Advances in Cryptology – EUROCRYPT 2014: 33rd Annual by Phong Q. Nguyen, Elisabeth Oswald

By Phong Q. Nguyen, Elisabeth Oswald

This ebook constitutes the lawsuits of the thirty third Annual foreign convention at the idea and purposes of Cryptographic concepts, EUROCRYPT 2014, held in Copenhagen, Denmark, in may well 2014. The 38 complete papers integrated during this quantity have been rigorously reviewed and chosen from 197 submissions. They care for public key cryptanalysis, identity-based encryption, key derivation and quantum computing, secret-key research and implementations, obfuscation and multi linear maps, authenticated encryption, symmetric encryption, multi-party encryption, side-channel assaults, signatures and public-key encryption, useful encryption, foundations and multi-party computation.

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This corresponds precisely to the case where these codes can be distinguished from random codes by square code considerations. The filtration attack has a polynomial time complexity and basically boils down to linear algebra. This is the first time in the 35 years of existence of the McEliece scheme based on Goppa codes that a polynomial time attack has been found on it. It questions the common belief that GRS codes are weak for a cryptographic use while Goppa codes are secure as soon as m 2 and that for the latter only generic information-set-decoding attacks apply.

Then, c c is of the form: c c = (y0 y0 f (x0 )g(x0 ), . . , yn−1 yn−1 f (xn−1 )g(xn−1 )) = (y0 y0 r(x0 ), . . , yn−1 yn−1 r(xn−1 )) where deg(r) k + k − 2. Conversely, any element (y0 y0 r(x0 ), . . , yn−1 yn−1 r(xn−1 )) where deg(r) k + k − 2, is a linear combination of star products of two elements of GRSk (x, y). Statement (ii) is a consequence of (i) by putting y = y and k = k. Since an alternant code is a subfield subcode of a GRS code, we might suspect that products of alternant codes have also an abnormal low dimension.

Without loss of generality, one can assume that the first two entries of x are x0 = 0 and x1 = 1. As explained further, this will in particular make possible 1 Recall that by (xa −xa )−(q+1) we mean the vector (xi − xa )−(q+1) . ,n−1}\{a} 28 A. Couvreur, A. –P. Tillich −(q+1) the computation of the vectors x0 and (x1 − 1)q+1 and we prove further that the knowledge of these two vectors provides that of x up to some Galois action. Let us now define precisely these codes C a (j). They are defined for any a ∈ {0, .

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