Abstract
Several factors (e.g., balancedness, good correlation immunity) are considered as important properties of Boolean functions for using in cryptographic primitives. A Boolean function is perfect algebraic immune if it is with perfect immunity against algebraic and fast algebraic attacks. There is an increasing interest in construction of Boolean function that is perfect algebraic immune combined with other characteristics, like resiliency. A resilient function is a balanced correlation-immune function. This paper uses bivariate representation of Boolean function and theory of finite field to construct a generalized and new class of Boolean functions on even variables by extending the Carlet-Feng functions. We show that the functions generated by this construction support cryptographic properties of 1-resiliency and (sub)optimal algebraic immunity and further propose the sufficient condition of achieving optimal algebraic immunity. Compared experimentally with Carlet-Feng functions and the functions constructed by the method of first-order concatenation existing in the literature on even (from 6 to 16) variables, these functions have better immunity against fast algebraic attacks. Implementation results also show that they are almost perfect algebraic immune functions.
Highlights
Boolean functions are one of the most important cryptographic primitives for stream ciphers, block ciphers, and hash functions in cryptography [1,2,3,4]
The new cryptographic property of Boolean functions-algebraic immunity (AI), the minimum algebraic degree of annihilators of f or f + 1, was introduced by Meier et al [11] to measure the ability of Boolean functions to resist algebraic attacks
We propose the sufficient condition of achieving optimal algebraic immunity
Summary
Boolean functions are one of the most important cryptographic primitives for stream ciphers, block ciphers, and hash functions in cryptography [1,2,3,4]. We take Boolean functions extensively as filter and combination generators of stream ciphers based on linear feedback shift registers [3]. Cryptographic criteria for Boolean functions include balancedness, algebraic degree, nonlinearity, and correlation immunity. An overview of cryptographic criteria for Boolean functions with extensive bibliography is given in [1]. The study of the cryptographic criteria of Boolean functions is essential because of the connections between known cryptanalytic attacks and these criteria [4]. An improperly chosen Boolean function will render the system open to various kinds of attacks. Take the property of balancedness (i.e., its Hamming weight = 2n−1), for example, the classical cryptographic criterion for designing Boolean function is useful in preventing the system from leaking statistical information on the plaintext when the ciphertext is known
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