scholarly journals An Algorithm for Improving Algebraic Degree of S-Box Coordinate Boolean Functions Based on Affine Equivalence Transformation

2018 ◽  
Vol 10 (1-2) ◽  
pp. 339-350
Author(s):  
Hoang Duc Tho ◽  
Nguyen Truong Thang ◽  
Nguyen Thi Thu Nga ◽  
Pham Quoc Hoang
Keyword(s):  
Boolean Functions ◽  
Algebraic Degree ◽  
Equivalence Transformation ◽  
Affine Equivalence

An Algorithm for Improving Algebraic Degree of S-Box Based on Affine Equivalence Transformation

2017 ◽  
Vol 8 (1) ◽  
pp. 53-64
Author(s):  
Luong The Dung ◽  
Hoang Duc Tho
Keyword(s):  
Block Cipher ◽  
Algebraic Degree ◽  
Equivalence Transformation ◽  
Substitution Box ◽  
Differential Uniformity ◽  
Affine Equivalence ◽  
Nonlinear Part ◽  
Cube Attacks ◽  
Differential Attacks ◽  
Efficient Software

The Substitution box (S-box) plays an important role in a block cipher as it is the only nonlinear part of the cipher in most cases. To avoid various attacks on the ciphers and for efficient software implementation, S-boxes are required to satisfy a lot of properties, for instance being a permutation defined on the fields with even degrees, with a high algebraic degree, a low differential uniformity and a high nonlinearity, etc. However, it seems very difficult to find an S-box to satisfy all the criteria. The S-box of low algebraic degree is vulnerable to many attacks such as linear and differential cryptanalysis, for instance higher-order differential attacks, algebraic attacks or cube attacks. In this paper the authors propose an algorithm for improving algebraic degree of the S-box while not affecting its other important properties. The algorithm is based on affine equivalence transformation of the S-boxes.

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.

Effects of the Internal Structure of H Boolean Functions on Algebraic Immunity and Algebraic Degree

2013 ◽  
Vol 774-776 ◽  
pp. 1721-1724
Author(s):  
Jing Lian Huang ◽  
Xiu Juan Yuan ◽  
Jian Hua Wang
Keyword(s):  
Boolean Function ◽  
Internal Structure ◽  
Boolean Functions ◽  
Algebraic Degree ◽  
Algebraic Immunity ◽  
Hamming Weight ◽  
Relationship Of ◽  
The Relationship

We go deep into the internal structure of the Boolean functions values, and study the relationship of algebraic immunity and algebraic degree of Boolean functions with the Hamming weight with the diffusion included. Then we get some theorems which relevance together algebraic immunity, annihilators and algebraic degree of H Boolean functions by the e-derivative which is a part of the H Boolean function. Besides, we also get the results that algebraic immunity and algebraic degree of Boolean functions can be linked together by the e-derivative of H Boolean functions and so on.

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