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

ISBN-10: 3540401113

ISBN-13: 9783540401117

ISBN-10: 3540448284

ISBN-13: 9783540448280

This e-book constitutes the refereed complaints of the fifteenth overseas 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 offered including 2 invited papers have been rigorously reviewed and chosen from forty submissions. one of the matters addressed are block codes; algebra and codes: earrings, fields, and AG codes; cryptography; sequences; interpreting algorithms; and algebra: buildings in algebra, Galois teams, differential algebra, and polynomials.

**Read Online or Download Applied Algebra, Algebraic Algorithms and Error-Correcting Codes: 15th International Symposium, AAECC-15, Toulouse, France, May 12–16, 2003 Proceedings PDF**

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**Sample text**

2) becomes X3 = T1 Y2 · X1 Z2 + Z1 X2 · Y1 T2 , Y3 = T1 Y2 · Y1 T2 − Z1 X2 · X1 Z2 , Z3 = T1 Z1 T2 Z2 − k 2 X1 Y1 X2 Y2 , T3 = (T1 Y2 )2 + (Z1 X2 )2 , (4) so that 16 (ﬁeld) multiplications (plus 1 multiplication by constant k 2 ) are required. The inverse of a point P1 = (X1 , Y1 , Z1 , T1 ) is −P1 = (−X1 , Y1 , Z1 , T1 ). Again, we see that (excluding the neutral element (0, 1, 1, 1)) there are three points of order 2, namely, (0, −1, 1, 1), (0, 1, −1, 1) and (0, 1, 1, −1). 2 37 Jacobi Quartic Jacobi also studied quartics of the form y 2 = (1 − x2 )(1 − k 2 x2 ) (5) with k = 0, ±1.

We use Mr to denote the random variables of the ﬁrst r messages, and H(E|Mr ) the conditional entropy of E given that the ﬁrst r messages have been observed. The following is called the information-theoretic bound [8], [10], [12]. Lemma 1. In any authentication code, 2 PI ≥ 2H(E|M)−H(E) , PS ≥ 2H(E|M )−H(E|M) . The following is called the combinatorial bound [9]. Lemma 2. [6,9] In any authentication system without splitting, PI ≥ |S| |S| − 1 and PS ≥ . |M| |M| − 1 If both equalities are achieved, then |E| ≥ |M|.

In this paper, we present three algebraic constructions of authentication codes with secrecy. The codes have simple algebraic structures and are easy to implement. They are asymptotically optimal with respect to certain bounds. 1 Introduction The authentication model introduced by Simmons involves three parties: a transmitter, a receiver, and an opponent. The transmitter wants to send a piece of information (called a source state) to the receiver through a public communication channel. A source state s is encoded to obtain a message m = Ek (s) with an encoding rule Ek shared by the transmitter and receiver, and is sent to the receiver through the channel.

### 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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