Construction and analysis of resilient five-valued spectra Boolean functions
Boolean functions with five-valued spectra have important applications in code division multiple access (CDMA) communication, coding and cryptography, true random number generators (TRNG), and combinatorial design. An approach based on the Walsh spectral neutralization technique for directly constru...
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| Main Authors: | , , , |
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| Format: | Article |
| Language: | zho |
| Published: |
Editorial Department of Journal on Communications
2025-07-01
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| Series: | Tongxin xuebao |
| Subjects: | |
| Online Access: | http://www.joconline.com.cn/zh/article/doi/10.11959/j.issn.1000-436x.2025116/ |
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| Summary: | Boolean functions with five-valued spectra have important applications in code division multiple access (CDMA) communication, coding and cryptography, true random number generators (TRNG), and combinatorial design. An approach based on the Walsh spectral neutralization technique for directly constructing a class of <italic>n</italic>‐variable resilient five-valued spectra Boolean functions was proposed. The nonlinearity of the resulting functions could reach up to <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M6"><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mi>n</mi><mo>-</mo><mn mathvariant="normal">1</mn></mrow></msup><mo>-</mo><msup><mrow><mn mathvariant="normal">2</mn></mrow><mrow><mfenced open="⌈" close="⌉" separators="|"><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></mrow></msup></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M006.jpg"><?fx-imagestate width="14.73200035" height="5.84200001"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M006c.jpg"><?fx-imagestate width="14.73200035" height="5.84200001"?></graphic></alternatives></inline-formula>, and it was proved that functions achieving this upper bound had the algebraic degree of <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M7"><mfenced open="⌊" close="⌋" separators="|"><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced><mo>+</mo><mn mathvariant="normal">1</mn></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M007.jpg"><?fx-imagestate width="10.15999985" height="7.11199999"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M007c.jpg"><?fx-imagestate width="10.15999985" height="7.11199999"?></graphic></alternatives></inline-formula>,the resiliency order of approximately <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M8"><mfenced open="⌊" close="⌋" separators="|"><mrow><mfrac><mrow><mi>n</mi><mo>+</mo><mn mathvariant="normal">1</mn></mrow><mrow><mn mathvariant="normal">4</mn></mrow></mfrac></mrow></mfenced></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M008.jpg"><?fx-imagestate width="10.15999985" height="7.11199999"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M008c.jpg"><?fx-imagestate width="10.15999985" height="7.11199999"?></graphic></alternatives></inline-formula>, and the correction order of <inline-formula><alternatives><math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M9"><mfenced open="⌈" close="⌉" separators="|"><mrow><mfrac><mrow><mi>n</mi></mrow><mrow><mn mathvariant="normal">2</mn></mrow></mfrac></mrow></mfenced></math><graphic specific-use="big" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M009.jpg"><?fx-imagestate width="4.65666676" height="7.11199999"?></graphic><graphic specific-use="small" xmlns:xlink="http://www.w3.org/1999/xlink" xlink:href="alternativeImage/91B2C2BE-4517-4c06-90F2-0DDC989C448C-M009c.jpg"><?fx-imagestate width="4.65666676" height="7.11199999"?></graphic></alternatives></inline-formula>. The results demonstrate that the constructed functions achieve a favorable balance among multiple cryptographic properties. |
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| ISSN: | 1000-436X |