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We study sums and products in a field. Let be a field with , where is the characteristic of . For any integer , we show that any can be written as with and , and that for any we can write every as with and . We also prove that for any and there are such that .
This work was presented in part at the 8th International Conference on Finite Fields and
Applications Fq^8 , Melbourne, Australia, 9-13 July, 2007.In this paper, we study the ratio θ(n) = λ2 (n) / ψ2 (n), where λ2 (n) is
the number of primitive polynomials and ψ2 (n) is the number of irreducible
polynomials in GF (2)[x] of degree n. Let n = ∏ pi^ri, i=1,..l
be the prime factorization of n. We show that, for fixed l and ri , θ(n) is close to 1 and θ(2n) is
not less than 2/3 for sufficiently large...
We consider an equation of the typeover the finite field . Carlitz obtained formulas for the number of solutions to this equation when and when and . In our earlier papers, we found formulas for the number of solutions when or or ; and when and is a power of modulo . In this paper, we obtain formulas for the number of solutions when , , or . For general case, we derive lower bounds for the number of solutions.
In the present paper we investigate distributional properties of sparse sequences modulo almost all prime numbers. We obtain new results for a wide class of sparse sequences which in particular find applications on additive problems and the discrete Littlewood problem related to lower bound estimates of the -norm of trigonometric sums.
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 with given joint 2-adic complexity.
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...
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