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Efficient algorithms for pairingbased cryptosystems
, 2002
"... Abstract. We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in ..."
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Cited by 291 (23 self)
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Abstract. We describe fast new algorithms to implement recent cryptosystems based on the Tate pairing. In particular, our techniques improve pairing evaluation speed by a factor of about 55 compared to previously known methods in characteristic 3, and attain performance comparable to that of RSA in larger characteristics. We also propose faster algorithms for scalar multiplication in characteristic 3 and square root extraction over Fpm, the latter technique being also useful in contexts other than that of pairingbased cryptography. 1
Improved Algorithms for Isomorphisms of Polynomials
 Advances in Cryptology – EUROCRYPT’98 (Kaisa Nyberg, Ed
, 1998
"... This paper is about the design of improved algorithms to solve Isomorphisms of Polynomials (IP) problems. These problems were first explicitly related to the problem of finding the secret key of some asymmetric cryptographic algorithms (such as Matsumoto and Imai's C # scheme of [13], or some variat ..."
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Cited by 20 (1 self)
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This paper is about the design of improved algorithms to solve Isomorphisms of Polynomials (IP) problems. These problems were first explicitly related to the problem of finding the secret key of some asymmetric cryptographic algorithms (such as Matsumoto and Imai's C # scheme of [13], or some variations of Patarin's HFE scheme of [15]). Moreover, in [15], it was shown that IP can be used in order to design an asymmetric authentication or signature scheme in a straightforward way. We also introduce the more general Morphisms of Polynomials problem (MP). As we see in this paper, these problems IP and MP have deep links with famous problems such as the Isomorphism of Graphs problem or the problem of fast multiplication of n n matrices. The complexities of our algorithms for IP are still not polynomial, but they are much more e#cient than the previously known algorithms. For example, for the IP problem of finding the two secret matrices of a MatsumotoImai C # scheme over K = F q , the complexity of our algorithms is O(q n/2 ) instead of O(q (n 2 ) ) for previous algorithms. (In [14], the C # scheme was broken, but the secret key was not found). Moreover, we have algorithms to achieve a complexity O(q 3 2 n ) on any system of n quadratic equations with n variables over K = F q (not only equations from C # ). We also show that the problem of deciding whether a polynomial isomorphism exists between two sets of equations is not NPcomplete (assuming the classical hypothesis about ArthurMerlin games), but solving IP is at least as di#cult as the Graph Isomorphism problem (GI) (and perhaps much more di#cult), so that IP ! is unl ikely to be solvable in polynomial time. Moreover, the more general Morphisms of Polynomials problem (MP) is NPhard. Finally, we suggest...
Factorization of Trinomials over Galois Fields of Characteristic 2
, 1997
"... We study the parity of the number of irreducible factors of trinomials over Galois fields of characteristic 2. As a consequence, some sufficient conditions for a trinomial being reducible are obtained. For example, x n + ax k + b 2 GF (2 t )[x] is reducible if both n, t are even, except possibly whe ..."
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Cited by 2 (1 self)
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We study the parity of the number of irreducible factors of trinomials over Galois fields of characteristic 2. As a consequence, some sufficient conditions for a trinomial being reducible are obtained. For example, x n + ax k + b 2 GF (2 t )[x] is reducible if both n, t are even, except possibly when n = 2k, k odd. The case t = 1 was treated by R.G. Swan [10], who showed that x n + x k + 1 is reducible over GF (2) if 8n.