Field Of Prime Order Research Articles

On the average number of elements in a finite field with order or index in a prescribed residue class

For any prime p the density of elements in F p ∗ having order, respectively index, congruent to a ( mod d) is being considered. These densities on average are determined, where the average is taken over all finite fields of prime order. Some connections between the two densities are established. It is also shown how to compute these densities with high numerical accuracy.

Estimates for the number of sums and products and for exponential sums over subgroups in fields of prime order

Our first result is a ‘sum–product’ theorem for subsets A of the finite field F p , p prime, providing a lower bound on max(| A+ A|,| A· A|). As corollary, the second and main result provides new bounds on exponential sums associated to subgroups of the multiplicative group F ∗ p . To cite this article: J. Bourgain, S.V. Konyagin, C. R. Acad. Sci. Paris, Ser. I 337 (2003).

A bound for the discrepancy of digital nets and its application to the analysis of certain pseudo-random number generators

method for the construction of low-discrepancy point sets in the s-dimensional unit cube. Furthermore, by recent work it turned out that digital nets also play an important role in the analysis of certain pseudo-random number generators. Until now the discrepancy of digital nets essentially was estimated by using discrepancy bounds valid for arbitrary nets. In this paper we give a more sensible—in some sense—discrepancy bound, especially for digital nets generated over a finite field of prime order, and we apply this bound for improving some results concerning the serial test of certain pseudo-random number generators. The serial test is a test for the statistical independence of successive

Bounds for multiplicative cosets over fields of prime order

Let m m be a positive integer and suppose that p p is an odd prime with p ≡ 1 mod m p \equiv 1 \bmod m . Suppose that a ∈ ( Z / p Z ) ∗ a \in (\mathbb {Z}/p\mathbb {Z})^* and consider the polynomial x m − a x^m-a . If this polynomial has any roots in ( Z / p Z ) ∗ (\mathbb {Z}/p\mathbb {Z})^* , where the coset representatives for Z / p Z \mathbb {Z}/p\mathbb {Z} are taken to be all integers u u with | u | > p / 2 |u|>p/2 , then these roots will form a coset of the multiplicative subgroup μ m \mu _m of ( Z / p Z ) ∗ (\mathbb {Z}/p\mathbb {Z})^* consisting of the m m th roots of unity mod p p . Let C C be a coset of μ m \mu _m in ( Z / p Z ) ∗ (\mathbb {Z}/p\mathbb {Z})^* , and define | C | = max u ∈ C | u | |C|=\max _{u \in C}{|u|} . In the paper “Numbers Having m m Small m m th Roots mod p p ” (Mathematics of Computation, Vol. 61, No. 203 (1993),pp. 393-413), Robinson gives upper bounds for M 1 ( m , p ) = min C ∈ ( Z / p Z ) ∗ / μ m | C | M_1(m,p)=\min _{\tiny C \in (\mathbb {Z}/p\mathbb {Z})^* /\mu _m }{|C|} of the form M 1 ( m , p ) > K m p 1 − 1 / ϕ ( m ) M_1(m,p)>K_mp^{1-1/\phi (m)} , where ϕ \phi is the Euler phi-function. This paper gives lower bounds that are of the same form, and seeks to sharpen the constants in the upper bounds of Robinson. The upper bounds of Robinson are proven to be optimal when m m is a power of 2 2 or when m = 6. m=6.

Isomorphisms of some graph coverings

Let G be a connected graph and Γ a group of automorphisms of G. We enumerate the number of Γ-isomorphism classes of derived graph coverings of G with voltages in a finite field of prime order p (>2).

Discrete logarithms and local units

Let K be a number field and (9 K its ring of integers. Let l be a prime number and e a positive integer. We give a method to construct l e th powers in (9 K using smooth algebraic integers. This method makes use of approximations of the l -adic logarithm to identify l e th powers. One version we give is successful if the class number of K is not divisible by l and if the units in C K which are congruent to 1 modulo l e +1 are l e th powers. A second version only depends on Leopoldt’s conjecture. We use the technique of constructing l e th powers to find discrete logarithms in a finite field of prime order. Our method for computing discrete logarithms is closely modelled after Gordon’s adaptation of the number field sieve to this problem. We conjecture th at the expected running time of our algorithm is L p [1/3; (64/9) 1/3 + o(1)] for p-> oo, where L p [ s; c ] = exp ( c (log q )s (log log q ) 1-8 ). This is the same running time as is conjectured for the number field sieve factoring algorithm.

Construction of orthogonal starters and corollaries

A method is proposed for constructing a system of (v−1)/2 pairwise disjoint orthogonal starters of order v for v≡6k+1≡7 (mod 12)≡p≡n2+n+1/t such that the number 3 is one of the primitive roots of the Galois field of prime order p (k is prime, k ≠ 2, and n and t are positive integers). The starters occurring in this system satisfy certain additional conditions. The construction of a series of combinatorial structures, including some not previously known, is a consequence of this result.

Cubic Character Sums of Cubic Polynomials

A complete evaluation is given of the sum over an arbitrary finite field of the values of a nontrivial cubic character of the field applied to an arbitrary polynomial of degree not greater than three in one variable defined over the field. Previous evaluations for the fields of prime order were given in theses of Friedman and Lagarias. The evaluation given below makes simple use of standard facts about equivalence of binary cubic forms. An interesting consequence of this evaluation is given connecting the values of these sums over the space of all polynomials of degree not greater than three over the finite field. The evaluation of these sums is of relevance to the theory of Shintani’s Dirichlet series associated to the space of binary cubic forms since they appear in the residues of the cubic twists of these series.

Cubic character sums of cubic polynomials

A complete evaluation is given of the sum over an arbitrary finite field of the values of a nontrivial cubic character of the field applied to an arbitrary polynomial of degree not greater than three in one variable defined over the field. Previous evaluations for the fields of prime order were given in theses of Friedman and Lagarias. The evaluation given below makes simple use of standard facts about equivalence of binary cubic forms. An interesting consequence of this evaluation is given connecting the values of these sums over the space of all polynomials of degree not greater than three over the finite field. The evaluation of these sums is of relevance to the theory of Shintani’s Dirichlet series associated to the space of binary cubic forms since they appear in the residues of the cubic twists of these series.

Indecomposable local maps of tessellation automata

Indecomposable local maps of one-dimensional tessellation automata are studied. The main results of this paper are the following. (1) For any alphabet ∑ containing two or more symbols and for anyn≥ 1, there exist indecomposable scope-n local maps over ∑. (2) If ∑ is a finite field of prime order, then a linear scope-n local map over ∑ is indecomposable if and only if its associated polynomial is an irreducible polynomial of degreen − 1 over ∑, except for a trivial case. (3) Result (2) is no longer true if ∑ is a finite field whose order is not prime.

The elementary symmetric functions in a finite field of prime order

The elementary symmetric functions in a finite field of prime order