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A class of permutation trinomials over finite fields

Xiang-dong Hou (2014)

Acta Arithmetica

Let q > 2 be a prime power and f = - x + t x q + x 2 q - 1 , where t * q . We prove that f is a permutation polynomial of q ² if and only if one of the following occurs: (i) q is even and T r q / 2 ( 1 / t ) = 0 ; (ii) q ≡ 1 (mod 8) and t² = -2.

À la recherche de petites sommes d'exponentielles

Étienne Fouvry, Philippe Michel (2002)

Annales de l’institut Fourier

Soit f ( x ) une fraction rationnelle à coefficients entiers, vérifiant des hypothèses assez générales. On prouve l’existence d’une infinité d’entiers n , ayant exactement deux facteurs premiers, tels que la somme d’exponentielles x = 1 n exp ( 2 π i f ( x ) / n ) soit en O ( n 1 2 - β f ) , où β f > 0 est une constante ne dépendant que de la géométrie de f . On donne aussi des résultats de répartition du type Sato-Tate, pour certaines sommes de Salié, modulo n , avec n entier comme ci- dessus.

A large family of Boolean functions

Huaning Liu, Min Zhang (2016)

Acta Arithmetica

In a series of papers many Boolean functions with good cryptographic properties were constructed using number-theoretic methods. We construct a large family of Boolean functions by using polynomials over finite fields, and study their cryptographic properties: maximum Fourier coefficient, nonlinearity, average sensitivity, sparsity, collision and avalanche effect.

A new approach to the ElGamal encryption scheme

Czesław Kościelny (2004)

International Journal of Applied Mathematics and Computer Science

The ElGamal encryption scheme can be used for both digital signatures and encryption, and its security results from the difficulty of calculating discrete logarithms in a finite field. This algorithm usually works in a multiplicative group of GF(p) and in this case the progress in the discrete logarithm problem forces the users of such a basic ElGamal public key cryptosystem to permanently increase a prime modulus p in order to ensure the desired security. But the task of finding a multiplicative...

A note on evaluations of some exponential sums

Marko J. Moisio (2000)

Acta Arithmetica

1. Introduction. The recent article [1] gives explicit evaluations for exponential sums of the form S ( a , p α + 1 ) : = x q χ ( a x p α + 1 ) where χ is a non-trivial additive character of the finite field q , q = p e odd, and a * q . In my dissertation [5], in particular in [4], I considered more generally the sums S(a,N) for all factors N of p α + 1 . The aim of the present note is to evaluate S(a,N) in a short way, following [4]. We note that our result is also valid for even q, and the technique used in our proof can also be used to evaluate certain...

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