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Research Directions in Number Theory pp 1–40Cite as

Ramanujan Graphs in Cryptography

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Part of the book series: Association for Women in Mathematics 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.

Brooke Feigon was partially supported by National Security Agency grant H98230-16-1-0017 and PSC-CUNY.

Maike Massierer was partially supported by Australian Research Council grant DP150101689.

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Notes

  1. 1.

    A similar construction exists for a more general \(\mathcal {O} .\) However, to relate the resulting graph to supersingular isogeny graphs, we require \(\mathcal {O}\) to be maximal.

  2. 2.

    If p is not a square modulo l, then the constructions described below result in bipartite Ramanujan graphs with twice as many vertices.

  3. 3.

    That is, the adjacency relation defined above is symmetric.

  4. 4.

    The definition here agrees with the choices in [14] as well as \(\Gamma (N)=\ker (G^{\prime }({\mathbb {Z}}[l^{-1}])\rightarrow G^{\prime }({\mathbb {Z}}[l^{-1}]/N{\mathbb {Z}}[l^{-1}]))\) in [15]. Here G ′ = B ×∕Z(B ×) as a \({\mathbb {Q}}\)-algebraic group. Note however that by (10) the center Z(B ×(R)) for \(R={\mathbb {Z}}[l^{-1}]/N{\mathbb {Z}}[l^{-1}],\ N=2M\) may not be spanned by \(1+N{\mathbb {Z}}[l^{-1}].\) In fact from (10) B ×(R) is commutative for M = 1 and for M = p we have Z(B ×(R)) = Z ⊕ [p]i + [p]j + [p]k. However the image of 〈S〉 in B ×(R) is trivial if M = 1 and intersects the center in Z when M = p.

  5. 5.

    In fact, since at every split place v we have \(B^{\times }({{\mathbb {Q}}_v}) \cong {\mathrm {GL}}_2({{\mathbb {Q}}_v})\) with the reduced norm on B × corresponding to the determinant on GL2 [26, p. 3] this is the “same argument at all but finitely many places.”

References

  1. Gora Adj, Omran Ahmadi, and Alfred Menezes, On isogeny graphs of supersingular elliptic curves over finite fields, Cryptology ePrint Archive, Report 2018/132, 2018, https://eprint.iacr.org/2018/132.

  2. Noga Alon, Eigenvalues and expanders, Combinatorica 6 (1986), no. 2, 83–96, Theory of computing (Singer Island, Fla., 1984). MR 875835

    Google Scholar 

  3. Denis X. Charles, Eyal Z. Goren, and Kristin E. Lauter, Cryptographic hash functions from expander graphs, J. Cryptology 22 (2009), no. 1, 93–113, available at https://eprint.iacr.org/2006/021.pdf. MR 2496385

  4. _________ , Families of Ramanujan graphs and quaternion algebras, Groups and symmetries, CRM Proc. Lecture Notes, vol. 47, Amer. Math. Soc., Providence, RI, 2009, pp. 53–80. MR 2500554

    Google Scholar 

  5. Gaëtan Chenevier, Lecture notes, 2010, http://gaetan.chenevier.perso.math.cnrs.fr/coursIHP/chenevier_lecture6.pdf, retrieved August 13, 2017.

  6. Pierre Deligne, Formes modulaires et représentationsl-adiques, Séminaire Bourbaki. Vol. 1968/69, vol. 179, Lecture Notes in Math., no. 355, Springer, Berlin, 1971, pp. 139–172.

    Google Scholar 

  7. _________ , La conjecture de Weil. I, Publications Mathématiques de l’Institut des Hautes Études Scientifiques 43 (1974), no. 1, 273–307.

    Google Scholar 

  8. Luca De Feo, David Jao, and Jérôme Plût, Towards quantum-resistant cryptosystems from supersingular elliptic curve isogenies, J. Math. Cryptol. 8 (2014), no. 3, 209–247. MR 3259113

    Google Scholar 

  9. Stephen S. Gelbart, Automorphic forms on adele groups, no. 83, Princeton University Press, 1975.

    Google Scholar 

  10. Yasutaka Ihara, Discrete subgroups of PL(2, k ℘), Algebraic Groups and Discontinuous Subgroups (Proc. Sympos. Pure Math., Boulder, Colo., 1965), Amer. Math. Soc., Providence, R.I., 1966, pp. 272–278. MR 0205952

    Google Scholar 

  11. David Jao, Stephen D Miller, and Ramarathnam Venkatesan, Do all elliptic curves of the same order have the same difficulty of discrete log?, International Conference on the Theory and Application of Cryptology and Information Security, Springer, 2005, pp. 21–40.

    Google Scholar 

  12. Wen-Ch’ing Winnie Li, A survey of Ramanujan graphs, Arithmetic, geometry and coding theory (Luminy, 1993), de Gruyter, Berlin, 1996, pp. 127–143. MR 1394930

    Google Scholar 

  13. Eyal Lubetzky and Yuval Peres, Cutoff on all Ramanujan graphs, Geometric and Functional Analysis 26 (2016), no. 4, 1190–1216.

    Article  MathSciNet  Google Scholar 

  14. Alexander Lubotzky, Richard L. Phillips, and Peter Sarnak, Ramanujan graphs, Combinatorica 8 (1988), no. 3, 261–277. MR 963118 (89m:05099)

    Article  MathSciNet  Google Scholar 

  15. Alexander Lubotzky, Discrete groups, expanding graphs and invariant measures, Modern Birkhäuser Classics, Birkhäuser Verlag, Basel, 2010, With an appendix by Jonathan D. Rogawski, Reprint of the 1994 edition. MR 2569682

    Google Scholar 

  16. Jean-Francois Mestre, La méthode des graphes. Exemples et applications, Proceedings of the International Conference on Class Numbers and Fundamental Units of Algebraic Number Fields (Katata, 1986), Nagoya Univ., Nagoya, 1986, pp. 217–242. MR 891898

    Google Scholar 

  17. Christophe Petit, Kristin Lauter, and Jean-Jacques Quisquater, Full cryptanalysis of LPS and Morgenstern hash functions, Security and Cryptography for Networks (Berlin, Heidelberg) (Rafail Ostrovsky, Roberto De Prisco, and Ivan Visconti, eds.), Springer Berlin Heidelberg, 2008, pp. 263–277.

    Google Scholar 

  18. Arnold Pizer, The representability of modular forms by theta series, Journal of the Mathematical Society of Japan 28 (1976), no. 4, 689–698.

    Article  MathSciNet  Google Scholar 

  19. _________ , An algorithm for computing modular forms on Γ0(N), Journal of Algebra 64 (1980), no. 2, 340–390.

    Article  MathSciNet  Google Scholar 

  20. _________ , Ramanujan graphs, Computational perspectives on number theory (Chicago, IL, 1995), AMS/IP Stud. Adv. Math., vol. 7, Amer. Math. Soc., Providence, RI, 1998, pp. 159–178. MR 1486836

    Google Scholar 

  21. Post-Quantum Cryptography Standardization, https://csrc.nist.gov/Projects/Post-Quantum-Cryptography/Post-Quantum-Cryptography-Standardization, Accessed: 2018-04-14.

  22. Naser T. Sardari, Diameter of Ramanujan graphs and random Cayley graphs, (2018). Combinatorica, 1–20. https://doi.org/10.1007/s00493-017-3605-0

    Article  MathSciNet  Google Scholar 

  23. Joseph H. Silverman, The arithmetic of elliptic curves, second ed., Graduate Texts in Mathematics, vol. 106, Springer, Berlin–Heidelberg–New York, 2009.

    Google Scholar 

  24. Jean-Pierre Tillich and Gilles Zémor, Collisions for the LPS expander graph hash function, Advances in Cryptology – EUROCRYPT 2008 (Nigel Smart, ed.), Springer, 2008, pp. 254–269.

    Google Scholar 

  25. Jacques Vélu, Isogénies entre courbes elliptiques, C. R. Acad. Sci. Paris Sér. A-B 273 (1971), A238–A241. MR 0294345

    Google Scholar 

  26. Marie-France Vignéras, Arithmétique des algèbres de quaternions, Lecture Notes in Mathematics, vol. 800, Springer, Berlin, 1980. MR 580949

    Google Scholar 

  27. John Voight, Quaternion algebras, 2018, https://math.dartmouth.edu/~jvoight/quat-book.pdf, retrieved October 20, 2017.

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Author information

Authors and Affiliations

  1. Department of Computer Science, University of Bristol, Bristol, UK

    Anamaria Costache

  2. Department of Mathematics, The City College of New York, CUNY, NAC 8/133, New York, NY, 10031, USA

    Brooke Feigon

  3. Microsoft Research, One Microsoft Way, Redmond, WA, 98052, USA

    Kristin Lauter

  4. School of Mathematics and Statistics, University of New South Wales, Sydney, NSW, 2052, Australia

    Maike Massierer

  5. Department of Mathematics & Statistics, University of Massachusetts, Amherst, MA, 01003, USA

    Anna Puskás

Authors
  1. Anamaria Costache
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  2. Brooke Feigon
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  4. Maike Massierer
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  5. Anna Puskás
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Corresponding author

Correspondence to Brooke Feigon .

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Editors and Affiliations

  1. Department of Mathematics and Statistics, Boston University, Boston, MA, USA

    Jennifer S. Balakrishnan

  2. Department of Mathematics and Statistics, Amherst College, Amherst, MA, USA

    Amanda Folsom

  3. Département de mathématiques et de statistique, Université de Montréal, Montréal, QC, Canada

    Matilde Lalín

  4. Department of Mathematics, University of Hawai‘i, Honolulu, HI, USA

    Michelle Manes

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Costache, A., Feigon, B., Lauter, K., Massierer, M., Puskás, A. (2019). Ramanujan Graphs in Cryptography. In: Balakrishnan, J., Folsom, A., Lalín, M., Manes, M. (eds) Research Directions in Number Theory. Association for Women in Mathematics Series, vol 19. Springer, Cham. https://doi.org/10.1007/978-3-030-19478-9_1

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