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An improved hybrid genetic algorithm to construct balanced Boolean function with optimal cryptographic properties

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Abstract

Boolean functions are used as nonlinear filter functions and combiner functions in several stream ciphers. The security of these stream ciphers largely depends upon cryptographic properties of Boolean functions. Finding a balanced Boolean function with optimal cryptographic properties is an open research problem in the cryptographic community. Since the number of n-variable Boolean functions is \(2^{2^n}\), it is not computationally feasible to search the entire space of such functions for cryptographically significant functions when \(n \ge 6\). In general, the construction of Boolean functions with optimal or near optimal cryptographically significant properties is formulated as combinatorial optimization problems. In this paper, we apply the Genetic algorithm with the integration of a local search procedure called hybrid GA (HGA) searching for the Boolean functions with high nonlinearity and low autocorrelation. The performance of the Genetic algorithm depends upon the tuning parameters such as crossover mechanism, mutation probability, selection criteria, and the choice of fitness/cost function. To achieve an optimal trade-off among two cryptographic properties, the choice of the cost function plays an important role for finding an optimal solution. So our main focus is to construct balanced Boolean functions using HGA with a new cost function and compare it with existing cost functions. Our experimental results shows that the HGA is more efficient than GA and the new cost function is more efficient than existing cost functions for reaching the optimal solutions.

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Authors and Affiliations

  1. Department of Computer Science and Engineering, Indian Institute of Technology Roorkee, Roorkee, India

    Pratap Kumar Behera & Sugata Gangopadhyay

Authors
  1. Pratap Kumar Behera
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  2. Sugata Gangopadhyay
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Correspondence to Pratap Kumar Behera.

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Appendix

Appendix

n = 6, Balanced: yes, nl: 26, ac: 16

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n = 8, Balanced: yes, nl: 116, ac: 24

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n = 10, Balanced: yes, nl: 488, ac: 40

0110000011100101011001001100010101100000110001011

001111100111010011000001100010101100000

11000101011001001100010111011111001110100110000011

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n = 12, Balanced: yes, nl: 2002, ac: 88

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01011100100001101110001101111010111000110111101011

100011011110010001110010000101000111

00100001010001110010000101000111011000011011100011

011110101110001101111010111000110111

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101110001101111010111000110111100100

01110010000110111000110111101011100011011110101110

001101111001000111001000010100011100

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001000011011100011011110101110001101

1110101110001101111010111000

110111100100011100100001

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Behera, P.K., Gangopadhyay, S. An improved hybrid genetic algorithm to construct balanced Boolean function with optimal cryptographic properties. Evol. Intel. 15, 639–653 (2022). https://doi.org/10.1007/s12065-020-00538-x

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