ABSTRACT
Finding balanced, highly nonlinear Boolean functions is a difficult problem where it is not known what nonlinearity values are possible to be reached in general. At the same time, evolutionary computation is successfully used to evolve specific Boolean function instances, but the approach cannot easily scale for larger Boolean function sizes. Indeed, while evolving smaller Boolean functions is almost trivial, larger sizes become increasingly difficult, and evolutionary algorithms perform suboptimally.
In this work, we ask whether genetic programming (GP) can evolve constructions resulting in balanced Boolean functions with high nonlinearity. This question is especially interesting as there are only a few known such constructions. Our results show that GP can find constructions that generalize well, i.e., result in the required functions for multiple tested sizes. Further, we show that GP evolves many equivalent constructions under different syntactic representations. Interestingly, the simplest solution found by GP is a particular case of the well-known indirect sum construction.
- Hernan Aguirre, Hiroyuki Okazaki, and Yasushi Fuwa. 2007. An Evolutionary Multiobjective Approach to Design Highly Non-linear Boolean Functions. In Genetic and Evolutionary Computation Conference (GECCO). 749--756.Google ScholarDigital Library
- Ruibin Bai, Edmund K. Burke, Graham Kendall, Jingpeng Li, and Barry McCollum. 2010. A Hybrid Evolutionary Approach to the Nurse Rostering Problem. IEEE Transactions on Evolutionary Computation 14, 4 (2010), 580--590. Google ScholarDigital Library
- Georg T. Becker. 2015. The Gap Between Promise and Reality: On the Insecurity of XOR Arbiter PUFs. In Cryptographic Hardware and Embedded Systems - CHES 2015, Tim Güneysu and Helena Handschuh (Eds.). Springer Berlin Heidelberg, Berlin, Heidelberg, 535--555.Google ScholarDigital Library
- Claude Carlet. 2021. Boolean Functions for Cryptography and Coding Theory. Cambridge University Press. Google ScholarCross Ref
- Claude Carlet, Domagoj Jakobovic, and Stjepan Picek. 2021. Evolutionary Algorithms-Assisted Construction of Cryptographic Boolean Functions. Association for Computing Machinery, New York, NY, USA, 565--573. Google ScholarDigital Library
- Claude Carlet and Sihem Mesnager. 2016. Four Decades of Research on Bent Functions. Des. Codes Cryptography 78, 1 (Jan. 2016), 5--50.Google ScholarDigital Library
- John A Clark and Jeremy L Jacob. 2000. Two-Stage Optimisation in the Design of Boolean Functions. In Information Security and Privacy. LNCS, Vol. 1841. Springer, 242--254.Google ScholarCross Ref
- John A. Clark, Jeremy L. Jacob, Subhamoy Maitra, and Pantelimon Stanica. 2004. Almost Boolean Functions: The Design of Boolean Functions by Spectral Inversion. Comput. Intell. 20, 3 (2004), 450--462.Google ScholarCross Ref
- J. Kennedy and R. Eberhart. 1995. Particle swarm optimization. In Proceedings of ICNN'95 - International Conference on Neural Networks, Vol. 4. 1942--1948 vol.4. Google ScholarCross Ref
- Zhichao Lu, Ian Whalen, Yashesh Dhebar, Kalyanmoy Deb, Erik D. Goodman, Wolfgang Banzhaf, and Vishnu Naresh Boddeti. 2021. Multiobjective Evolutionary Design of Deep Convolutional Neural Networks for Image Classification. IEEE Transactions on Evolutionary Computation 25, 2 (2021), 277--291. Google ScholarCross Ref
- Luca Mariot and Alberto Leporati. 2015. A Genetic Algorithm for Evolving Plateaued Cryptographic Boolean Functions. In Theory and Practice of Natural Computing - Fourth International Conference, TPNC 2015, Mieres, Spain, December 15--16, 2015. Proceedings (Lecture Notes in Computer Science, Vol. 9477), Adrian-Horia Dediu, Luis Magdalena, and Carlos Martín-Vide (Eds.). Springer, 33--45.Google ScholarDigital Library
- Luca Mariot and Alberto Leporati. 2015. Heuristic Search by Particle Swarm Optimization of Boolean Functions for Cryptographic Applications. In Genetic and Evolutionary Computation Conference, GECCO, Companion Material Proceedings. 1425--1426.Google ScholarDigital Library
- Luca Mariot, Stjepan Picek, Alberto Leporati, and Domagoj Jakobovic. 2019. Cellular automata based S-boxes. Cryptography and Communications 11, 1 (2019), 41--62. Google ScholarDigital Library
- Luca Mariot, Martina Saletta, Alberto Leporati, and Luca Manzoni. 2021. Heuristic Search of (Semi-)Bent Functions based on Cellular Automata. CoRR abs/2111.13248 (2021).Google Scholar
- William Millan, Andrew Clark, and Ed Dawson. 1997. An Effective Genetic Algorithm for Finding Highly Nonlinear Boolean Functions. In First Int. Conference on Information and Communication Security (ICICS '97). Springer, 149--158.Google Scholar
- William Millan, Andrew Clark, and Ed Dawson. 1998. Heuristic Design of Cryptographically Strong Balanced Boolean Functions. In Advances in Cryptology - EUROCRYPT '98. 489--499.Google Scholar
- Julian F. Miller. 1999. An Empirical Study of the Efficiency of Learning Boolean Functions Using a Cartesian Genetic Programming Approach. In Proceedings of the 1st Annual Conference on Genetic and Evolutionary Computation - Volume 2 (Orlando, Florida) (GECCO'99). Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, 1135--1142.Google Scholar
- Alberto Moraglio, Krzysztof Krawiec, and Colin G. Johnson. 2012. Geometric Semantic Genetic Programming. In Parallel Problem Solving from Nature - PPSN XII - 12th International Conference, Taormina, Italy, September 1--5, 2012, Proceedings, Part I (Lecture Notes in Computer Science, Vol. 7491), Carlos A. Coello Coello, Vincenzo Cutello, Kalyanmoy Deb, Stephanie Forrest, Giuseppe Nicosia, and Mario Pavone (Eds.). Springer, 21--31.Google Scholar
- Stjepan Picek and Domagoj Jakobovic. 2016. Evolving Algebraic Constructions for Designing Bent Boolean Functions. In Proceedings of the 2016 on Genetic and Evolutionary Computation Conference, Denver, CO, USA, July 20--24, 2016, Tobias Friedrich, Frank Neumann, and Andrew M. Sutton (Eds.). ACM, 781--788.Google ScholarDigital Library
- Stjepan Picek and Domagoj Jakobovic. 2020. Evolutionary Computation and Machine Learning in Cryptology. In Proceedings of the 2020 Genetic and Evolutionary Computation Conference Companion (Cancún, Mexico) (GECCO '20). Association for Computing Machinery, New York, NY, USA, 1147--1173. Google ScholarDigital Library
- Stjepan Picek, Domagoj Jakobovic, and Marin Golub. 2013. Evolving Crypto-graphically Sound Boolean Functions. In Genetic and Evolutionary Computation Conference (GECCO) (GECCO '13 Companion). ACM, 191--192.Google Scholar
- Stjepan Picek, Dominik Sisejkovic, and Domagoj Jakobovic. 2017. Immunological algorithms paradigm for construction of Boolean functions with good cryptographic properties. Eng. Appl. of AI 62 (2017), 320--330.Google ScholarDigital Library
- Riccardo Poli, William B. Langdon, and Nicholas Freitag McPhee. 2008. A field guide to genetic programming. Published via http://lulu.com and freely available at http://www.gp-field-guide.org.uk. http://www.gp-field-guide.org.uk (With contributions by J. R. Koza).Google Scholar
- Richard L. Rudell and Alberto L. Sangiovanni-Vincentelli. 1987. Multiple-Valued Minimization for PLA Optimization. IEEE Trans. Comput. Aided Des. Integr. Circuits Syst. 6, 5 (1987), 727--750.Google ScholarDigital Library
- Sayandeep Saha, Rajat Subhra Chakraborty, Srinivasa Shashank Nuthakki, Anshul, and Debdeep Mukhopadhyay. 2015. Improved Test Pattern Generation for Hardware Trojan Detection Using Genetic Algorithm and Boolean Satisfiability. In Cryptographic Hardware and Embedded Systems - CHES 2015 - 17th International Workshop, Saint-Malo, France, September 13--16, 2015, Proceedings. 577--596. Google ScholarDigital Library
- Zhenbin Zhang, Liji Wu, An Wang, Zhaoli Mu, and Xiangmin Zhang. 2015. A novel bit scalable leakage model based on genetic algorithm. Security and Communication Networks 8, 18 (2015), 3896--3905. arXiv:https://onlinelibrary.wiley.com/doi/pdf/10.1002/sec.1308 Google ScholarDigital Library
Index Terms
Evolving constructions for balanced, highly nonlinear boolean functions
Recommendations
Evolving Algebraic Constructions for Designing Bent Boolean Functions
GECCO '16: Proceedings of the Genetic and Evolutionary Computation Conference 2016The evolution of Boolean functions that can be used in cryptography is a topic well studied in the last decades. Previous research, however, has focused on evolving Boolean functions directly, and not on general methods that are capable of generating ...
The Design of Boolean Functions by Modified Hill Climbing Method
ITNG '09: Proceedings of the 2009 Sixth International Conference on Information Technology: New GenerationsWith cryptographic investigations, the design of Boolean functions is a wide area. The Boolean functions play important role in the construction of a symmetric cryptosystem. In this paper the modified hill climbing method is considered. Using hill ...
On the constructions of resilient Boolean functions with five-valued Walsh spectra and resilient semi-bent functions
AbstractBoolean functions with five-valued Walsh spectra (shortened as five-valued functions) and semi-bent functions are two classes of Boolean functions with high nonlinearity. They have useful applications in cryptography and ...
Comments