scholarly journals Construction of bent functions and balanced Boolean functions with high nonlinearity

1995 ◽  
pp. 61-74 ◽  
Author(s):  
Hans Dobbertin
Keyword(s):  
Boolean Functions ◽  
Bent Functions ◽  
High Nonlinearity

scholarly journals Two Boolean Functions with Five-Valued Walsh Spectra and High Nonlinearity

2015 ◽  
Vol 26 (05) ◽  
pp. 537-556 ◽  
Author(s):  
Xiwang Cao ◽  
Lei Hu
Keyword(s):  
Finite Fields ◽  
Boolean Functions ◽  
Exponential Sums ◽  
New Method ◽  
High Nonlinearity ◽  
Walsh Spectra ◽  
And Diffusion ◽  
Confusion And Diffusion

For cryptographic systems the method of confusion and diffusion is used as a fundamental technique to achieve security. Confusion is reflected in nonlinearity of certain Boolean functions describing the cryptographic transformation. In this paper, we present two Boolean functions which have low Walsh spectra and high nonlinearity. In the proof of the nonlinearity, a new method for evaluating some exponential sums over finite fields is provided.

scholarly journals New Constructions of Balanced Boolean Functions with Maximum Algebraic Immunity, High Nonlinearity and Optimal Algebraic Degree

2020 ◽  
Vol 17 (7) ◽  
pp. 639-654
Author(s):  
Dheeraj Kumar SHARMA ◽  
Rajoo PANDEY
Keyword(s):  
Lower Bound ◽  
Boolean Function ◽  
Boolean Functions ◽  
Primitive Element ◽  
Galois Field ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Greatest Common Divisor ◽  
Primitive Elements ◽  
High Nonlinearity

This paper consists of proposal of two new constructions of balanced Boolean function achieving a new lower bound of nonlinearity along with high algebraic degree and optimal or highest algebraic immunity. This construction has been made by using representation of Boolean function with primitive elements. Galois Field,  used in this representation has been constructed by using powers of primitive element such that greatest common divisor of power and  is 1. The constructed balanced  variable Boolean functions achieve higher nonlinearity, algebraic degree of , and algebraic immunity of   for odd ,  for even . The nonlinearity of Boolean function obtained in the proposed constructions is better as compared to existing Boolean functions available in the literature without adversely affecting other properties such as balancedness, algebraic degree and algebraic immunity.

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