Applied Algebra, Algebraic Algorithms and Error-Correcting by Jacques Stern (auth.), Marc Fossorier, Tom Høholdt, Alain

By Jacques Stern (auth.), Marc Fossorier, Tom Høholdt, Alain Poli (eds.)

This publication constitutes the refereed lawsuits of the fifteenth overseas Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-15, held in Toulouse, France, in could 2003.

The 25 revised complete papers provided including 2 invited papers have been conscientiously reviewed and chosen from forty submissions. one of the topics addressed are block codes; algebra and codes: earrings, fields, and AG codes; cryptography; sequences; deciphering algorithms; and algebra: buildings in algebra, Galois teams, differential algebra, and polynomials.

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Additional info for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 15th International Symposium, AAECC-15, Toulouse, France, May 12–16, 2003 Proceedings

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3 Isomorphisms Two elliptic curves E and E , respectively given by the Weierstraß equations E/K : y 2 = x3 + ax + b and E/K : y 2 = x3 + a x + b , are isomorphic over K if and only if there exists a nonzero element u ∈ K such that u4 a = a and u6 b = b. Moreover, the isomorphism is given by ∼ φ : E −→ E , (x, y) −→ (u−2 x, u−3 y) O −→ O . The elliptic curve E/K : y 2 = x3 + ax + b can thus be made isomorphic to the elliptic curve E/K : y 2 = x3 − 3x + b if and only if a = −3u4 for some u ∈ K \ {0}.

It appears that only 13 (field) multiplications (plus 3 multiplications by constants) are required. We insist that the same procedure equally applies for doubling a point. We further note that the neutral element is (0 : 1 : 1) and that the procedure remains valid for it too. When constant δ (resp. ) is small, the cost of a multiplication by δ (resp. ) can be neglected. A good choice consists in imposing a small value for since this removes two multiplications by constants – as shown in Fig. 1, there are 2 multiplications by and 1 multiplication by δ.

We have Π(s) + Π(m1 − s) = m2 . When m1 = 0 and q is even, the number of solutions to this equation is q 2t−1 . In other cases, the minimum number of solutions to this equation is q 2t−1 − q t−1 . Hence the secrecy protection for the source state is at least log2 (q 2t−1 − q t−1 ) bits. 3 Second Specific Construction of Codes in This Family Define Π(x) = TrGF(qn )/GF(q) (x2 ), where n is a positive integer, q is an odd prime, and TrGF(qn )/GF(q) is the trace function. Since x2 is a perfect nonlinear mapping from GF(q n ) to itself, Π is a perfect nonlinear mapping from GF(q n ) to GF(q).

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