Paper 2021/1106
Primary Elements in Cyclotomic Fields with Applications to Power Residue Symbols, and More
Eric Brier and Rémi Géraud-Stewart and Marc Joye and David Naccache
Abstract
Higher-order power residues have enabled the construction of numerous public-key encryption schemes, authentication schemes, and digital signatures. Their explicit characterization is however challenging; an algorithm of Caranay and Scheidler computes $p$-th power residue symbols, with $p \le 13$ an odd prime, provided that primary elements in the corresponding cyclotomic field can be efficiently found. In this paper, we describe a new, generic algorithm to compute primary elements in cyclotomic fields; which we apply for $p=3,5,7,11,13$. A key insight is a careful selection of fundamental units as put forward by Dénes. This solves an essential step in the Caranay--Scheidler algorithm. We give a unified view of the problem. Finally, we provide the first efficient deterministic algorithm for the computation of the 9-th and 16-th power residue symbols.
Metadata
- Available format(s)
- Category
- Foundations
- Publication info
- Preprint. MINOR revision.
- Keywords
- PrimarityCyclotomic fieldPower residue symbolCryptography
- Contact author(s)
- david naccache @ ens fr
- History
- 2021-08-31: received
- Short URL
- https://ia.cr/2021/1106
- License
-
CC BY