Cryptology ePrint Archive: Report 2021/984

On the Use of the Legendre Symbol in Symmetric Cipher Design

Alan Szepieniec

Abstract: This paper proposes the use of Legendre symbols as component gates in the design of ciphers tailored for use in cryptographic proof systems. Legendre symbols correspond to high-degree maps, but can be evaluated much faster. As a result, a cipher that uses Legendre symbols can offer the same security as one that uses high-degree maps but without incurring the penalty of a comparatively slow evaluation time.

After discussing the design considerations induced by the use of Legendre symbol gates, we present a concrete design that follows this strategy, along with an elaborate security analysis thereof. This cipher is called Grendel.

Category / Keywords: secret-key cryptography / arithmetization-oriented, hash functions, zero knowledge

Date: received 22 Jul 2021, last revised 29 Nov 2021

Contact author: alan szepieniec at gmail com

Available format(s): PDF | BibTeX Citation

Note: add root finding attack to security analysis

Version: 20211129:130416 (All versions of this report)

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