Diophantine Equations With Binomial Coefficients and Perturbations of Symmetric Boolean Functions

Abstract

Abstract. This work presents a study of perturbations of symmetric Boolean functions. In particular, it establishes a connection between exponential sums of these perturbations and Diophantine equations of the form To be specific, it is proved that, excluding the trivial cases, balanced perturbations of fixed degree do not exist when the number of variables grows. Some sporadic balanced perturbations are presented. Finally, a beautiful but unexpected identity between perturbations of two very different symmetric Boolean functions is also included in this work.

Closed formulas for exponential sums of symmetric polynomials over Galois fields

“…In other words, A natural problem to explore is to see how solutions to (3.14) given by exponential sums of linear combinations of elementary symmetric polynomials look like as n grows. Perhaps something similar to the study presented in [5] holds true in this case. This is part of future research.…”

Value distribution of elementary symmetric polynomials and its perturbations over finite fields

“…Fine proposed as problems the veracity of the other properties for general p. O. Aberth [1] disproved (2) and (5) by showing that p (5) 6 (0) = 1/5 and p (5) 6 (2) = 26/125. He also showed that p (5) 30 (0) = 15749/78125 > 1/5 = p (5) 6 (0) and therefore (4) is also false. In [26], J. D. Smith generalized Aberth's example and showed that if p > 3 is prime, then As mentioned at the beginning of Section 2, one of the reasons the asymptotic behavior of exponential sums of symmetric polynomials was calculated over the binary field was to provide an asymptotic proof of Conjecture 2.1 (see [6]).…”

“…Many problems in number theory and combinatorics, as well as in their applications, can be formulated in terms of exponential sums. In cryptography, for example, exponential sums can be used to detect when a particular function is balanced (a property very useful in cryptographic applications) [5,6,7,8,9,12,13]. Some classical examples of exponential sums include the number-theoretical Gauss sums, Kloosterman sums, and Weyl sums.…”

“…The exponential sum over F q of F is defined as where ξ p = exp(2πi/p) and Tr = Tr Fq/Fp is the field trace function. These exponential sums have been extensively studied when the characteristic of the field is 2 because of their cryptographic applications, see [3,4,5,8,9,12,13,24]. Recently, some cryptographic applications when the characteristic of the field is different than 2 has been found.…”

A variation on bisecting the binomial coefficients

“…The type of non-constant symmetric Boolean functions, which can be represented by a polynomial of only one variable, defined on {0, 1, ..., n}, with the degree less than the expected one, namely n (using Lagrange interpolation), became of special interest for obtaining various cryptographic properties (see Gopalakrishnan et al [8], Cusick and Li [3], Mitchell [9], Sarkar and Maitra [11] and more recent works such as Castro, Gonzalez and Medina [1]). A symmetric Boolean function with this special property is now referred to as balanced.…”