Download e-book for iPad: 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.)

ISBN-10: 3540401113

ISBN-13: 9783540401117

ISBN-10: 3540448284

ISBN-13: 9783540448280

This booklet constitutes the refereed complaints of the fifteenth foreign Symposium on utilized Algebra, Algebraic Algorithms and Error-Correcting Codes, AAECC-15, held in Toulouse, France, in may perhaps 2003.

The 25 revised complete papers awarded including 2 invited papers have been conscientiously reviewed and chosen from forty submissions. one of the matters addressed are block codes; algebra and codes: jewelry, 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

Example text

3 = 2161748259540728720113669865088143302633269781215144746593 An application of transformation given by Eq. (9) shows that the Weierstraß curve E is equivalent to the Jacobi curve Y 2 = X 4 − 2δ X 2 Z 2 + Z 4 with = 439238437583428445099508669973297609255723032614505577652 and δ = 294835057741119445319427130851297392990523848101510040967. Further, 42 O. Billet and M. Joye since there are three points of order 2, can be rescaled to the case = 1 via the additional transformation X ← 2X/ξ with √ ξ = θ2 − θ3 = 2362324240509570404961221823945617479743113384215829517748 .

Let N (f ) be the number of affine solutions in Fn+1 to the equation q Z p − Z = f (X1 , . . , Xn ). Theorem 1. There exists an explicit deterministic algorithm with the following input, output and complexity. The input is a polynomial f ∈ Fq [X1 , . . , Xn ] with a diagonal leading form of degree d not divisible by the characteristic p, where p > 2. The output is the number N (f ) of rational solutions to the affine equation Z p − Z = f . The running time is ˜ n dmin(4n+1,3n+3) log(q)3 p2 ) O(c bit operations, where cn depends only on n.

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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Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 15th International Symposium, AAECC-15, Toulouse, France, May 12–16, 2003 Proceedings by Jacques Stern (auth.), Marc Fossorier, Tom Høholdt, Alain Poli (eds.)


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