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Post-quantum cryptography: lattice signatures

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Abstract

This survey provides a comparative overview of lattice-based signature schemes with respect to security and performance. Furthermore, we explicitly show how to construct a competitive and provably secure Merkle tree signature scheme solely based on worst-case lattice problems.

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References

  1. Ajtai M (1996) Generating hard instances of lattice problems (extended abstract). In: Proceedings of the annual symposium on the theory of computing (STOC) 1996. ACM Press, pp 99–108

  2. Ajtai M (1999) Generating hard instances of the short basis problem. In: International colloquium on automata, languages and programming (ICALP). Lecture notes in computer science. Springer, New York, pp 1–9

  3. Babai L (1986) On Lovász’ lattice reduction and the nearest lattice point problem. Combinatorica 6(1): 1–13

    Article  MATH  MathSciNet  Google Scholar 

  4. Bernstein, DJ, Buchmann, J, Dahmen, E (eds) (2008) Post-quantum cryptography. Springer, New York

    Google Scholar 

  5. Buchmann J, Dahmen E, Klintsevich E, Okeya K, Vuillaume C (2007) Merkle signatures with virtually unlimited signature capacity. In: International conference on applied cryptography and network security—ACNS 2007. Lecture notes in computer science. Springer, New York, pp 31–45

  6. Buchmann J, Dahmen E, Schneider M (2008) Merkle tree traversal revisited. In: Post-quantum cryptography (PQCrypto) 2008. Lecture notes in computer science. Springer, New York, pp 63–78

  7. Buchmann J, Lindner R, Rückert M (2008) Explicit hard instances of the shortest vector problem. In: Post-quantum cryptography (PQCrypto) 2008. Lecture notes in computer science. Springer, New York, pp 79–94

  8. Buchmann J, Lindner R, Rückert M, Schneider M (2008) Explicit hard instances of the shortest vector problem (extended version). Technical Report 2008/333, Cryptology ePrint Archive

  9. Dinur I (2002) Approximating SVP∞ to within almost-polynomial factors is NP-hard. Theor Comput Sci 285(1): 55–71

    Article  MATH  MathSciNet  Google Scholar 

  10. Gama N, Howgrave-Graham N, Nguyen PQ (2006) Symplectic lattice reduction and NTRU. In: Vaudenay S(eds) EUROCRYPT. Lecture notes in computer science, vol 4004. Springer, New York, pp 233–253

    Google Scholar 

  11. Gentry C, Jonsson J, Stern J, Szydlo M (2001) Cryptanalysis of the ntru signature scheme (nss) from eurocrypt 2001. In: Advances in cryptology—Asiacrypt 2001, pp 1–20

  12. Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard lattices and new cryptographic constructions. In: Proceedings of the annual symposium on the theory of computing (STOC) 2008. ACM Press, pp 197–206

  13. Goldreich O, Goldwasser S, Halevi S (1997) Public-key cryptosystems from lattice reduction problems. In: Advances in cryptology—crypto 1997. Lecture notes in computer science. Springer, New York, pp 112–131

  14. Hoffstein J, Howgrave-Graham N, Pipher J, Silverman JH, Whyte W (2005) Performance improvements and a baseline parameter generation algorithm for NTRUsign. http://grouper.ieee.org/groups/1363/lattPK/submissions.html

  15. Hoffstein J, Pipher J, Silverman JH (2001) NSS: an ntru lattice-based signature scheme. In: Advances in cryptology—Eurocrypt 2001. Lecture notes in computer science. Springer, New York, pp 211–228

  16. Jakobsson M, Leighton T, Micali S, Szydlo M (2003) Fractal merkle tree representation and traversal. In: Topics in cryptology—cryptographer’s track, RSA Conference (CT-RSA) 2003. Lecture notes in computer science. Springer, New York, pp 314–326

  17. Khot S (2005) Hardness of approximating the shortest vector problem in lattices. J ACM 52(5): 789–808

    Article  MathSciNet  Google Scholar 

  18. Klein PN (2000) Finding the closest lattice vector when it’s unusually close. In: Proceedings of the annual symposium on discrete algorithms (SODA) 2000. ACM Press, pp 937–941

  19. Lenstra A, Lenstra H, Lovász L (1982) Factoring polynomials with rational coefficients. Math Ann 261(4): 515–534

    Article  MATH  MathSciNet  Google Scholar 

  20. Lyubashevsky V, Micciancio D (2006) Generalized compact knapsacks are collision resistant. In: International colloquium on automata, languages and programming (ICALP) 2006. Lecture notes in computer science. Springer, New York, pp 144–155

  21. Lyubashevsky V, Micciancio D (2008) Asymptotically efficient lattice-based digital signatures. In: Theory of cryptography conference (TCC) 2008. Lecture notes in computer science. Springer, New York, pp 37–54

  22. Lyubashevsky V, Micciancio D, Peikert C, Rosen A (2008) Swifft: a modest proposal for fft hashing. In: Fast software encryption (FSE) 2008. Lecture notes in computer science. Springer, New York, pp 54–72

  23. Merkle R (1990) A certified digital signature. In: Advances in cryptology—crypto 1989. Lecture notes in computer science. Springer, New York, pp 218–238

  24. Micciancio D (2001) Improving lattice based cryptosystems using the Hermite normal form. In: Cryptography and lattices (CaLC) 2001. Lecture notes in computer science. Springer, New York, pp 126–145

  25. Micciancio D (2002) Generalized compact knapsacks, cyclic lattices, and efficient one-way functions from worst-case complexity assumptions. In: Proceedings of the 43rd annual IEEE symposium on foundations of computer science, 2002, pp 356–365

  26. Micciancio D, Goldwasser S (2002) Complexity of lattice problems: a cryptographic perspective. The Kluwer international series in engineering and computer science, vol 671. Kluwer, Boston

  27. Micciancio D, Regev O (2007) Worst-case to average-case reductions based on gaussian measures. SIAM J Comput 37(1): 267–302

    Article  MATH  MathSciNet  Google Scholar 

  28. Nguyen PQ (1999) Cryptanalysis of the Goldreich-Goldwasser-Halevi cryptosystem from crypto’97. In: Advances in cryptology—crypto 1999. Lecture notes in computer science. Springer, New York, pp 288–304

  29. Nguyen PQ, Regev O (2006) Learning a parallelepiped: cryptanalysis of GGH and NTRU signatures. In: Advances in cryptology—Eurocrypt 2006. Lecture notes in computer science. Springer, New York, pp 215–233

  30. Peikert C, Rosen A (2006) Efficient collision-resistant hashing from worst-case assumptions on cyclic lattices. In: Theory of cryptography conference (TCC) 2006. Lecture notes in computer science. Springer, New York, pp 145–166

  31. Regev O (2007) On the complexity of lattice problems with polynomial approximation factors. In: A survey for the LLL+25 conference

  32. Regev O, Rosen R (2006) Lattice problems and norm embeddings. In: Proceedings of the annual symposium on the theory of computing (STOC) 2006. ACM Press, pp 447–456

  33. Shor PW (1997) Polynomial-time algorithms for prime factorization and discrete logarithms on a quantum computer. SIAM J Comput 26(5): 1484–1509

    Article  MATH  MathSciNet  Google Scholar 

  34. Szydlo M (2003) Merkle tree traversal in log space and time (preprint version). http://www.szydlo.com

  35. Szydlo M (2004) Merkle tree traversal in log space and time. In: Advances in cryptology—Eurocrypt 2004. Lecture notes in computer science. Springer, New York, pp 541–554

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Authors and Affiliations

  1. Department of Computer Science, Technische Universität Darmstadt, Hochschulstraße 10, 64289, Darmstadt, Germany

    Johannes Buchmann, Richard Lindner, Markus Rückert & Michael Schneider

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  1. Johannes Buchmann
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  2. Richard Lindner
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  3. Markus Rückert
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  4. Michael Schneider
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Correspondence to Richard Lindner.

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Buchmann, J., Lindner, R., Rückert, M. et al. Post-quantum cryptography: lattice signatures. Computing 85, 105–125 (2009). https://doi.org/10.1007/s00607-009-0042-y

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