| Summary: | 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>.
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