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On the statistical properties of Diffie–Hellman distributions
 MR 2001k:11258 Zbl 0997.11066
"... Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that giv ..."
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Cited by 29 (10 self)
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Let p be a large prime such that p−1 has some large prime factors, and let ϑ ∈ Z ∗ p be an rth power residue for all small factors of p − 1. The corresponding DiffieHellman (DH) distribution is (ϑ x, ϑ y, ϑ xy) where x, y are randomly chosen from Z ∗ p. A recently formulated assumption is that given p, ϑ of the above form it is infeasible to distinguish in reasonable time between DH distribution and triples of numbers chosen
On certain exponential sums and the distribution of DiffieHellman triples
 J. London Math. Soc
, 1999
"... Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy ..."
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Cited by 26 (14 self)
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Let g be a primitive root modulo a prime p. It is proved that the triples (gx,gy,gxy), x,y�1,…,p�1, are uniformly distributed modulo p in the sense of H. Weyl. This result is based on the following upper bound for double exponential sums. Let ε�0 be fixed. Then p−� x,y=� exp0 2πiagx�bgy�cgxy
Normal Bases over Finite Fields
, 1993
"... Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to repr ..."
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Cited by 9 (0 self)
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Interest in normal bases over finite fields stems both from mathematical theory and practical applications. There has been a lot of literature dealing with various properties of normal bases (for finite fields and for Galois extension of arbitrary fields). The advantage of using normal bases to represent finite fields was noted by Hensel in 1888. With the introduction of optimal normal bases, large finite fields, that can be used in secure and e#cient implementation of several cryptosystems, have recently been realized in hardware. The present thesis studies various theoretical and practical aspects of normal bases in finite fields. We first give some characterizations of normal bases. Then by using linear algebra, we prove that F q n has a basis over F q such that any element in F q represented in this basis generates a normal basis if and only if some groups of coordinates are not simultaneously zero. We show how to construct an irreducible polynomial of degree 2 n with linearly i...
Constructing nonresidues in finite fields and the extended Riemann hypothesis
 Math. Comp
, 1991
"... Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in pol ..."
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Cited by 8 (0 self)
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Abstract. We present a new deterministic algorithm for the problem of constructing kth power nonresidues in finite fields Fpn,wherepis prime and k is a prime divisor of pn −1. We prove under the assumption of the Extended Riemann Hypothesis (ERH), that for fixed n and p →∞, our algorithm runs in polynomial time. Unlike other deterministic algorithms for this problem, this polynomialtime bound holds even if k is exponentially large. More generally, assuming the ERH, in time (n log p) O(n) we can construct a set of elements
A SkolemMahlerLech theorem in positive characteristic . . .
, 2005
"... Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the SkolemMahlerLech theorem. Lech gave a counterexample to a similar statement in pos ..."
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Cited by 5 (1 self)
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Lech proved in 1953 that the set of zeroes of a linear recurrence sequence in a field of characteristic 0 is the union of a finite set and finitely many infinite arithmetic progressions. This result is known as the SkolemMahlerLech theorem. Lech gave a counterexample to a similar statement in positive characteristic. We will present some more pathological examples. We will state and prove a correct analog of the SkolemMahlerLech theorem in positive characteristic. The zeroes of a
On Singularity of Generalized Vandermonde Matrices over Finite Fields
"... We use an upper bound on the number of zeros of sparse polynomials over a finite field IF q to estimate the number of singular matrices of the form # u j i m i,j=1 , where # 1 , . . . , #m # IF # q are fixed nonzero elements, taken over all (q  1) m integer mtuples (u 1 , . . . , um ..."
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Cited by 3 (0 self)
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We use an upper bound on the number of zeros of sparse polynomials over a finite field IF q to estimate the number of singular matrices of the form # u j i m i,j=1 , where # 1 , . . . , #m # IF # q are fixed nonzero elements, taken over all (q  1) m integer mtuples (u 1 , . . . , um ) # [0, q  2] m . 2000 Mathematics Subject Classification. Primary 11T30, 12E20. Secondary 15A15. 1 1 Introduction Let IF q denote a finite field of q elements. Let us fix m nonzero elements # 1 , . . . , #m # IF # q We consider matrices of the form # u j i m i,j=1 , with integer u 1 , . . . , um # [0, q2]. The choice u j = j1, j = 1, . . . , m, corresponds to the Vandermonde matrix, which is known to be nonsingular, provided that # 1 , . . . , #m are pairwise distinct. Here show that for m fixed almost all matrices of the above form are nonsingular. More precisely, let us denote by V (# 1 , . . . , #m ) the number of mtuples (u 1 , . . . , um ) # [0, q  2] m...