Balanced symmetric functions over GF(p)

Thomas W. Cusick; Yuan Li; Pantelimon Stǎnicǎ

doi:10.1109/TIT.2007.915920

Balanced symmetric functions over GF(p)

Thomas W. Cusick
, Yuan Li
, Pantelimon Stǎnicǎ

Mathematics

Alcorn State University
Naval Postgraduate School

Research output: Contribution to journal › Article › peer-review

39 Scopus citations

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 prove that X(2^t, 2^t+1ℓ - 1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d, n) = ∑_{1≤i1<i1<⋯<id≤n} xi₁ xi₂⋯x_id.

Original language	English
Pages (from-to)	1304-1307
Number of pages	4
Journal	IEEE Transactions on Information Theory
Volume	54
Issue number	3
DOIs	https://doi.org/10.1109/TIT.2007.915920
State	Published - Mar 2008

Keywords

Balancedness
Cryptography
Finite fields
Multinomial coefficients
Symmetric polynomials

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10.1109/TIT.2007.915920

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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 prove that X(2t, 2t+1ℓ - 1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d, n) = ∑1≤i1 xi1 xi2⋯xid.",

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AU - Stǎnicǎ, Pantelimon

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N2 - 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 prove that X(2t, 2t+1ℓ - 1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d, n) = ∑1≤i1 xi1 xi2⋯xid.

AB - 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 prove that X(2t, 2t+1ℓ - 1) are balanced and conjecture that these are the only balanced symmetric polynomials over GF(2), where X(d, n) = ∑1≤i1 xi1 xi2⋯xid.

KW - Balancedness

KW - Cryptography

KW - Finite fields

KW - Multinomial coefficients

KW - Symmetric polynomials

UR - https://www.scopus.com/pages/publications/40949135342

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SP - 1304

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JO - IEEE Transactions on Information Theory

JF - IEEE Transactions on Information Theory

IS - 3

ER -

Balanced symmetric functions over GF(p)

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