool Elliptic Curves for the IKE Group Description Registry
draft-harkins-brainpool-ike-groups-00
Abstract
This memo allocates code points for fourteen new elliptic curve
domain parameter sets over finite prime fields into a registry
established by The Internet Key Exchange (IKE).
Status of this Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
Internet-Drafts are working documents of the Internet Engineering
Task Force (IETF). Note that other groups may also distribute
working documents as Internet-Drafts. The list of current Internet-
Drafts is at http://datatracker.ietf.org/drafts/current/.
Internet-Drafts are draft documents valid for a maximum of six months
and may be updated, replaced, or obsoleted by other documents at any
time. It is inappropriate to use Internet-Drafts as reference
material or to cite them other than as "work in progress."
This Internet-Draft will expire on February 9, 2013.
Copyright Notice
Copyright (c) 2012 IETF Trust and the persons identified as the
document authors. All rights reserved.
This document is subject to BCP 78 and the IETF Trust's Legal
Provisions Relating to IETF Documents
(http://trustee.ietf.org/license-info) in effect on the date of
publication of this document. Please review these documents
carefully, as they describe your rights and restrictions with respect
to this document. Code Components extracted from this document must
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.
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.1. Requirements Language . . . . . . . . . . . . . . . . . . 3
2. Brainpool Elliptic Curves . . . . . . . . . . . . . . . . . . 3
2.1. Domain Parameters for 160-bit Curves . . . . . . . . . . . 4
2.1.1. Curve-ID: brainpoolP160r1 . . . . . . . . . . . . . . 4
2.1.2. Twisted Curve-ID: brainpoolP160t1 . . . . . . . . . . 4
2.2. Domain Parameters for 192-bit Curves . . . . . . . . . . . 4
2.2.1. Curve-ID: brainpoolP192r1 . . . . . . . . . . . . . . 5
2.2.2. Twisted Curve-ID: brainpoolP192t1 . . . . . . . . . . 5
2.3. Domain Parameters for 224-bit Curves . . . . . . . . . . . 5
2.3.1. Curve-ID: brainpoolP224r1 . . . . . . . . . . . . . . 6
2.3.2. Twisted Curve-ID: brainpoolP224t1 . . . . . . . . . . 6
2.4. Domain Parameters for 256-bit Curves . . . . . . . . . . . 6
2.4.1. Curve-ID: brainpoolP256r1 . . . . . . . . . . . . . . 7
2.4.2. Twisted Curve-ID: brainpoolP256t1 . . . . . . . . . . 7
2.5. Domain Parameters for 320-bit Curves . . . . . . . . . . . 7
2.5.1. Curve-ID: brainpoolP320r1 . . . . . . . . . . . . . . 8
2.5.2. Twisted Curve-ID: brainpoolP320t1 . . . . . . . . . . 8
2.6. Domain Parameters for 384-bit Curves . . . . . . . . . . . 9
2.6.1. Curve-ID: brainpoolP384r1 . . . . . . . . . . . . . . 9
2.6.2. Twisted Curve-ID: brainpoolP384t1 . . . . . . . . . . 9
2.7. Domain Parameters for 512-bit Curves . . . . . . . . . . . 10
2.7.1. Curve-ID: brainpoolP512r1 . . . . . . . . . . . . . . 10
2.7.2. Twisted Curve-ID: brainpoolP512t1 . . . . . . . . . . 11
3. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 11
4. Security Considerations . . . . . . . . . . . . . . . . . . . 12
5. Insecurity Considerations . . . . . . . . . . . . . . . . . . 12
6. References . . . . . . . . . . . . . . . . . . . . . . . . . . 12
6.1. Normative References . . . . . . . . . . . . . . . . . . . 12
6.2. Informative References . . . . . . . . . . . . . . . . . . 13
Appendix A. Appendix A: Test Data . . . . . . . . . . . . . . . . 13
A.1. brainpoolP160r1 . . . . . . . . . . . . . . . . . . . . . 14
A.2. brainpoolP160t1 . . . . . . . . . . . . . . . . . . . . . 14
A.3. brainpoolP192r1 . . . . . . . . . . . . . . . . . . . . . 15
A.4. brainpoolP192t1 . . . . . . . . . . . . . . . . . . . . . 15
A.5. brainpoolP224r1 . . . . . . . . . . . . . . . . . . . . . 16
A.6. brainpoolP224t1 . . . . . . . . . . . . . . . . . . . . . 17
A.7. brainpoolP256r1 . . . . . . . . . . . . . . . . . . . . . 18
A.8. brainpoolP256t1 . . . . . . . . . . . . . . . . . . . . . 19
A.9. brainpoolP320r1 . . . . . . . . . . . . . . . . . . . . . 20
A.10. brainpoolP320t1 . . . . . . . . . . . . . . . . . . . . . 21
A.11. brainpoolP384r1 . . . . . . . . . . . . . . . . . . . . . 22
A.12. brainpoolP384t1 . . . . . . . . . . . . . . . . . . . . . 23
A.13. brainpoolP512r1 . . . . . . . . . . . . . . . . . . . . . 24
A.14. brainpoolP512t1 . . . . . . . . . . . . . . . . . . . . . 25
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . . 25
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1. Introduction
[RFC5639] defines new elliptic curve domain parmaeters for curves
over seven different prime fields, each with a "twisted" varient,
making a total of fourteen additional domain parameter sets. These
curves have a number of interesting security properties (as described
in [EBP]) that make them desirable to use.
IANA maintains a registry for [RFC2409] to map complete domain
parameter sets to numbers. Other pprotocols, for example
[IEEE802.11], refer to this registry for its convenience. Therefore,
this memo instructs IANA to allocate new code points for the
Brainpool curves defined in [RFC5639] to the registry established by
[RFC2409].
1.1. Requirements Language
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 RFC 2119 [RFC2119].
2. Brainpool Elliptic Curves
[RFC5639] defines several elliptic curves over finite prime fields
(ECP, in the parlance of [RFC2409]). The domain parameter sets for
each of the elliptic curves defined in [RFC5639] and are copied here
for convenient reference.
The equation for all elliptic curves defined here is:
y^2 = x^3 + ax + b (mod p)
Domain parameter sets consist of:
o p: the prime
o A, B: parameters to the equation of the curve
o x, y: the coordinates of the generator for the group
o q: the order of the group formed by the generator
o h: the co-factor
o z: the "twist" (for twisted curves only)
The prime used with the twisted curve is the same as its untwisted
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sibling-- that is, brainpoolP256t1 uses "p" from brainpoolP256r1.
Parameter z is used for the quadratic twist to generate the
parameters for the twisted varients. See [EBP] for more information.
2.1. Domain Parameters for 160-bit Curves
2.1.1. Curve-ID: brainpoolP160r1
p = E95E4A5F737059DC60DFC7AD95B3D8139515620F
A = 340E7BE2A280EB74E2BE61BADA745D97E8F7C300
B = 1E589A8595423412134FAA2DBDEC95C8D8675E58
x = BED5AF16EA3F6A4F62938C4631EB5AF7BDBCDBC3
y = 1667CB477A1A8EC338F94741669C976316DA6321
q = E95E4A5F737059DC60DF5991D45029409E60FC09
h = 1
2.1.2. Twisted Curve-ID: brainpoolP160t1
z = 24DBFF5DEC9B986BBFE5295A29BFBAE45E0F5D0B
A = E95E4A5F737059DC60DFC7AD95B3D8139515620C
B = 7A556B6DAE535B7B51ED2C4D7DAA7A0B5C55F380
x = B199B13B9B34EFC1397E64BAEB05ACC265FF2378
y = ADD6718B7C7C1961F0991B842443772152C9E0AD
q = E95E4A5F737059DC60DF5991D45029409E60FC09
h = 1
2.2. Domain Parameters for 192-bit Curves
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2.2.1. Curve-ID: brainpoolP192r1
p = C302F41D932A36CDA7A3463093D18DB78FCE476DE1A86297
A = 6A91174076B1E0E19C39C031FE8685C1CAE040E5C69A28EF
B = 469A28EF7C28CCA3DC721D044F4496BCCA7EF4146FBF25C9
x = C0A0647EAAB6A48753B033C56CB0F0900A2F5C4853375FD6
y = 14B690866ABD5BB88B5F4828C1490002E6773FA2FA299B8F
q = C302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1
h = 1
2.2.2. Twisted Curve-ID: brainpoolP192t1
z = 1B6F5CC8DB4DC7AF19458A9CB80DC2295E5EB9C3732104CB
A = C302F41D932A36CDA7A3463093D18DB78FCE476DE1A86294
B = 13D56FFAEC78681E68F9DEB43B35BEC2FB68542E27897B79
x = 3AE9E58C82F63C30282E1FE7BBF43FA72C446AF6F4618129
y = 097E2C5667C2223A902AB5CA449D0084B7E5B3DE7CCC01C9
q = C302F41D932A36CDA7A3462F9E9E916B5BE8F1029AC4ACC1
h = 1
2.3. Domain Parameters for 224-bit Curves
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2.3.1. Curve-ID: brainpoolP224r1
p = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FF
A = 68A5E62CA9CE6C1C299803A6C1530B514E182AD8B0042A59CAD29F43
B = 2580F63CCFE44138870713B1A92369E33E2135D266DBB372386C400B
x = 0D9029AD2C7E5CF4340823B2A87DC68C9E4CE3174C1E6EFDEE12C07D
y = 58AA56F772C0726F24C6B89E4ECDAC24354B9E99CAA3F6D3761402CD
q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F
h = 1
2.3.2. Twisted Curve-ID: brainpoolP224t1
z = 2DF271E14427A346910CF7A2E6CFA7B3F484E5C2CCE1C8B730E28B3F
A = D7C134AA264366862A18302575D1D787B09F075797DA89F57EC8C0FC
B = 4B337D934104CD7BEF271BF60CED1ED20DA14C08B3BB64F18A60888D
x = 6AB1E344CE25FF3896424E7FFE14762ECB49F8928AC0C76029B4D580
y = 0374E9F5143E568CD23F3F4D7C0D4B1E41C8CC0D1C6ABD5F1A46DB4C
q = D7C134AA264366862A18302575D0FB98D116BC4B6DDEBCA3A5A7939F
h = 1
2.4. Domain Parameters for 256-bit Curves
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2.4.1. Curve-ID: brainpoolP256r1
p = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5377
A = 7D5A0975FC2C3057EEF67530417AFFE7FB8055C126DC5C6CE94A4B44F330B5D9
B = 26DC5C6CE94A4B44F330B5D9BBD77CBF958416295CF7E1CE6BCCDC18FF8C07B6
x = 8BD2AEB9CB7E57CB2C4B482FFC81B7AFB9DE27E1E3BD23C23A4453BD9ACE3262
y = 547EF835C3DAC4FD97F8461A14611DC9C27745132DED8E545C1D54C72F046997
q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7
h = 1
2.4.2. Twisted Curve-ID: brainpoolP256t1
z = 3E2D4BD9597B58639AE7AA669CAB9837CF5CF20A2C852D10F655668DFC150EF0
A = A9FB57DBA1EEA9BC3E660A909D838D726E3BF623D52620282013481D1F6E5374
B = 662C61C430D84EA4FE66A7733D0B76B7BF93EBC4AF2F49256AE58101FEE92B04
x = A3E8EB3CC1CFE7B7732213B23A656149AFA142C47AAFBC2B79A191562E1305F4
y = 2D996C823439C56D7F7B22E14644417E69BCB6DE39D027001DABE8F35B25C9BE
q = A9FB57DBA1EEA9BC3E660A909D838D718C397AA3B561A6F7901E0E82974856A7
h = 1
2.5. Domain Parameters for 320-bit Curves
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2.5.1. Curve-ID: brainpoolP320r1
p = D35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC
28FCD412B1F1B32E27
A = 3EE30B568FBAB0F883CCEBD46D3F3BB8A2A73513F5EB79DA66190EB085FFA9
F492F375A97D860EB4
B = 520883949DFDBC42D3AD198640688A6FE13F41349554B49ACC31DCCD884539
816F5EB4AC8FB1F1A6
x = 43BD7E9AFB53D8B85289BCC48EE5BFE6F20137D10A087EB6E7871E2A10A599
C710AF8D0D39E20611
y = 14FDD05545EC1CC8AB4093247F77275E0743FFED117182EAA9C77877AAAC6A
C7D35245D1692E8EE1
q = D35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658
E98691555B44C59311
h = 1
2.5.2. Twisted Curve-ID: brainpoolP320t1
z = 15F75CAF668077F7E85B42EB01F0A81FF56ECD6191D55CB82B7D861458A18F
EFC3E5AB7496F3C7B1
A = D35E472036BC4FB7E13C785ED201E065F98FCFA6F6F40DEF4F92B9EC7893EC
28FCD412B1F1B32E24
B = A7F561E038EB1ED560B3D147DB782013064C19F27ED27C6780AAF77FB8A547
CEB5B4FEF422340353
x = 925BE9FB01AFC6FB4D3E7D4990010F813408AB106C4F09CB7EE07868CC136F
FF3357F624A21BED52
y = 63BA3A7A27483EBF6671DBEF7ABB30EBEE084E58A0B077AD42A5A0989D1EE7
1B1B9BC0455FB0D2C3
q = D35E472036BC4FB7E13C785ED201E065F98FCFA5B68F12A32D482EC7EE8658
E98691555B44C59311
h = 1
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2.6. Domain Parameters for 384-bit Curves
2.6.1. Curve-ID: brainpoolP384r1
p = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB711
23ACD3A729901D1A71874700133107EC53
A = 7BC382C63D8C150C3C72080ACE05AFA0C2BEA28E4FB22787139165EFBA91F9
0F8AA5814A503AD4EB04A8C7DD22CE2826
B = 04A8C7DD22CE28268B39B55416F0447C2FB77DE107DCD2A62E880EA53EEB62
D57CB4390295DBC9943AB78696FA504C11
x = 1D1C64F068CF45FFA2A63A81B7C13F6B8847A3E77EF14FE3DB7FCAFE0CBD10
E8E826E03436D646AAEF87B2E247D4AF1E
y = 8ABE1D7520F9C2A45CB1EB8E95CFD55262B70B29FEEC5864E19C054FF99129
280E4646217791811142820341263C5315
q = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425
A7CF3AB6AF6B7FC3103B883202E9046565
h = 1
2.6.2. Twisted Curve-ID: brainpoolP384t1
z = 41DFE8DD399331F7166A66076734A89CD0D2BCDB7D068E44E1F378F41ECBAE
97D2D63DBC87BCCDDCCC5DA39E8589291C
A = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B412B1DA197FB711
23ACD3A729901D1A71874700133107EC50
B = 7F519EADA7BDA81BD826DBA647910F8C4B9346ED8CCDC64E4B1ABD11756DCE
1D2074AA263B88805CED70355A33B471EE
x = 18DE98B02DB9A306F2AFCD7235F72A819B80AB12EBD653172476FECD462AAB
FFC4FF191B946A5F54D8D0AA2F418808CC
y = 25AB056962D30651A114AFD2755AD336747F93475B7A1FCA3B88F2B6A208CC
FE469408584DC2B2912675BF5B9E582928
q = 8CB91E82A3386D280F5D6F7E50E641DF152F7109ED5456B31F166E6CAC0425
A7CF3AB6AF6B7FC3103B883202E9046565
h = 1
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2.7. Domain Parameters for 512-bit Curves
2.7.1. Curve-ID: brainpoolP512r1
p = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308
717D4D9B009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A
48F3
A = 7830A3318B603B89E2327145AC234CC594CBDD8D3DF91610A83441CAEA9863
BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117A72BF2C7B9E7C1AC4D77FC
94CA
B = 3DF91610A83441CAEA9863BC2DED5D5AA8253AA10A2EF1C98B9AC8B57F1117
A72BF2C7B9E7C1AC4D77FC94CADC083E67984050B75EBAE5DD2809BD638016
F723
x = 81AEE4BDD82ED9645A21322E9C4C6A9385ED9F70B5D916C1B43B62EEF4D009
8EFF3B1F78E2D0D48D50D1687B93B97D5F7C6D5047406A5E688B352209BCB9
F822
y = 7DDE385D566332ECC0EABFA9CF7822FDF209F70024A57B1AA000C55B881F81
11B2DCDE494A5F485E5BCA4BD88A2763AED1CA2B2FA8F0540678CD1E0F3AD8
0892
q = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308
70553E5C414CA92619418661197FAC10471DB1D381085DDADDB58796829CA9
0069
h = 1
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2.7.2. Twisted Curve-ID: brainpoolP512t1
z = 12EE58E6764838B69782136F0F2D3BA06E27695716054092E60A80BEDB212B
64E585D90BCE13761F85C3F1D2A64E3BE8FEA2220F01EBA5EEB0F35DBD29D9
22AB
A = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308
717D4D9B009BC66842AECDA12AE6A380E62881FF2F2D82C68528AA6056583A
48F0
B = 7CBBBCF9441CFAB76E1890E46884EAE321F70C0BCB4981527897504BEC3E36
A62BCDFA2304976540F6450085F2DAE145C22553B465763689180EA2571867
423E
x = 640ECE5C12788717B9C1BA06CBC2A6FEBA85842458C56DDE9DB1758D39C031
3D82BA51735CDB3EA499AA77A7D6943A64F7A3F25FE26F06B51BAA2696FA90
35DA
y = 5B534BD595F5AF0FA2C892376C84ACE1BB4E3019B71634C01131159CAE03CE
E9D9932184BEEF216BD71DF2DADF86A627306ECFF96DBB8BACE198B61E00F8
B332
q = AADD9DB8DBE9C48B3FD4E6AE33C9FC07CB308DB3B3C9D20ED6639CCA703308
70553E5C414CA92619418661197FAC10471DB1D381085DDADDB58796829CA9
0069
h = 1
3. IANA Considerations
IANA is instructed to update the [IANA-IKE] registry entitled "Group
description" to include the fourteen domain parameter sets defined in
Section 2 and to add a note for each in the "Note" column of the
registry saying "not for IKEv1".
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+-------+--------------------------------------------+
| Value | Group Description |
+-------+--------------------------------------------+
| TBD1 | 160-bit Brainpool: brainpoolP160r1 |
| TBD2 | Twisted 160-bit Brainpool: brainpoolP160t1 |
| TBD3 | 192-bit Brainpool: brainpoolP192r1 |
| TBD4 | Twisted 192-bit Brainpool: brainpoolP192t1 |
| TBD5 | 224-bit Brainpool: brainpoolP224r1 |
| TBD6 | Twisted 224-bit Brainpool: brainpoolP224t1 |
| TBD7 | 256-bit Brainpool: brainpoolP256r1 |
| TBD8 | Twisted 256-bit Brainpool: brainpoolP256t1 |
| TBD9 | 320-bit Brainpool: brainpoolP320r1 |
| TBD10 | Twisted 320-bit Brainpool: brainpoolP320t1 |
| TBD11 | 384-bit Brainpool: brainpoolP384r1 |
| TBD12 | Twisted 384-bit Brainpool: brainpoolP384t1 |
| TBD13 | 512-bit Brainpool: brainpoolP512r1 |
| TBD14 | Twisted 512-bit Brainpool: brainpoolP512t1 |
+-------+--------------------------------------------+
Table 1: Group Description Updates
4. Security Considerations
[EBP] describes the security properties of the curves referenced
here. The curves support security levels of 80 (Section 2.1), 112
(Section 2.3), 128 (Section 2.4), 160 (Section 2.5), and 256
(Section 2.7).
5. Insecurity Considerations
The reason for the note in Section 3 is because [RFC2409] has been
deprecated but the protocol that deprecated it is not being used in
any significant fashion and the fear is that using these elliptic
curves with [RFC2409] will be further disincentive from switching
over to the new, but seldom used, protocol.
6. References
6.1. Normative References
[IANA-IKE]
IANA, "Internet Key Exchange (IKE) Attributes", Registry
Name: Group Description (Value 4), 2012.
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
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Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC5639] Lochter, M. and J. Merkle, "Elliptic Curve Cryptography
(ECC) Brainpool Standard Curves and Curve Generation",
RFC 5639, March 2010.
6.2. Informative References
[EBP] The Brainpool Workgroup, "ECC Brainpool Standard Curves
and Curge Generation", Brainpool: http://
www.ecc-brainpool.org/download/Domain-parameters.pdf,
October 2005.
[IEEE802.11]
IEEE Computer Society, "Telecommunications and information
exchange between systems Local and metropolitan area
networks--", Part 11: Wireless LAN Medium Access Control
(MAC) and Physical Layer (PHY) Specifications IEEE Std
802.11-2012.
[RFC2409] Harkins, D. and D. Carrel, "The Internet Key Exchange
(IKE)", RFC 2409, November 1998.
Appendix A. Appendix A: Test Data
This section provides some test vectors for example Diffie-Hellman
key exchanges using each of the curves defined in Section 2. In all
of the following sections the following notation is used:
o dA: the secret key of party A
o x_qA: the x-coordinate of the public key of party A
o y_qA: the y-coordinate of the public key of party A
o dB: the secret key of party B
o x_qB: the x-coordinate of the public key of party B
o y_qB: the y-coordinate of the public key of party B
o x_Z: the x-coordinate of the shared secret that results from
completion of the Diffie-Hellman computation
o y_Z: the y-coordinate of the shared secret that results from
completion of the Diffie-Hellman computation
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A.1. brainpoolP160r1
dA =
65636E6F 74776879 696B6576 32697361 6661696C
x_qA =
B3F0660C AEBD7850 C528DF37 744088B1 5DE756FD
y_qA =
66CB8CCB FDAFA97B D3C76174 F52BA816 D32E10E1
dB =
7082A67E 033CF4CB A09E386B 22A09355 6718AB85
x_qB =
26CB345F DC1B9FC6 C8437AB7 E67864B8 90BDD5E3
y_qB =
7AAD7218 09F1E4FD BACA6EF0 A09BDF4A 32CED089
x_Z =
857E1AE5 F2FD06F5 04A8971E 2D7FA9CB 6A8EDD82
y_Z =
A3D8FBE0 EBA79835 BA656C4F 1CCA4061 ABB54649
A.2. brainpoolP160t1
dA =
9C260CB5 65A6FDAD D31D46BF B0033CDE A548DF20
x_qA =
9C367FE5 DB9F3DE8 8324EF59 C3237ED0 FCEBA61C
y_qA =
C7ADAB09 87FFF508 80B2BB66 FADAF093 619477B9
dB =
DC9EF1EB 6DE455A9 33008043 EAABAE91 82D6C314
x_qB =
55A74F96 A017C17A 43923892 85F320B6 117DC757
y_qB =
C10F30B4 894487B4 81B1D674 7284AD9A C2F878AE
x_Z =
E956CA60 E3205B78 E19EFB3A 35841CE1 C1DFE0B1
y_Z =
5916C962 484738B4 5B9F8923 40D12D65 C3FF6908
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A.3. brainpoolP192r1
dA =
4B32E699 290E3F05 2A99C7F0 CFFD1C57 07898015 C743FE2A
x_qA =
7E245B17 DB3674A5 239ADBDC FC267340 ED741635 C6D3C551
y_qA =
6967C744 E1444EAC AC984115 801539E0 C65CAA6F 8A65ADB1
dB =
6916B2A3 36B3305A 256238EA 2BAB549D 9C893F47 D07964B8
x_qB =
3AA86ED0 6F3E289A 309970C7 A3C4F45D 18C03549 77080718
y_qB =
9EE978D8 2CCC8DBB B97BDBA0 9DBCDE63 35721DD7 ACD4207A
x_Z =
3BC21456 3BEB3A8A 27DB2CCE 9940B866 1F8E914B E2B9EA82
y_Z =
AC733793 D588A775 17A6152A D6EDBB65 D5A3FAE0 164BA6EA
A.4. brainpoolP192t1
dA =
BE1C4EBE 4947E7BD 1FA6A8A8 133BE0DB 9CF6EED9 50994E2F
x_qA =
374C9F83 4719EF30 14DBECBE 68FC6AEE A3105298 85F9EFCC
y_qA =
576861EE ABC2AEA5 B4565F5E 0DFB4796 021CBE35 F8ACEC3C
dB =
B9BB399F 24855569 2BBD8C86 F2C217F2 91140AC0 F239579F
x_qB =
883BD1A6 8D90B6C5 A3ACB0A4 04751060 0E9D4E80 4ACAD4DC
y_qB =
6C9D0045 6AF2A879 185BB355 674468AC 5D1552F2 65ED9DA3
x_Z =
5E4FF4B6 5F2530E7 648E4A33 9467EFC8 01D069BA 10774E41
y_Z =
9761DF2 CE9A9186 14530862 C48F3C26 B5C20FDD B7B19510
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A.5. brainpoolP224r1
dA =
39F15548 3CEE191F BECFE9C8 1D8AB1A0 3CDA6790 E7184ACE
44BCA161
x_qA =
A9C21A56 9759DA95 E0387041 18426144 0327AFE3 3141CA04
B82DC92E
y_qA =
98A0F75F BBF61D8E 58AE5511 B2BCDBE8 E549B31E 37069A28
25F590C1
dB =
60605523 03899E21 40715816 C45B57D9 B42204FB 6A5BF5BE
AC10DB00
x_qB =
034A56C5 50FF8805 6144E6DD 56070F54 B0135976 B5BF7782
7313F36B
y_qB =
75165AD9 9347DC86 CAAB1CBB 579E198E AF88DC35 F927B358
AA683681
x_Z =
1A4BFE70 5445120C 8E3E0266 99054104 510D1197 57B74D5F
E2462C66
y_Z =
BB6802AC 01F8B7E9 1B1A1ACF B9830A95 C079CEC4 8E52805D
FD7D2AFE
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A.6. brainpoolP224t1
dA =
36FC6A84 AE6E2336 1E440D3F CFB1AAC5 0580C348 47159554
FC136A7C
x_qA =
A0133727 466F725A 3BA76FF1 E358CA3A 622C006F E34D9C3B
4E83AAAF
y_qA =
5794548A EE38A029 655C8035 DA3A0AEE FDEBAB16 7AECCFA9
C7BDEE42
dB =
AE43C2F0 255E6C48 1C0C0A97 D6DBC2AF 6AF6E6BC 96B650B5
B98E1821
x_qB =
89CC5C34 DB58FA72 3E013BE6 3C1D5A97 85C9CF4A 8A48AE72
1029430B
y_qB =
AF814269 83C1D3CF A35CF297 66761B8F F5F66C6C 2BA52879
E0C87203
x_Z =
C259CE78 D4A9EECE DF24DE82 E826E9F3 7461D0AD A44EAD40
3316D01F
y_Z =
45386727 1508DC3D 86815165 8E77FA42 70263DA2 77653E93
3455E404
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A.7. brainpoolP256r1
dA =
81DB1EE1 00150FF2 EA338D70 8271BE38 300CB542 41D79950
F77B0630 39804F1D
x_qA =
44106E91 3F92BC02 A1705D99 53A8414D B95E1AAA 49E81D9E
85F929A8 E3100BE5
y_qA =
8AB4846F 11CACCB7 3CE49CBD D120F5A9 00A69FD3 2C272223
F789EF10 EB089BDC
dB =
55E40BC4 1E37E3E2 AD25C3C6 654511FF A8474A91 A0032087
593852D3 E7D76BD3
x_qB =
8D2D688C 6CF93E11 60AD04CC 4429117D C2C41825 E1E9FCA0
ADDD34E6 F1B39F7B
y_qB =
990C5752 0812BE51 2641E470 34832106 BC7D3E8D D0E4C7F1
136D7006 547CEC6A
x_Z =
89AFC39D 41D3B327 814B8094 0B042590 F96556EC 91E6AE79
39BCE31F 3A18BF2B
y_Z =
49C27868 F4ECA217 9BFD7D59 B1E3BF34 C1DBDE61 AE129316
48F43E59 632504DE
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A.8. brainpoolP256t1
dA =
203B977B 00441D77 7C1FDE7F 2A74E9D6 D4DF83A4 97270C66
4B24C0D3 7693F16A
x_qA =
75097578 80446CDF 64425D0A 66316FFA 2150AC44 E231834A
A2EFD0C0 E41393B3
y_qA =
2CB29D33 7940E4B0 20C0FBB4 0829971B EB14E892 E409A4B7
6C835470 828E22FE
dB =
2ACA3258 FC1567CA A31E2024 925B71AA 61723B92 53B62756
389655AE 403983BA
x_qB =
637C6099 C5641550 DCD1EFA5 04E3758B BBC0A7E8 918B6C6F
30ABF8F7 24837470
y_qB =
9A5B0F5B E7A5C87B 25D77D84 2278BBF5 DB6618FE B4E8E40E
CD180662 B75FD8CC
x_Z =
58D52959 1F883614 6B14BEBB B1A68CDD 76FDD216 FAD3493C
1A80E2E1 A077A58B
y_Z =
97187640 47B1B195 2331ADF5 F28D6312 B0203F5F 75DC91DC
AD2F5765 397F9E89
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A.9. brainpoolP320r1
dA =
7CD9C454 BA907F76 17E262A7 FD73764C 4A3157C1 3F82279E
F9F062BE 5D49A8E3 90B66A4D CEDFA867
x_qA =
BC43666C 00E4B943 FE1C785D D8AA842A 42AB54B0 B49819F9
60F77694 193CD3AF A71B6B3C 826C7734
y_qA =
69E99889 2C076446 8023C8E3 A7B8F219 A1446042 BE175D44
76B2FDFD 85B22EAD 2F29101A 1242A578
dB =
B832A73D A5F671E8 0D87F093 72544801 F6812224 B19A4BC1
B37AA7DB 0842E6DD 3CA11DE0 F802BFED
x_qB =
B1246229 429354D1 D687BCA4 8BCCD6FC 733B146D AC03642A
0AD4B896 F5D8BCBD 2F4BCA16 776E4526
y_qB =
A4168389 8F9A76EF 36EA2DC7 B74D419E 55CF3664 721890D6
A2B2FB8C EB7C1131 67ED137A 358EE37F
x_Z =
730314D9 06B2F21D C11BE050 31B028D6 65696BEE C7139328
CDF70C71 8BE5D208 659BB967 43A88067
y_Z =
C338B5B7 A3FB62ED E9BAA9C0 6DF9BC36 D4B5F0D3 5EFDF792
49913E6D C4DB6DBC 7BA9B74E 59C840F1
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A.10. brainpoolP320t1
dA =
A09C32F8 4AD534BF 7FA0BC5F 548F0EB6 C1BA8127 3B0762B4
137D969F C840AD2F 160F1612 060D229A
x_qA =
47B55315 38B66BA3 9B6E382D F8E8C37F E0BBF04C C9739087
4F990D78 4A099156 60586CCE 140D59C4
y_qA =
775B5482 E9316063 EDAEB987 319A470A BD0AFE29 CEA29138
28AB3A7B BFF3701D B0B4E08C CE754B53
dB =
8BCD3AFD 75A73D02 E673DE41 E5EBFE3E 4CE53955 A959F064
0A5194F4 F66BA6E9 6D7A4069 76C91392
x_qB =
3EA750DB 5FD4B1C9 A62F5029 CF7ABBFC 25916EFF 5AECB06A
258B8E45 4E565290 76A987E6 D9DDFB41
y_qB =
4CA5E9D4 0778AE8E FFC96B78 A5654DA0 7669339B 6BE9F7D5
C55B0FB6 3842B808 B040C40A 2A7003C0
x_Z =
6F41BC68 4ECFAA3B E38307E9 FF88FF5B 5077CDBE F84CF8FF
D761DCAC ED7EFE68 9856D75E EE65AC7E
y_Z =
74D32A6D 3F89A215 95FD6BEE 7F75AAA9 90182A04 539C9EAD
77EE18C0 D77B54DA 16748498 E208C4DA
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A.11. brainpoolP384r1
dA =
1E20F5E0 48A5886F 1F157C74 E91BDE2B 98C8B52D 58E5003D
57053FC4 B0BD65D6 F15EB5D1 EE1610DF 87079514 3627D042
x_qA =
68B665DD 91C19580 0650CDD3 63C625F4 E742E813 4667B767
B1B47679 3588F885 AB698C85 2D4A6E77 A252D638 0FCAF068
y_qA =
55BC91A3 9C9EC01D EE36017B 7D673A93 1236D2F1 F5C83942
D049E3FA 20607493 E0D038FF 2FD30C2A B67D15C8 5F7FAA59
dB =
032640BC 6003C592 60F7250C 3DB58CE6 47F98E12 60ACCE4A
CDA3DD86 9F74E01F 8BA5E032 4309DB6A 9831497A BAC96670
x_qB =
4D44326F 269A597A 5B58BBA5 65DA5556 ED7FD9A8 A9EB76C2
5F46DB69 D19DC8CE 6AD18E40 4B15738B 2086DF37 E71D1EB4
y_qB =
62D69213 6DE56CBE 93BF5FA3 188EF58B C8A3A0EC 6C1E151A
21038A42 E9185329 B5B27590 3D192F8D 4E1F32FE 9CC78C48
x_Z =
0BD9D3A7 EA0B3D51 9D09D8E4 8D0785FB 744A6B35 5E6304BC
51C229FB BCE239BB ADF64037 15C35D4F B2A5444F 575D4F42
y_Z =
0DF21341 7EBE4D8E 40A5F76F 66C56470 C489A347 8D146DEC
F6DF0D94 BAE9E598 157290F8 75606697 5F1DB34B 2324B7BD
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A.12. brainpoolP384t1
dA =
69533063 89403A3B EE6FB415 DCDFF2F8 48CD4083 F3E9FD73
8562860D 0D64FBF8 89CD3D18 A166422A AB85A1C7 E079940A
x_qA =
6AA8A43B 53CEC913 E3482BBE 6AD2EC8E 70E0D45B E73DD7DB
C9CA3684 E614E592 2C754053 C6271DA2 9D351682 DF38094D
y_qA =
37B30062 3120B879 11D0AF24 FBDAAF34 252255FA 07D17EB2
F97165E8 E76B161F 15EF94F8 23192D2F B0841EF7 394F32D5
dB =
4BE33D42 65A66BDD D47E32B5 91C7AD6A 7C5702BA A8A6C0BE
5D816D9E 9734ED06 6B46D9F6 6CD48FC5 3180E8E4 0D08DFF0
x_qB =
57D03114 173727A6 AE945310 C2DD4960 EEF33D41 BB12C83D
4210C417 D92AD1C2 FD5B7B64 5F81BA9A 664D8FD5 30F00990
y_qB =
3476261B 06DF1BD6 2EB99F3B AAC0805B CCD240EA 29457F9D
C2DD3CD1 FBFB9492 2353E18F 10257BCB 6C41E67C 53B425FF
x_Z =
065F24E9 5E1AC20A DAFEB7F9 ABBE5182 A31935F8 34D62DCA
6F6C8936 55774606 52A476E8 4B67C6E4 308A0E5F F8516B61
y_Z =
70AA22D8 978A67D7 EE1971B5 EDD5E046 691425FD F719AB17
8B997C69 60B32F36 D66236E4 852FA93B 0CCAB352 B1AF52CB
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A.13. brainpoolP512r1
dA =
16302FF0 DBBB5A8D 733DAB71 41C1B45A CBC87159 39677F6A
56850A38 BD87BD59 B09E8027 9609FF33 3EB9D4C0 61231FB2
6F92EEB0 4982A5F1 D1764CAD 57665422
x_qA =
7047CD66 D78504B0 D68DDB0E 1332FFA5 531CF755 73D4597A
49EF681C A6237038 94144CDF 91810E49 53098E35 FC0CAFD0
35A323AB F102FF49 5C3BC9F5 558A1168
y_qA =
779D9BA5 73DF5D9E B09568CA 183F86B0 11865236 31A88414
A0F91D71 E1B7D87F F306C5D9 9DD1C6BA 8C0C3AD9 05A6DB6B
13969D44 4A11DFC7 06DD3E3D C1613E2A
dB =
230E18E1 BCC88A36 2FA54E4E A3902009 292F7F80 33624FD4
71B5D8AC E49D12CF ABBC1996 3DAB8E2F 1EBA00BF FB29E4D7
2D13F222 4562F405 CB805036 66B25429
x_qB =
4ABC9477 AB47FC42 835EFD60 7720F148 A6008042 619BEC79
C1DAB3A3 21B7D910 A8F0D633 40C13CA8 E436CE4F 1FA55DDF
14822477 BE9F5DA8 F11B172A 90FF9B9F
y_qB =
63E6A8DF 47897EAF 1E083208 655326CF A2924238 2CC36334
25E61405 6EFB85CE D2BE217D 4B4FDEFA 9EDEDB3B 15989868
BB82974D 4ACA9639 EAEDDB0D 8E9DD518
x_Z =
1EE7EC58 878D26B0 B28AC2D8 46D0780D 431E6A34 0E5DBE51
7CDBDD71 8909C782 DAEECBA8 493B194B 3E51172A 7AD79354
1171A132 47C77D2B CEFECAA8 D09F4E3F
y_Z =
5A005DAE 91F82175 01EDDEF2 A37F9E6B 9F2A7983 E436E67C
C82E79F6 0AC083C8 521FC5D5 0698F36C B987563D 61289781
4E9F6265 AE3A039C BD7467B1 F1B4C391
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A.14. brainpoolP512t1
dA =
25F04FBE 4EB90853 46CE85D7 614D711A 1830EFA5 54135D80
CFE1CB1D 414C6C17 86A1D740 364C8487 EC89E52A F46C7C80
5C378AE2 D985D7BC 48B3A776 D759A4F9
x_qA =
2FCAF73E E7801B12 D06DDBDF 5DC446A1 AC820B4D 5C45C928
E0999A36 1A7E5429 616FA8DD 9F0C60BB 831783E2 3706C8C8
57EF9B75 4B84A53E C973960F D4C77BD3
y_qA =
061CDCDB 1AEF90A0 80E24508 C7AC7075 CA6BCEC8 2BE33FEF
0A8B80C0 EEB24A0C 6A528FF5 BD6CC4C3 417C0061 C75D9933
3B536F13 4C5ECE9D 4976FD57 FD55C6F8
dB =
9F4EB0AE 43E7D541 0B1F908D 8EBE3F20 3EC4278C EAF58967
538A9A48 DFF3A131 A708C1C7 175CF7FB 36209FAE 4024DD0B
B8C4D182 6406642D 67D88197 9EF32100
x_qB =
3C5865BB 0763806B 268640AE 62E25E56 D9AC75A3 5E860D64
5C19FFFA D625F028 59BCC688 CFC8F112 F8EC9607 BEAC0A7A
431CEDC4 D43025F9 5A522FEE 91AB0A2F
y_qB =
81CDE7F9 F6E29AC7 1903494C DEF49CB6 A79F0CE9 EC0C5E98
266F5158 6B676130 2C15E9AC 073451B2 F1AD6418 AD25823E
2A893F5F A76860D4 5DC8E935 61CE96F3
x_Z =
077FEC42 01645930 04E07CDF A19FAFF1 0BCF3813 602DB890
FCA12805 03C2EAAA B530216D 31797177 F9E41890 B7179FEC
41189F95 805AF93B 3C508A67 EF6FFDD9
y_Z =
1D196A9D 32F26EA5 AA81C1C3 987DDC2E 540D33E9 03C9F8A4
EA5D08FD 7E82245F A00F692F 9EF8A9A3 97B3BEEA E528BAD2
B035109C C9B63188 50C4019C A00E53D2
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Author's Address
Dan Harkins (editor)
Aruba Networks
1322 Crossman avenue
Sunnyvale, Californaia 94089
United States of America
Phone: +1 408 227 4500
Email: dharkins@arubanetworks.com
Harkins Expires February 9, 2013 [Page 26]