On Katz’s (A,B)-exponential sums

2020 ◽  
Author(s):  
Lei FU ◽  
Daqing WAN
Keyword(s):  
Finite Fields ◽  
Arithmetic Progression ◽  
Betti Number ◽  
Exponential Sums ◽  
Decomposition Theorem ◽  
Characteristic P ◽  
Universal Family ◽  
Degenerate Cases

Abstract We deduce Katz’s theorems for (A, B)-exponential sums over finite fields using $\ell$-adic cohomology and a theorem of Denef–Loeser, removing the hypothesis that A + B is relatively prime to the characteristic p. In some degenerate cases, the Betti number estimate is improved using toric decomposition and Adolphson–Sperber’s bounds for degrees of L-functions. Applying the facial decomposition theorem, we prove that the universal family of (A, B)-polynomials is generically ordinary for its L-function when p is in certain arithmetic progression.

scholarly journals T-adic exponential sums over finite fields

2009 ◽  
Vol 3 (5) ◽  
pp. 489-509 ◽  
Author(s):  
Chunlei Liu ◽  
Daqing Wan
Keyword(s):  
Finite Fields ◽  
Exponential Sums

scholarly journals Two Boolean Functions with Five-Valued Walsh Spectra and High Nonlinearity

2015 ◽  
Vol 26 (05) ◽  
pp. 537-556 ◽  
Author(s):  
Xiwang Cao ◽  
Lei Hu
Keyword(s):  
Finite Fields ◽  
Boolean Functions ◽  
Exponential Sums ◽  
New Method ◽  
High Nonlinearity ◽  
Walsh Spectra ◽  
And Diffusion ◽  
Confusion And Diffusion

For cryptographic systems the method of confusion and diffusion is used as a fundamental technique to achieve security. Confusion is reflected in nonlinearity of certain Boolean functions describing the cryptographic transformation. In this paper, we present two Boolean functions which have low Walsh spectra and high nonlinearity. In the proof of the nonlinearity, a new method for evaluating some exponential sums over finite fields is provided.

EXPANSION OF ORBITS OF SOME DYNAMICAL SYSTEMS OVER FINITE FIELDS

2010 ◽  
Vol 82 (2) ◽  
pp. 232-239 ◽  
Author(s):  
JAIME GUTIERREZ ◽  
IGOR E. SHPARLINSKI
Keyword(s):  
Dynamical Systems ◽  
Lower Bound ◽  
Finite Field ◽  
Rational Function ◽  
Finite Fields ◽  
Exponential Sums ◽  
Short Interval ◽  
Additive Combinatorics ◽  
Distribution Of Elements ◽  
Image Position

AbstractGiven a finite field 𝔽p={0,…,p−1} of p elements, where p is a prime, we consider the distribution of elements in the orbits of a transformation ξ↦ψ(ξ) associated with a rational function ψ∈𝔽p(X). We use bounds of exponential sums to show that if N≥p1/2+ε for some fixed ε then no N distinct consecutive elements of such an orbit are contained in any short interval, improving the trivial lower bound N on the length of such intervals. In the case of linear fractional functions we use a different approach, based on some results of additive combinatorics due to Bourgain, that gives a nontrivial lower bound for essentially any admissible value of N.

DEGREE MATRICES AND ESTIMATES FOR EXPONENTIAL SUMS OF POLYNOMIALS OVER FINITE FIELDS

2013 ◽  
Vol 12 (07) ◽  
pp. 1350030 ◽  
Author(s):  
WEI CAO
Keyword(s):  
Finite Field ◽  
Finite Fields ◽  
Exponential Sums ◽  
Elementary Approach ◽  
Multivariate Polynomial ◽  
Elementary Divisors ◽  
Polynomials Over Finite Fields ◽  
Degree Matrix

Let f be a multivariate polynomial over a finite field and its degree matrix be composed of the degree vectors appearing in f. In this paper, we provide an elementary approach to estimating the exponential sums of the polynomials with positive square degree matrices in terms of the elementary divisors of the degree matrices.

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