By Florian Hess (auth.), Ferruh Özbudak, Francisco Rodríguez-Henríquez (eds.)
This e-book constitutes the refereed court cases 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 offered have been conscientiously reviewed and chosen from 29 submissions. The papers are prepared in topical sections on coding conception and code-based cryptography, Boolean services, finite box mathematics, equations and services, and polynomial factorization and permutation polynomial.
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Additional info for Arithmetic of Finite Fields: 4th International Workshop, WAIFI 2012, Bochum, Germany, July 16-19, 2012. Proceedings
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Matsumoto – Let P1 , . . , PN −1 , Q be rational places of an algebraic function field over Fq √ and denote by g the genus. Assume N = ( q − 1)g. That is, we assume that the function field attains the Drinfeld-Vladut bound. Using codes CL (U = P1 + · · · + PN −1 , mQ) with g < m as outer codes one gets for all and k (see Section 2 for a discussion) Ω(k) X ⊆ F2 |X | = O , 3 k log(1/ ) . 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 .
Let Σ be a permutation of {1, . . , n} and γ = (γ1 , . . , γn ) ∈ Fnq such that ∀i, γi = 0. The transformation Πγ,Σ is defined as follows: Πγ,Σ : Fnq −→ Fnq v → (γΣ(1) vΣ(1) , . . , γΣ(n) vΣ(n) ) Notice that ∀α ∈ Fq , ∀v ∈ Fnq , Πγ,Σ (αv) = αΠγ,Σ (v), and wt(Πγ,Σ (v)) = wt(v). The q-SD identification scheme is comprised of two algorithms: key generation and identification protocol. Given the security parameter κ, the key generation algorithm described in Fig. 1 picks a random (r × n) q-ary matrix H common to all users, a random vector s (secret key) with Hamming weight ω.
The multiset X is made into a probability space by adjoining to each element x ∈ X the probability p(x) = i(x)/|X | where i(x) denotes the number of times x appears in X . As a measure for describing how close a given space X is to the above situation with respect to randomization, Naor and Naor [15], and Alon et. al. [1] introduced the concept of -biasness [15, Def. 3]. (See also [14]). ¨ F. Ozbudak and F. ): WAIFI 2012, LNCS 7369, pp. 29–44, 2012. c Springer-Verlag Berlin Heidelberg 2012 30 O.