An Explicit Construction of Orthogonal Basis in $p$-adic Fields
Chi Zhang, Yingpu Deng
TL;DR
The paper tackles the problem of constructing an explicit orthogonal basis in $p$-adic fields to enable larger residue degrees and thereby improve the security prospects of $p$-adic cryptosystems. It develops a concrete framework that decomposes a ramified extension into an unramified part of degree $f$ and a totally ramified part of degree $e$, yielding an orthogonal basis of the form $( heta^{i} au^{j})$ for $0\, ext{≤}\,i\, ext{≤}\,f-1$ and $0\, ext{≤}\,j\, ext{≤}\,e-1$ under suitable irreducibility conditions modulo $p$. The work provides both a theoretical criterion for orthogonality and a practical construction via roots of unity, Eisenstein polynomials, and resultant-based primitive element techniques, accompanied by complexity analyses. The results offer a blueprint to tailor $p$-adic lattices for cryptographic primitives and inform secure parameter choices for modified $p$-adic signature schemes and public-key cryptosystems, while acknowledging security caveats and directions for future refinement.
Abstract
In 2021, the $p$-adic signature scheme and public-key encryption cryptosystem were introduced. These schemes have good efficiency but are shown to be not secure. The attack succeeds because the extension fields used in these schemes are totally ramified. In order to avoid this attack, the extension field should have a large residue degree. In this paper, we propose a method of constructing a kind of specific orthogonal basis in $p$-adic fields with a large residue degree, which would be helpful to modify the $p$-adic signature scheme and public-key encryption cryptosystem.