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Ramanujan Graphs in Cryptography

  • Anamaria Costache
  • Brooke FeigonEmail author
  • Kristin Lauter
  • Maike Massierer
  • Anna Puskás
Conference paper
Part of the Association for Women in Mathematics Series book series (AWMS, volume 19)

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 SSIG path-finding problem introduced in Charles et al. (2009). 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.

Keywords

Post-Quantum Cryptography Supersingular isogeny graphs Ramanujan graphs 

2010 Mathematics Subject Classification

Primary: 14G50, 11F70 Secondary: 05C75, 11R52 

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Copyright information

© The Author(s) and The Association for Women in Mathematics 2019

Authors and Affiliations

  • Anamaria Costache
    • 1
  • Brooke Feigon
    • 2
    Email author
  • Kristin Lauter
    • 3
  • Maike Massierer
    • 4
  • Anna Puskás
    • 5
  1. 1.Department of Computer ScienceUniversity of BristolBristolUK
  2. 2.Department of MathematicsThe City College of New YorkNew YorkUSA
  3. 3.Microsoft ResearchOne Microsoft WayRedmondUSA
  4. 4.School of Mathematics and StatisticsUniversity of New South WalesSydneyAustralia
  5. 5.Department of Mathematics & StatisticsUniversity of MassachusettsAmherstUSA

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