## Cryptology ePrint Archive: Report 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.

Category / Keywords: foundations / Primarity, Cyclotomic field, Power residue symbol, Cryptography

Date: received 28 Aug 2021, last revised 28 Aug 2021

Contact author: david naccache at ens fr

Available format(s): PDF | BibTeX Citation

Short URL: ia.cr/2021/1106

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