Cryptography Reference
In-Depth Information
3.9.5 Tate Pairing
Let
E
be an elliptic curve defined over the finite field
F
q
, and let
n
be an integer such
that
n
is divisible by
q-
1. Let th
e
elements of
F
q
whose order is divisible by
n
be denoted
as
(
[]
=Î =
. Let us assume that
()
n
μ
EF
contains elements
of order
n
. Then there exists a
modified
Ta t e - Licht e nba um pa ir in g ,
EF n
and let
{
zKz
|
1}
q
n
q
(
[]
(
[]
(
[]
τ
´
μ
(3.63)
:
EFnEFnnEFn
/
n
q
q
q
n
3.10 Summary
We have introduced some basic concepts of modern algebra and looked into some basic
definitions in field theory. We discussed the importance of elliptic curve cryptogra-
phy and then looked into the elliptic curve discrete log problem. We then looked into
pairing-based cryptography and, in particular, Weil and Tate pairing.
3.11 References
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Boneh, D., and M. Franklin. Identity-based encryption from theWeil pairing.
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ACM Conference
on Computer and Communications Security.
ACM, 2004, 168-177.
Boneh, D., B. Lynn, and H. Shacham. Short signatures from the Weil pairing.
Journal of
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(2004): 297-319.
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ElGamal, T. A public-key cryptosystem and a signature scheme based on discrete logarithms.
IEEE Transactions on Information Theory
(IEEE) 31, no. 4 (1984): 469-472.
Joux, A. A one round protocol for tripartite Diffie-Hellman.
Lecture Notes in Computer Science
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Koblitz, N. Elliptic curve cryptosystems.
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48, no. 177 (1987):
203-209.
Lenstra, H. W. Factoring integers with elliptic curves.
Annals of Mathematics
126, no. 3 (1987):
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Menezes, A.
Elliptic Curve Public Key Cryptosystems.
Kluwer Academic, 1993.
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Communications of the ACM
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Sakai, R., K. Ohgishi, and M. Kasahara. Cryptosystems based on pairing.
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