Cryptology ePrint Archive: Report 2018/593

Ramanujan graphs in cryptography

Anamaria Costache and Brooke Feigon and Kristin Lauter and Maike Massierer and Anna Puskas

Abstract: In this paper we study the security of a proposal for Post-Quantum Cryptography from both a number theoretic and cryptographic perspective. Charles-Goren-Lauter in 2006 proposed two hash functions based on the hardness of finding paths in Ramanujan graphs. One is based on Lubotzky--Phillips--Sarnak (LPS) graphs and the other one is based on Supersingular Isogeny Graphs. A 2008 paper by Petit-Lauter-Quisquater breaks the hash function based on LPS graphs. On the Supersingular Isogeny Graphs proposal, recent work has continued to build cryptographic applications on the hardness of finding isogenies between supersingular elliptic curves. A 2011 paper by De Feo-Jao-Plût proposed a cryptographic system based on Supersingular Isogeny Diffie--Hellman as well as a set of five hard problems. In this paper we show that the security of the SIDH proposal relies on the hardness of the SIG path-finding problem introduced in [CGL06]. In addition, similarities between the number theoretic ingredients in the LPS and Pizer constructions suggest that the hardness of the path-finding problem in the two graphs may be linked. By viewing both graphs from a number theoretic perspective, we identify the similarities and differences between the Pizer and LPS graphs.

Category / Keywords: public-key cryptography / Post-Quantum Cryptography, supersingular isogeny graphs, Ramanujan graphs

Original Publication (in the same form): Research Directions in Number Theory: Women in Numbers IV, AWM Springer Series

Date: received 8 Jun 2018, last revised 18 Dec 2018

Contact author: klauter at microsoft com

Available format(s): PDF | BibTeX Citation

Note: This revised version is now accepted for publication.

Version: 20181219:025615 (All versions of this report)

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