Balanced Symmetric Functions Over ${\hbox{GF}}(p)$

Abstract

Under mild conditions on n, p, we give a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF (p), where p is a prime number. The existence of nonlinear balanced symmetric polynomials is an immediate corollary of this bound. Furthermore, we conjecture that X(2 t , 2 t+1 l − 1) are the only nonlinear balanced elementary symmetric polynomials over GF (2), whereand we prove various results in support of this conjecture.

On a conjecture for balanced symmetric Boolean functions

“…Then, if j is odd, we get cos((n − 2 s )A + jπ/2) sin(jπ/2) = (cos((n − 2 s )A) cos(jπ/2) − sin((n − 2 s )A) sin(jπ/2)) sin(jπ/2) = − sin((n − 2 s )A). t + 1; further, if n ≥ 2(d − 1), then n ≥ 2 t+1 and if 2 t+1 does not divide n, then n = 2 t+1 + r with r ≡ 0 (mod 2 t+1 ) and so Lemma 3.17 applies; if 2 t+1 does divide n, then we already know from Theorem 1, which summarizes the results of [5] that X(d, n) is balanced).…”

“…Symmetry is also often required [2,3], and naturally, one would ask when the two features will intersect. In [5], we conjectured that the polynomials X(2 t , 2 t+1 − 1) are the only nonlinear balanced elementary symmetric polynomials, where X(d, n) = 1≤i 1 <···<i d ≤n…”

Recent Results on Balanced Symmetric Boolean Functions

“…Conjecture 2: [3] There are no nonlinear balanced elementary symmetric Boolean functions except for σ l·2 t+1 −1,2 t , where t and l are any positive integers.…”

“…Since the number of nontrivial balanced functions seems to be very small, they conjectured that balanced symmetric functions of fixed degree do not exist when the number of variables grows. For elementary symmetric Boolean functions, Cusick et al proposed a conjecture in [3] about the nonexistence of nonlinear balanced elementary symmetric Boolean functions σ n,d except for n = l · 2 t+1 − 1 and d = 2 t , where t and l are any positive integers. They also obtained many results towards the conjecture in [4].…”

Generalization of Boolean Functions Properties to Functions Defined over GF(p)

“…The author in (Li, 2008) generalized the counting results of rotation symmetric Boolean functions to the rotation symmetric polynomials over finite fields GF(p). Cusick et al (2008) gave a lower bound on the number of n-variable balanced symmetric polynomials over finite fields GF(p). Recently, functions defined over GF(p) have been used to propose a new a group re-keying protocol based on modular polynomial arithmetic (Sudha et al, 2009).…”