2008
DOI: 10.1109/tit.2007.915920
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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.

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Cited by 38 publications
(57 citation statements)
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“…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).…”
Section: The Case Wt(d) ≥mentioning
confidence: 72%
See 1 more Smart Citation
“…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).…”
Section: The Case Wt(d) ≥mentioning
confidence: 72%
“…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…”
mentioning
confidence: 99%
“…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.…”
Section: The Equivalence Of Cusick's Conjecturementioning
confidence: 99%
“…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].…”
Section: Introductionmentioning
confidence: 99%
“…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).…”
Section: Introductionmentioning
confidence: 99%