Download Arithmetic of Finite Fields: 4th International Workshop, by Florian Hess (auth.), Ferruh Özbudak, Francisco PDF

By Florian Hess (auth.), Ferruh Özbudak, Francisco Rodríguez-Henríquez (eds.)

This booklet constitutes the refereed lawsuits of the 4th foreign Workshop at the mathematics of Finite box, WAIFI 2012, held in Bochum, Germany, in July 2012. The thirteen revised complete papers and four invited talks provided have been rigorously reviewed and chosen from 29 submissions. The papers are geared up in topical sections on coding idea and code-based cryptography, Boolean features, finite box mathematics, equations and capabilities, and polynomial factorization and permutation polynomial.

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Additional resources for Arithmetic of Finite Fields: 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012. Proceedings

Example text

Our proposed scheme benefits of a performance gain as a result of the reduction in the soundness error from 2/3 for Stern’s scheme to 1/2 per round for the q-SD scheme. Our threshold ring signature scheme uses random linear codes over the field Fq , secure in the random oracle model and its security relies on the hardness of an error-correcting codes problem (namely the q-ary syndrome decoding problem). In this paper we also provide implementation results of the Aguilar et al. scheme and our proposal, this is the first efficient implementation of this type of code-based schemes.

This result which is in the folklore is known as the AG-bound. 1 – Using Hermitian codes with m < g as outer codes one achieves [2] for ≥ k − 2 Ω(k) X ⊆ F2 |X | = O , k 2 log(1/ ) 5 4 . (3) This we call the BT-bound after the authors of [2], Ben-Aroya and Ta-Shma. – Using in larger generality Norm-Trace codes of low dimension as outer codes − √1 one achieves [12] for l = 4, 5, . . and ≥ k l (see Section 5) ⎛ l+1 ⎞ l k Ω(k) ⎠. √ X ⊆ F2 , |X | = O ⎝ l− l log(1/ ) Here, l = 4 corresponds to the Hermitian case described in [2].

G } ∪ {2g, . . , n − 1} ∪ {λn−g+1 , . . , λn }, (7) where λi ≤ g − 1 + i for i = 1, . . , g. This is a general result for Weierstrass semigroups and not particular for the Hermitian function field. Having described the Hermitian codes as affine variety codes we are now ready to introduce the combination of codes on which our construction of small-bias spaces rely. Consider the ideal 2 (2) 2 2 2 Iq2 := X1q+1 − Y1q − Y1 , X2q+1 − Y2q − Y2 , X1q − X1 , Y1q − Y1 , X2q − X2 , Y2q − Y2 and the corresponding variety (2) VFq2 (Iq2 ) = VFq2 (Iq2 ) × VFq2 (Iq2 ) = {Q1 , .

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