Cubic power functions with optimal second-order differential uniformity
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arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866914961119248384 |
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| author | O'Reilly, Connor Sălăgean, Ana |
| author_facet | O'Reilly, Connor Sălăgean, Ana |
| contents | We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_03467 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Cubic power functions with optimal second-order differential uniformity O'Reilly, Connor Sălăgean, Ana Information Theory Cryptography and Security Number Theory 94D10 (Primary) 11T06, 94A60 (Secondary) We discuss the second-order differential uniformity of vectorial Boolean functions. The closely related notion of second-order zero differential uniformity has recently been studied in connection to resistance to the boomerang attack. We prove that monomial functions with univariate form $x^d$ where $d=2^{2k}+2^k+1$ and $\gcd(k,n)=1$ have optimal second-order differential uniformity. Computational results suggest that, up to affine equivalence, these might be the only optimal cubic power functions. We begin work towards generalising such conditions to all monomial functions of algebraic degree 3. We also discuss further questions arising from computational results. |
| title | Cubic power functions with optimal second-order differential uniformity |
| topic | Information Theory Cryptography and Security Number Theory 94D10 (Primary) 11T06, 94A60 (Secondary) |
| url | https://arxiv.org/abs/2409.03467 |