{"ID":2896565,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2507.06808","arxiv_id":"2507.06808","title":"A Note on the Walsh Spectrum of Power Residue S-Boxes","abstract":"Let $\\mathbb{F}_q$ be a prime field with $q \\geq 3$, and let $d, m \\geq 1$ be integers such that $\\gcd \\left( d, q \\right) = 1$ and $m \\mid (q - 1)$. In this paper we bound the absolute values of the Walsh spectrum of S-Boxes $S (x) = x^d \\cdot T \\left( x^\\frac{q - 1}{m} \\right)$, where $T$ is a function with $T (x) \\neq 0$ if $x \\neq 0$. Such S-Boxes have been proposed for the Zero-Knowledge-friendly hash functions Grendel and Polocolo. In particular, we prove the conjectured correlation of the Polocolo S-Box.","short_abstract":"Let $\\mathbb{F}_q$ be a prime field with $q \\geq 3$, and let $d, m \\geq 1$ be integers such that $\\gcd \\left( d, q \\right) = 1$ and $m \\mid (q - 1)$. In this paper we bound the absolute values of the Walsh spectrum of S-Boxes $S (x) = x^d \\cdot T \\left( x^\\frac{q - 1}{m} \\right)$, where $T$ is a function with $T (x) \\n...","url_abs":"https://arxiv.org/abs/2507.06808","url_pdf":"https://arxiv.org/pdf/2507.06808v2","authors":"[\"Matthias Johann Steiner\"]","published":"2025-07-09T12:57:15Z","proceeding":"math.NT","tasks":"[\"math.NT\",\"cs.CR\"]","methods":"[]","has_code":false}