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This paper proposes a novel scheme which consists of a weight-counting algorithm, the combinatorial designs of the Assmus-Mattson theorem, and the weight polynomial of Gleasonpsilas theorem to determine the weight distributions of binary extended quadratic residue codes. As a consequence, the weight distribution of binary (168, 84, 24) extended quadratic residue code is given.
Kocarev gave the first cryptosystems based on the semi-group property of Chebyshev polynomials, which seemed excellent but actually insecure. Due to the inherent periodicity of trigonometric function, an attack can easily get plaintext given ciphertext. In this paper, we extend Chebyshev polynomials from real number to finite fields to avoid the attack and present the corresponding key exchange scheme,...
Let Mq(n) denote the number of multiplications required to compute the coefficients of the product of two polynomials of degree n over a q-element field by means of bilinear algorithms. It is shown that Mq(n) ≥ 3n - o(n). In particular, if q/2 ≪ n ≤ q + 1, we establish the tight bound Mq(n) = 3n + 1 - ⌊q/2⌋. The technique we use can be applied to analysis of algorithms for multiplication of polynomials...
Let d = d(n) be the minimum d such that for every sequence of n subsets F1, F2, . . . , Fn of {1, 2, . . . , n} there exist n points P1, P2, . . . , Pn and n hyperplanes H1, H2 .... , Hn in Rd such that Pj lies in the positive side of Hi iff j ∈ Fi. Then n/32 ≤ d(n) ≤ (1/2 + 0(1)) ?? n. This implies that the probabilistic unbounded-error 2-way complexity of almost all the Boolean functions of 2p variables...
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