By Jacques Stern (auth.), Marc Fossorier, Tom Høholdt, Alain Poli (eds.)
This ebook 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 possibly 2003.
The 25 revised complete papers provided including 2 invited papers have been rigorously reviewed and chosen from forty submissions. one of the matters addressed are block codes; algebra and codes: jewelry, fields, and AG codes; cryptography; sequences; interpreting algorithms; and algebra: buildings in algebra, Galois teams, differential algebra, and polynomials.
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Extra info for Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 15th International Symposium, AAECC-15, Toulouse, France, May 12–16, 2003 Proceedings
Suitable for fast implementations; see Sect. 2. We do not consider the zero isogeny φ = . 46 E. Brier and M. 4], such that φˆ ◦ φ = [m] and φ ◦ φˆ = [m] where m = deg φ and [m] (resp. [m] ) is the multiplication-by-m isogeny on E (resp. E ). Interestingly, this leads to a diﬀerent way for computing Q = [rm]P as Q = φˆ [r] φ(P ) . [rm] P ∈ E(K) −−−−−→ Q = [rm]P ∈ E(K) ˆ φ φ [r] P ∈ E (K) −−−−−→ Q = [r] P ∈ E (K) Fig. 1. Computing Q = [rm]P through isogenies Isogenies have been intensively studied in order to improve point counting algorithms.
Walker and P. Wild, The combinatorics of perfect authentication schemes, SIAM J. Disc. Math. 7 (1994), 102–107. 8. D. Pei, Information-theoretic bounds for authentication codes and block designs, J. Cryptography 8 (1995), 177–188. 9. S. R. Stinson, Combinatorial characterizations of authentication codes, Designs, Codes and Cryptography 7 (1996), 239–259. 10. U. Rosenbaum, A lower bound on authentication after having observed a sequence of messages, J. Cryptography 6 (1993), 135–156. 11. R, Safavi-Naini and L.
The authentication code of (3) provides at least log2 q n − 1 − 2(q − 1)q n/2 q bits of secrecy protection. Furthermore we have 1 2(q − 1)q n/2 1 2(q − 1)q n/2 − ≤ PI ≤ + n q q(q − 1) q q(q n − 1) and 2(q 2 − 1)q n/2 + q n − 1 −2(q 2 − 1)q n/2 + q n − 1 ≤ PS ≤ n/2 n q[2(q − 1)q + q − 1] q[−2(q − 1)q n/2 + q n − 1] 28 C. Ding et al. Optimality of the Codes We now prove that PI and PS of authentication code (3) meet the lower bound of Lemma 2 asymptotically. In the authentication code of (3), |S| = q n − 1, |M| = q(q n − 1) So the bound on PI given in Lemma 2 is P = Since lim n→∞ 1 .