Advances in Cryptology — CRYPTO '97: 17th Annual - download pdf or read online

By Mikael Goldmann, Mats NÄslund (auth.), Burton S. Kaliski Jr. (eds.)

ISBN-10: 3540633847

ISBN-13: 9783540633846

This publication constitutes the refereed complaints of the seventeenth Annual overseas Cryptology convention, CRYPTO'97, held in Santa Barbara, California, united states, in August 1997 less than the sponsorship of the overseas organization for Cryptologic examine (IACR).
The quantity provides 35 revised complete papers chosen from a hundred and sixty submissions bought. additionally integrated are invited displays. The papers are equipped in sections on complexity thought, cryptographic primitives, lattice-based cryptography, electronic signatures, cryptanalysis of public-key cryptosystems, info idea, elliptic curve implementation, number-theoretic platforms, dispensed cryptography, hash capabilities, cryptanalysis of secret-key cryptosystems.

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Extra info for Advances in Cryptology — CRYPTO '97: 17th Annual International Cryptology Conference Santa Barbara, California, USA August 17–21, 1997 Proceedings

Example text

Let V be a vector space over k and let S be a subset of V . We say S is linearly independent if whenever α1 , . . , αn ∈ k and v1 , . . , vn ∈ S satisfy α1 v1 + . . αn vn = 0, it must be true that α1 = · · · = αn = 0. We say S spans V if for any w ∈ V there exist α1 , . . , αn ∈ k and v1 , . . , vn ∈ S such that α1 v1 + . . αn vn = w. We say S is a basis for V if S is linearly independent and spans V . In this case, the number of elements of S is called the dimension of V . In general, there are several linearly independent subsets S which span the vector space V , but they all 52 A.

17. Prove that elements of Rn are in one-to-one correspondence with polynomials over k of degree at most n − 1. Hint: Use the Division Algorithm. 17, we can think of the elements of Rn as actually being polynomials over k, as long as we always replace xn with 1 when doing computations. 18. 19. Let k be any field, n a positive integer, and let a0 , . . , an−1 ∈ k. Compute x(a0 + a1 x + · · · + an−1 xn−1 ) in Rn . 3. 20. Let k be a field. A vector space V over k is an abelian group which admits a scalar multiplication by elements of k.

If we set Pi = (αi : 1) (using the numbering of the elements of F× q as above), we have the following alternate description of the Reed-Solomon code: RS(k, q) = {(f (P1 ), . . , f (Pq−1 )) | f ∈ L((k − 1)P∞ )} Goppa’s idea [Go] was to generalize this. Let X be a projective, nonsingular plane curve over Fq , and let D be a divisor on X. Let P = {P1 , . . , Pn } ⊂ X(Fq ) be a set of n distinct Fq -rational points on X. If we assume that P ∩ suppD = ∅, then no Pi can be a pole of any f ∈ L(D), and, in fact, f (Pi ) ∈ Fq for any f ∈ L(D) and any Pi ∈ P.

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Advances in Cryptology — CRYPTO '97: 17th Annual International Cryptology Conference Santa Barbara, California, USA August 17–21, 1997 Proceedings by Mikael Goldmann, Mats NÄslund (auth.), Burton S. Kaliski Jr. (eds.)

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