Improved Lower Bounds on Second Order Non-linearities of Cubic Boolean Functions

2021 ◽  
pp. 31-43
Author(s):  
Ruchi Telang Gode ◽  
Shahab Faruqi ◽  
Ashutosh Mishra
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order

On the lower bounds of the second order nonlinearities of some Boolean functions

2010 ◽  
Vol 180 (2) ◽  
pp. 266-273 ◽  
Author(s):  
Sugata Gangopadhyay ◽  
Sumanta Sarkar ◽  
Ruchi Telang
Keyword(s):  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order

LOWER BOUNDS ON THE SECOND ORDER NONLINEARITY OF BOOLEAN FUNCTIONS

2011 ◽  
Vol 22 (06) ◽  
pp. 1331-1349 ◽  
Author(s):  
XUELIAN LI ◽  
YUPU HU ◽  
JUNTAO GAO
Keyword(s):  
Lower Bound ◽  
Positive Integer ◽  
Lower Bounds ◽  
Boolean Functions ◽  
Second Order ◽  
Algebraic Degree ◽  
Positive Integers ◽  
Second Order Nonlinearity ◽  
Better Than ◽  
Monomial Functions

It is a difficult task to compute the r-th order nonlinearity of a given function with algebraic degree strictly greater than r > 1. Though lower bounds on the second order nonlinearity are known only for a few particular functions, the majority of which are cubic. We investigate lower bounds on the second order nonlinearity of cubic Boolean functions [Formula: see text], where [Formula: see text], dl = 2il + 2jl + 1, m, il and jl are positive integers, n > il > jl. Furthermore, for a class of Boolean functions [Formula: see text] we deduce a tighter lower bound on the second order nonlinearity of the functions, where [Formula: see text], dl = 2ilγ + 2jlγ + 1, il > jl and γ ≠ 1 is a positive integer such that gcd(n,γ) = 1. Lower bounds on the second order nonlinearity of cubic monomial Boolean functions, represented by fμ(x) = Tr(μx2i+2j+1), [Formula: see text], i and j are positive integers such that i > j, were obtained by Gode and Gangopadhvay in 2009. In this paper, we first extend the results of Gode and Gangopadhvay from monomial Boolean functions to Boolean functions with more trace terms. We further generalize and improve the results to a wider range of n. Our bounds are better than those of Gode and Gangopadhvay for monomial functions fμ(x). Especially, our lower bounds on the second order nonlinearity of some Boolean functions F(x) are better than the existing ones.

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