scholarly journals Linear Recurrences and Asymptotic Behavior of Exponential Sums of Symmetric Boolean Functions

10.37236/2004 ◽  
2011 ◽  
Vol 18 (2) ◽  
Author(s):  
Francis N. Castro ◽  
Luis A. Medina
Keyword(s):  
Asymptotic Behavior ◽  
Boolean Functions ◽  
Exponential Sums ◽  
Exponential Sum ◽  
Linear Recurrence ◽  
Symmetric Boolean Function ◽  
Linear Recurrences ◽  
Symmetric Boolean Functions ◽  
Integer Coefficients ◽  
Homogeneous Linear

In this paper we give an improvement of the degree of the homogeneous linear recurrence with integer coefficients that exponential sums of symmetric Boolean functions satisfy. This improvement is tight. We also compute the asymptotic behavior of symmetric Boolean functions and provide a formula that allows us to determine if a symmetric boolean function is asymptotically not balanced. In particular, when the degree of the symmetric function is a power of two, then the exponential sum is much smaller than $2^n$.

The Estimation of Complete Exponential Sums

1985 ◽  
Vol 28 (4) ◽  
pp. 440-454 ◽  
Author(s):  
J. H. Loxton ◽  
R. C. Vaughan
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Integer Coefficients ◽  
Complex Zeros

AbstractThis paper proves a conjecture of Loxton and Smith about the size of the exponential sum S(f;q) formed by summing exp (2πif(x)/q) over x mod q, where f is a polynomial of degree n with integer coefficients. It is shown that |S(f;q)| ≤ Cfdn(q)qe/(e+1), where e is the maximum of the orders of the complex zeros of f'. An estimate is also obtained for Cf in terms of n, e and the different of f, and a number of examples are given to show that the estimate is best possible.

scholarly journals Properties of Symmetric Boolean functions

2016 ◽  
Vol 12 (1) ◽  
pp. 5-24
Author(s):  
L. Haviarová ◽  
E. Toman
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Normal Forms ◽  
Special Property ◽  
Interval Graphs ◽  
Vertex Degree ◽  
Symmetric Boolean Function ◽  
Symmetric Boolean Functions

Abstract In the present paper we consider symmetric Boolean functions with special property. We study properties of the maximal intervals of these functions. Later we show characteristics of corresponding interval graphs and simplified interval graphs. Specifically we prove, that these two graphs are isomorphic for symmetric Boolean function. Then we obtain the vertex degree of these graphs. We discuss also disjunctive normal forms.

scholarly journals The Weight and Nonlinearity of 2-rotation Symmetric Cubic Boolean Function

2015 ◽  
Vol 7 (2) ◽  
pp. 187 ◽  
Author(s):  
Hongli Liu
Keyword(s):  
Boolean Function ◽  
Boolean Functions ◽  
Symmetric Functions ◽  
Recursive Formula ◽  
Symmetric Boolean Function ◽  
Symmetric Boolean Functions ◽  
Rotation Symmetric ◽  
Rotation Symmetric Boolean Function ◽  
Rotation Symmetric Boolean Functions

The conceptions of $\chi$-value and K-rotation symmetric Boolean functions are introduced by Cusick. K-rotation symmetric Boolean functions are a special rotation symmetric functions, which are invariant under the $k-th$ power of $\rho$.In this paper, we discuss cubic 2-value 2-rotation symmetric Boolean function with $2n$ variables, which denoted by $F^{2n}(x^{2n})$. We give the recursive formula of weight of $F^{2n}(x^{2n})$, and prove that the weight of $F^{2n}(x^{2n})$ is the same as its nonlinearity.

CLASS OF BOOLEAN FUNCTIONS CONSTRUCTED USING SIGNIFICANT BITS OF LINEAR RECURRENCES OVER THE RING ℤ2n

2019 ◽  
Vol 6 (2) ◽  
pp. 90-94
Author(s):  
Hernandez Piloto Daniel Humberto
Keyword(s):  
Linear Function ◽  
Boolean Functions ◽  
Hamming Distance ◽  
Linear Recurrences ◽  
Maximum Period ◽  
Non Linear ◽  
The Given ◽  
Class Of Functions

In this work a class of functions is studied, which are built with the help of significant bits sequences on the ring ℤ2n. This class is built with use of a function ψ: ℤ2n → ℤ2. In public literature there are works in which ψ is a linear function. Here we will use a non-linear ψ function for this set. It is known that the period of a polynomial F in the ring ℤ2n is equal to T(mod 2)2α, where α∈ , n01- . The polynomials for which it is true that T(F) = T(F mod 2), in other words α = 0, are called marked polynomials. For our class we are going to use a polynomial with a maximum period as the characteristic polyomial. In the present work we show the bounds of the given class: non-linearity, the weight of the functions, the Hamming distance between functions. The Hamming distance between these functions and functions of other known classes is also given.

scholarly journals A note on two-term exponential sum and the reciprocal of the quartic Gauss sums

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Wenpeng Zhang ◽  
Xingxing Lv
Keyword(s):  
Exponential Sums ◽  
Exponential Sum ◽  
Gauss Sums ◽  
Power Mean ◽  
Hybrid Power ◽  
Analytic Methods ◽  
Hybrid Power Mean

AbstractThe main purpose of this article is by using the properties of the fourth character modulo a prime p and the analytic methods to study the calculating problem of a certain hybrid power mean involving the two-term exponential sums and the reciprocal of quartic Gauss sums, and to give some interesting calculating formulae of them.

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