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On the joint 2-adic complexity of binary multisequences∗

Lu Zhao, Qiao-Yan Wen (2012)

RAIRO - Theoretical Informatics and Applications

Joint 2-adic complexity is a new important index of the cryptographic security for multisequences. In this paper, we extend the usual Fourier transform to the case of multisequences and derive an upper bound for the joint 2-adic complexity. Furthermore, for the multisequences with pn-period, we discuss the relation between sequences and their Fourier coefficients. Based on the relation, we determine a lower bound for the number of multisequences...

On the number of zero trace elements in polynomial bases for F2n.

Igor E. Shparlinski (2005)

Revista Matemática Complutense

Let Fq denote the finite field of q elements. O. Ahmadi and A. Menezes have recently considered the question about the possible number of elements with zero trace in polynomial bases of F2n over F2. Here we show that the Weil bound implies that there is such a basis with n + O(log n) zero-trace elements.

On the restricted Waring problem over 2 n [ t ]

Luis Gallardo (2000)

Acta Arithmetica

1. Introduction. The Waring problem for polynomial cubes over a finite field F of characteristic 2 consists in finding the minimal integer m ≥ 0 such that every sum of cubes in F[t] is a sum of m cubes. It is known that for F distinct from ₂, ₄, 16 , each polynomial in F[t] is a sum of three cubes of polynomials (see [3]). If a polynomial P ∈ F[t] is a sum of n cubes of polynomials in F[t] such that each cube A³ appearing in the decomposition has degree < deg(P)+3, we say that P is a restricted...

On the value set of small families of polynomials over a finite field, II

Guillermo Matera, Mariana Pérez, Melina Privitelli (2014)

Acta Arithmetica

We obtain an estimate on the average cardinality (d,s,a) of the value set of any family of monic polynomials in q [ T ] of degree d for which s consecutive coefficients a = ( a d - 1 , . . . , a d - s ) are fixed. Our estimate asserts that ( d , s , a ) = μ d q + ( q 1 / 2 ) , where μ d : = ∑ r = 1 d ( ( - 1 ) r - 1 ) / ( r ! ) . We also prove that ₂ ( d , s , a ) = μ ² d q ² + ( q 3 / 2 ) , where ₂(d,s,a) is the average second moment of the value set cardinalities for any family of monic polynomials of q [ T ] of degree d with s consecutive coefficients fixed as above. Finally, we show that ₂ ( d , 0 ) = μ ² d q ² + ( q ) , where ₂(d,0) denotes the average second moment for all monic polynomials...

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