Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions
Recently, balanced Boolean functions with an even number <i>n</i> of variables achieving very good autocorrelation properties have been obtained for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow...
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MDPI AG
2023-08-01
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| Online Access: | https://www.mdpi.com/2073-431X/12/8/159 |
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| author | Erol Özçekiç Selçuk Kavut Hakan Kutucu |
| author_facet | Erol Özçekiç Selçuk Kavut Hakan Kutucu |
| author_sort | Erol Özçekiç |
| collection | DOAJ |
| description | Recently, balanced Boolean functions with an even number <i>n</i> of variables achieving very good autocorrelation properties have been obtained for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>12</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>26</mn></mrow></semantics></math></inline-formula>. These functions attain the maximum absolute value in the autocorrelation spectra (without considering the zero point) less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mfrac><mi>n</mi><mn>2</mn></mfrac></msup></semantics></math></inline-formula> and are found by using a heuristic search algorithm that is based on the design method of an infinite class of such functions for a higher number of variables. Here, we consider balanced Boolean functions that are closest to the bent functions in terms of the Hamming distance and perform a genetic algorithm efficiently aiming to optimize their cryptographic properties, which provides better absolute indicator values for all of those values of <i>n</i> for the first time. We also observe that among our results, the functions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>16</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>26</mn></mrow></semantics></math></inline-formula> have nonlinearity greater than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mn>2</mn><mfrac><mi>n</mi><mn>2</mn></mfrac></msup></mrow></semantics></math></inline-formula>. In the process, our search strategy produces balanced Boolean functions with the best-known nonlinearity for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>8</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>16</mn></mrow></semantics></math></inline-formula>. |
| first_indexed | 2024-03-11T00:01:50Z |
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| spelling | doaj.art-f6bf2a6a57594757bf12a5c658f217732023-11-19T00:43:22ZengMDPI AGComputers2073-431X2023-08-0112815910.3390/computers12080159Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent FunctionsErol Özçekiç0Selçuk Kavut1Hakan Kutucu2Computer Technologies Department, Vocational School, Balıkesir University, 10145 Balıkesir, TurkeyComputer Engineering Department, Faculty of Engineering, Balıkesir University, 10145 Balıkesir, TurkeySoftware Engineering Department, Faculty of Engineering, Karabük University, 78050 Karabük, TurkeyRecently, balanced Boolean functions with an even number <i>n</i> of variables achieving very good autocorrelation properties have been obtained for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>12</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>26</mn></mrow></semantics></math></inline-formula>. These functions attain the maximum absolute value in the autocorrelation spectra (without considering the zero point) less than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><msup><mn>2</mn><mfrac><mi>n</mi><mn>2</mn></mfrac></msup></semantics></math></inline-formula> and are found by using a heuristic search algorithm that is based on the design method of an infinite class of such functions for a higher number of variables. Here, we consider balanced Boolean functions that are closest to the bent functions in terms of the Hamming distance and perform a genetic algorithm efficiently aiming to optimize their cryptographic properties, which provides better absolute indicator values for all of those values of <i>n</i> for the first time. We also observe that among our results, the functions for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>16</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>26</mn></mrow></semantics></math></inline-formula> have nonlinearity greater than <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msup><mn>2</mn><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo>−</mo><msup><mn>2</mn><mfrac><mi>n</mi><mn>2</mn></mfrac></msup></mrow></semantics></math></inline-formula>. In the process, our search strategy produces balanced Boolean functions with the best-known nonlinearity for <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mn>8</mn><mo>≤</mo><mi>n</mi><mo>≤</mo><mn>16</mn></mrow></semantics></math></inline-formula>.https://www.mdpi.com/2073-431X/12/8/159absolute indicatorBoolean functiongenetic algorithmnonlinearity |
| spellingShingle | Erol Özçekiç Selçuk Kavut Hakan Kutucu Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions Computers absolute indicator Boolean function genetic algorithm nonlinearity |
| title | Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions |
| title_full | Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions |
| title_fullStr | Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions |
| title_full_unstemmed | Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions |
| title_short | Genetic Approach to Improve Cryptographic Properties of Balanced Boolean Functions Using Bent Functions |
| title_sort | genetic approach to improve cryptographic properties of balanced boolean functions using bent functions |
| topic | absolute indicator Boolean function genetic algorithm nonlinearity |
| url | https://www.mdpi.com/2073-431X/12/8/159 |
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