Internet Engineering Task Force D. Atkins
Internet-Draft SecureRF Corporation
Intended status: Standards Track December 08, 2014
Expires: June 11, 2015
Using Algebraic Eraser in OpenPGP
draft-atkins-openpgp-algebraic-eraser-03
Abstract
The Algebraic Eraser(TM) is an encryption engine that supports, among
other configurations, a Diffie-Hellman-like key agreement protocol.
This draft specifies how to encode, store, share, and use Algebraic
Eraser Key Agreement Protocol keys in OpenPGP.
Status of This Memo
This Internet-Draft is submitted in full conformance with the
provisions of BCP 78 and BCP 79.
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material or to cite them other than as "work in progress."
This Internet-Draft will expire on June 11, 2015.
Copyright Notice
Copyright (c) 2014 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
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Table of Contents
1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . 2
2. The Algebraic Eraser . . . . . . . . . . . . . . . . . . . . 2
2.1. E-Multiplication . . . . . . . . . . . . . . . . . . . . 3
2.2. AEDH Keyset Parameters . . . . . . . . . . . . . . . . . 3
2.3. Generating Key Pairs . . . . . . . . . . . . . . . . . . 4
2.4. Computing the Shared Secret . . . . . . . . . . . . . . . 5
3. Encoding of Public and Private Keys . . . . . . . . . . . . . 5
3.1. Encoding Bit-Strings . . . . . . . . . . . . . . . . . . 5
3.1.1. Encoding Techniques . . . . . . . . . . . . . . . . . 6
3.1.2. Multi-Byte Entries . . . . . . . . . . . . . . . . . 7
3.2. Encoding Public Keys . . . . . . . . . . . . . . . . . . 7
3.3. Encoding Private Keys . . . . . . . . . . . . . . . . . . 7
4. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . 8
5. IANA Considerations . . . . . . . . . . . . . . . . . . . . . 8
6. Security Considerations . . . . . . . . . . . . . . . . . . . 8
7. References . . . . . . . . . . . . . . . . . . . . . . . . . 9
7.1. Normative References . . . . . . . . . . . . . . . . . . 9
7.2. Informative References . . . . . . . . . . . . . . . . . 9
Appendix A. Test Vectors . . . . . . . . . . . . . . . . . . . . 9
A.1. Sample key . . . . . . . . . . . . . . . . . . . . . . . 10
A.2. Sample key agreement . . . . . . . . . . . . . . . . . . 11
Author's Address . . . . . . . . . . . . . . . . . . . . . . . . 12
1. Introduction
The OpenPGP specification in [RFC4880] defines the use of RSA,
Elgamal, and DSA public key algorithms. [RFC6637] adds support for
Elliptic Curve Cryptography and specifies the ECDSA and ECDH
algorithms.
The Algebraic Eraser was first introduced in Key agreement, the
Algebraic Eraser, and lightweight cryptography [AAGL] published by
the American Mathematical Society in 2004. It describes "a new key
agreement protocol suitable for implementation on low-cost platforms
which constrain the use of computational resources." It is further
compared to other algorithims in [AEIntro]. This document specifies
how to encode, store, and use the Algebraic Eraser(TM) Key Agreement
Protocol (AEKAP, also called AEDH) in OpenPGP.
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. The Algebraic Eraser
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The Algebraic Eraser brings together the Braid Group, Matrices, and
operations over small Finite Fields to produce an algorithm that
executes linear in time with the increase in key size.
A complete description of the Algebraic Eraser is available in
[AAGL].
2.1. E-Multiplication
The Algebraic Eraser defines an operation called "E-Multiplication"
upon which the algorithm is based (see [AAGL]). E-Multiplication
(denoted herein by *) takes one matrix (M0) and permutation (S0) and
operates on a second matrix (M1) and permutation (S1), resulting in
another matrix (M2) and permutation (S2). In other words: (M0,S0) *
(M1,S1) = (M2,S2).
The secret to E-multiplication is that you can take a Braid element
(an "Artin generator") and map that to a matrix and permutation upon
which you can operate. Specifically, each Artin generator is
associated to a specific Colored Burau (CB) matrix and permutation
(see [AAGL]). The E-multiplication process involves permuting the
variables in the CB matrix using the current permutation,
substituting those variables with the TValues from the Keyset (see
Section 2.2), then multiplying that against the starting matrix
resulting in the another matrix. Then you multiply the permutations,
resulting in a new permutation. This new matrix and permutation is
the the result of the E-Multiplication.
2.2. AEDH Keyset Parameters
AEDH Keyset Parameters are similar to Diffie-Hellman cyclic groups of
prime order or ECC curves. Just as users must choose the same DH
prime or ECC curve in order to communicate, similarly participants in
the AEDH must be using the same Keyset Parameters.
The first basic set of parameters is the chosen Braid Group and Field
Size, BnFq, where n is the number of strands in the chosen braid
(also called the braid index) and q is the size of the field in use.
The field size, q, must be a power of a prime. Generally it is 2^r
(where r is a small integer) although this is not a requirement. For
example, one might choose B10F8 or B16F32. This is like choosing how
many bits to use when generating a prime for Diffie-Hellman.
Once the BnFq space is chosen then the Keyset Parameters can be
generated by a trusted third party (TTP). First they generate an
n-by-n matrix (M) where each entry in the matrix is a member of the
field Fq. Second, the TTP generates a set of TValues, which is an
array of n invertable entries within the field Fq (i.e., values 1 to
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Fq-1). Finally, the TTP generates at least two sets of braid
conjugates, Ca and Cb, where each conjugate in Ca commutes with each
conjugate in Cb. The conjugates are lists of "braid words", or
"Artin generators" within the Bn braid group. The TTP generates La
conjugates for set Ca and Lb conjugates for set Cb, where the numbers
La and Lb MAY be different.
The public Keyset Parameters are the Matrix (M), TValues array, and
conjugate sets and must be available to generate keys that can
communicate. Moreover, the TValue array must be available to anyone
using the Keyset. These Keysets MAY be published and named, but MUST
be numbered with an OID.
For two users to execute the AEDH they MUST generate keys from the
same Keyset and they MUST choose from different conjugate sets within
that Keyset. I.e., for Alice and Bob to complete the AEDH Alice must
generate her key from Ca and Bob must generate his key from Cb.
This document does not specify any particular Keyset Parameters that
MUST be implemented.
2.3. Generating Key Pairs
The Algebraic Eraser has a two-part Private Key and a two-part Public
Key. The Public key is then generated from the two Private Keys.
To generate the 1st private key you generate a random polynomial and
apply that to the public matrix from the keyset within the keyset
field. This results in an n-by-n matrix where each entry in the
matrix is a member of the field Fq (although the last row of the
matrix contains n-1 zeros). The key search space for the 1st private
key is q^(n-1). Note that the 1st private key permutation is always
the identity permutation, so there is no need to store it.
To generate the 2nd private key you choose a random set of conjugates
(and inverses) and string them together. This results in a long
string of Artin generators (and inverses). You MAY reduce the string
if you so choose using the Dehornoy reduction [Dehornoy]. The search
space of the 2nd private key is (2l)^k (where l is the number of
published conjugates (La or Lb), and k the number of chosen
conjugates and inverses).
The Public Key is computed by an E-Multiplication of the 1st private
key and the 2nd private key, where the 2nd private key is iteratively
processed (see Section 2.1. The result (the public key) is an n-by-n
matrix of Fq and another permutation.
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Note that the last row of the Public Key Matrix is all zero except
for the last entry. When encoding the Public Key you SHOULD ignore
those zeros.
2.4. Computing the Shared Secret
To compute the shared secret you first perform regular matrix
multiplication of the 1st private key against the matrix component of
the public key you receive from the other person. This results in
another matrix. The permutation of the 1st private key is the
identity, hence the result of multiplying it against the permutation
of the public key leaves the permutation of the public key unchanged
and you can just use it directly. Then you perform E-multiplication
of this matrix/permutation result against the 2nd private key. The
resulting matrix/permutation is the shared secret.
3. Encoding of Public and Private Keys
Each portion of a key can be reduced to a byte-string (or, more
accurately, multiple byte strings). Each matrix can be encoded by
stringing together each field element in each row and then stringing
each row together. A permutation can be encoded by stringing
together each element in the list. The conjugates are also encoded
by stringing together each element.
The following public key algorithm IDs are added to expand section
9.1 of [RFC4880], "Public-Key Algorithms":
+------+---------------------------+
| ID | Description of Algorithm |
+------+---------------------------+
| TBD1 | AEDH public key algorithm |
+------+---------------------------+
Encoding of Public and Private keys MUST use the version 4 packet
format (or newer).
3.1. Encoding Bit-Strings
The Algebraic eraser uses matrices, fields, and braids that are
denoted in bits, particular strings of bits. These objects need to
be encoded into bit strings for storage and transmission. The most
simplistic method of encoding is to take each field as a byte (or
multi-byte word) and string them together. The following sections
detail multiple (alternate) ways these bit strings can be encoded to
possibly reduce the space used.
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3.1.1. Encoding Techniques
Depending on the number of bits used per element (which is defined by
the braid index and field size), using different encodings of these
strings may result in reducing storage space by dropping extra bits
and combining elements.
For example, when using the finite field F16 each entry can be
encoded in exactly one nibble of four (4) bits, so you can easily
combine two entries into a single 8-bit byte (called nibble-
encoding). This technique could also be used for entries smaller
than a nibble, although then you would still have extra (unused)
bits. When using the nibble-encoding of an odd number of nibbles the
encoding rules MUST specify whether the extra nibble is at the
leading or trailing byte.
Another encoding option is bit-stealing. This merges all bits
together and then cuts it up into 8-bit bytes. For example if the
entries are 5 bits each you might steal 3 bits from the second entry
to merge into the first, then shift the remaining 2 bits of the
second entry, combine with the next 5 bits from the third, and then
steal one bit from the fourth entry, and so on, until you've reached
the end. This could end up with unused bits at the end of the
string.
Yet another option is the reverse-bit-stealing, where you start at
the end of the string and work your way to the front. This could
leave you with unused bits a the front of the string.
Assume you require five (5) bits to encode your numbers, the
following table shows how you could could use bit stealing and
reverse bit stealing to encode them (where a, b, c, and d are the
bits in the first, second, third, and fourth entries)
+-----------------------+----------+----------+----------+----------+
| Full Bytes: | 000aaaaa | 000bbbbb | 000ccccc | 000ddddd |
+-----------------------+----------+----------+----------+----------+
| Bit stealing: | aaaaabbb | bbcccccd | dddd0000 | |
+-----------------------+----------+----------+----------+----------+
| Reverse bit stealing: | 0000aaaa | abbbbbcc | cccddddd | |
+-----------------------+----------+----------+----------+----------+
Any unused bits MUST be left as zero (and MUST be checked to be
zero).
The actual encoding method MUST be defined by the Keyset parameter
definition and may change from one keyset parameter to another.
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The row of zeros in the matrix SHOULD be assumed to "not exist".
When using these encoding techniques you SHOULD just tack the last
entry of the final row onto the end of the list of entries of the
rest of the matrix. This could result in an odd number of entries
depending on your n and q choices potentially requiring passing at
the start or end of the bit string.
3.1.2. Multi-Byte Entries
In the case of entries wider than 8 bits (e.g. a Field parameter
greater than 256), the bits are combined in network byte order.
However they can still be merged together using the same encoding
algorithms from Section 3.1.1 in the case of entries that are not
8-bit multiples. For example, a 12-bit field (F4096) could be
combined a nibble at a time, or a 10-bit field (F1024) could use bit-
stealing.
3.2. Encoding Public Keys
The following algorithm specific packets are added to Section 5.5.2
of [RFC4880], "Public-Key Packet Formats", to support AEDH:
o a variable length field containing a keyset parameter OID,
formatted as follows (see [RFC6637] for a full description of the
OID encoding method):
* a one-octet size of the following field; values 0 and 0xFF are
reserved for future extensions,
* octets representing a keyset parameter OID
o one byte denoting from which set of conjugates in the keyset this
key was generated (e.g. the Alice set or the Bob set)
o MPI of the public key matrix
o MPI of the public key permutation
3.3. Encoding Private Keys
The following algorithm specific packets are added to Section 5.5.3
of [RFC4880], "Secret-Key Packet Formats", to support AEDH:
o MPI of the 1st private key (matrix)
o MPI of the 2nd private key (conjugate string)
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4. Acknowledgements
The term "Algebraic Eraser" is a trademark of SecureRF Corporation
and is used herein with permission.
The author would like to thank Paul Gunnells, Dorian Goldfeld, and
Iris Anshel for their tireless efforts to review this document,
suggest improvements, and explain to me how to improve my description
of how AE works. Big thanks also to Werner Koch and Vedaal for their
comments and suggestions.
5. IANA Considerations
IANA is requested to assign an algorithm number from the OpenPGP
Public-Key Algorithms range, or the "namespace" in the terminology of
[RFC5226], that was created by [RFC4880]. See Section 3.
+------+---------------------------+-----------+
| ID | Algorithm | Reference |
+------+---------------------------+-----------+
| TBD1 | AEDH public key algorithm | This doc |
+------+---------------------------+-----------+
[Notes to RFC-Editor: Please remove the table above on publication.
It is desirable not to reuse old or reserved algorithms because some
existing tools might print a wrong description. A higher number is
also an indication for a newer algorithm. As of now 22 is the next
free number, but we request the selection of 23.]
6. Security Considerations
The security considerations of [RFC4880] apply accordingly.
AEDH will generate the same session key when used with the same two
public/private key pairs. The authors of AE generally recommend that
at least one party use an ephemeral key pair in order to prevent the
same session key being generated every time.
AEDH is an encryption-only algorithm, therefore it cannot self-
certify a key. To have an AEDH master key you MUST implement
[I-D.atkins-openpgp-device-certificates].
When using the generated session key, you MUST only use the bits
included in the protocol. You should MUST NOT use any always-zero
bits, including those in the last row of the matrix.
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7. References
7.1. Normative References
[AAGL] Anshel, I., Anshel, M., Goldfeld, D., and S. Lemieux, "Key
agreement, the Algebraic Eraser, and lightweight
cryptography", Contemporary Mathematics 418, 2004, .
[RFC2119] Bradner, S., "Key words for use in RFCs to Indicate
Requirement Levels", BCP 14, RFC 2119, March 1997.
[RFC4880] Callas, J., Donnerhacke, L., Finney, H., Shaw, D., and R.
Thayer, "OpenPGP Message Format", RFC 4880, November 2007.
[RFC5226] Narten, T. and H. Alvestrand, "Guidelines for Writing an
IANA Considerations Section in RFCs", BCP 26, RFC 5226,
May 2008.
[RFC6637] Jivsov, A., "Elliptic Curve Cryptography (ECC) in
OpenPGP", RFC 6637, June 2012.
7.2. Informative References
[AEIntro] SecureRF Corporation, SRF., "An Introduction to
Cryptographic Security Methods and Their Role in Securing
Low Resource Computing Devices", 2011, .
[Dehornoy]
Dehornoy, P., "A fast method for comparing braids",
Advances in Mathematics 123, 1997,
.
[I-D.atkins-openpgp-device-certificates]
Atkins, D., "OpenPGP Extensions for Device Certificates",
draft-atkins-openpgp-device-certificates-00 (work in
progress), August 2014.
Appendix A. Test Vectors
To help implementing this specification a non-normative example is
provided. This example assumes:
o the algorithm id for AEDH will be 23
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o the keyset OID 1.3.6.1.4.1.44196.1.0.0, which defines:
* the braid/field as B10F8
* the public key packing is nibble-packed with trailing zeros
* the 2nd private key is not bit-packed; it uses bit 7 to define
"inverse" and bits 3-0 to define the Artin generator.
and gets encoded with length 11 and the following hex bytes: 2B 06
01 04 01 82 D9 24 01 00 00
A.1. Sample key
The secret key used for this example is:
1st Private Key Matrix:
4 2 7 4 1 2 7 7 3 5
1 1 5 4 0 5 0 0 3 1
2 7 5 3 4 0 6 0 0 4
6 1 0 7 4 7 7 4 1 1
1 1 7 6 6 2 4 6 5 7
7 5 4 1 7 3 7 5 0 7
1 6 0 7 3 6 4 2 5 6
7 2 3 6 6 6 4 2 7 7
3 7 5 2 2 2 0 7 5 2
6
2nd Private key (in hex):
060481820304050384840506028304050682838485810203048506878807880984
858384828384858383838485068708070809868788888887880586078809080987
880788090809878809030485850384848583838483848506070809878809060506
070809878788860788090687080983040506070809030405060708090203840506
070405060708838484858583848384858586870687080102030405060708098586
878888858586838485858182828282828181828203048586878809080907080607
080984858586860283040586868601820304858586878787870808098102038485
860102030405860708878787878787088181828384858687080607070606060706
070809090607880988090687880907088787080986878809090607080987888687
878888078806078809868788888886878788888787888807880987880607880909
068708070809098686878807078809090987888686878809880988090906878807
880906878888078806060687880987080808080708098687880909888788090987
880607060607070788090809878809060708098787888686868787878809880909
090607080906878809888888090987880909078809878807880909098788090708
098687880607880908098788888888880909878886078888090607080986878886
868787888809090809878888090607080987888806878809870809868788090607
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080987878809880907080987068708090708098686878788888686868686860788
880987880987870687888788860788068788078886078806878888078809868787
878788078806078886070707070788098708080986868607088787870886870886
878708090707088687878787878787080806070886868708090906070809868788
078809090809870809870809870809868788060788090906878809090707880986
878888888788090807080907860788090987080808090908080808080987880707
860707880909098788090986878888880909868788090981828384858687880906
070506050606078607070707048304858607070782038485860781028384850606
070707080809098401828384050607880909040506870881828384858687088708
080102030485060701020384050607080802020101020202020283848586870182
838485868708038485860706070809888809098788098687880485860788090905
060708090708868708098182838485868788078809878807880987880788090403
040303040405068708080985868788090607080806070583040303030304058602
03040584058683048582038485010203040485860582838483840201
The key was created on 2014-09-09 16:35:36 from the tag conjugates
(type 1), and thus the fingerprint of the OpenPGP key is:
8020 A772 BA18 EA47 3501 B8CE EABE 1082 56BB 5D64
and the entire public key packet is:
98 4a 04 54 0f 64 98 17 0b 2b 06 01 04 01 82 d9
24 01 00 00 01 01 6e 26 44 05 46 10 02 50 43 37
56 66 37 42 40 10 72 06 14 44 16 67 13 02 70 73
11 00 30 27 47 21 75 35 76 13 13 31 00 60 52 75
24 50 57 23 60 00 25 12 35 76 a8 94
A.2. Sample key agreement
The key agreement is created using the sample key against a second
(reader) public key. The reader public key has the following data:
Matrix (in nibbled-packed hex with trailing zeros):
24 14 13 22 14 67 30 02 20 23 11 26 26 51 20 11
67 40 56 57 60 77 01 04 66 56 71 35 21 27 57 00
55 75 16 40 07 75 05 12 31 35 75 45 66 40
Permutation (in nibble-packed hex): 32 14 56 78 9a
Which results in the following shared secret:
Matrix:
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4 0 6 5 2 3 0 5 6 0
6 5 5 0 2 0 1 7 5 5
2 0 2 1 1 1 2 7 2 0
4 0 1 2 5 6 6 6 1 2
5 0 1 0 7 4 3 3 3 4
5 1 2 5 3 3 5 5 7 1
1 0 7 1 6 3 4 0 2 1
2 7 5 4 6 7 1 4 7 4
7 1 5 5 3 6 1 4 1 6
5
Permutation (decimal): 3 2 1 5 7 6 10 8 9 4
Author's Address
Derek Atkins
SecureRF Corporation
100 Beard Sawmill Rd, Suite 350
Shelton, CT 06484
US
Phone: +1 617 623 3745
Email: datkins@securerf.com, derek@ihtfp.com
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