Post-quantum Cryptography: An Introduction

Part of the IITK Directions book series (IITKD, volume 4)


We present a brief introduction to post-quantum cryptography. This note introduces the concept of post-quantum cryptography, discusses its importance and provides a short overview of the mathematical techniques that are currently used to develop this field.


Post-quantum cryptography Lattices 


  1. 1.
    Goldreich O (2000) Foundations of cryptography: basic tools. Cambridge University Press, New YorkzbMATHGoogle Scholar
  2. 2.
    Shor PW (1994) Algorithms for quantum computation: discrete logarithms and factoring. In: 35th annual symposium on foundations of computer science, Santa Fe, New Mexico, USA, 20–22 Nov 1994, pp 124–134Google Scholar
  3. 3.
    Chen L, Jordan S, Liu YK, Moody D, Peralta R, Perlner R, Smith-Tone D. Report on post-quantum cryptography.
  4. 4.
    Ajtai M (1996) Generating hard instances of lattice problems (extended abstract). In: Proceedings of the twenty-eighth annual ACM symposium on theory of computing, STOC ’96Google Scholar
  5. 5.
    Matsumoto T, Imai H (1988) Public quadratic polynomial-tuples for efficient signature-verification and message-encryption. In: Advances in cryptology — EUROCRYPT ’88. Springer, Berlin, Heidelberg, pp 419–453Google Scholar
  6. 6.
    Bardet M, Faugere JC, Salvy B, Spaenlehauer PJ (2013) On the complexity of solving quadratic boolean systems. J Complex 29(1):53–75.
  7. 7.
    Wolf C (2005) Multivariate quadratic polynomials in public key cryptography. PhD thesis, Katholieke Universiteit LeuvenGoogle Scholar
  8. 8.
    Ding J, Yang BY (2009) Multivariate public key cryptography. Springer, Berlin, pp 193–241zbMATHGoogle Scholar
  9. 9.
    Patarin J (1996) Hidden fields equations (hfe) and isomorphisms of polynomials (ip): two new families of asymmetric algorithms. In: Maurer U (ed) Advances in cryptology — EUROCRYPT ’96Google Scholar
  10. 10.
    Kipnis A, Patarin J, Goubin L (1999) Unbalanced oil and vinegar signature schemes. In: Stern J (ed) Advances in cryptology — EUROCRYPT ’99Google Scholar
  11. 11.
    Hashimoto Y (2018) Multivariate public key cryptosystems. In: Mathematical modelling for next-generation cryptography. Springer, pp 17–42Google Scholar
  12. 12.
    McEliece RJ (1978) A Public-key cryptosystem based on algebraic coding theory. Deep space network progress report, vol 44, pp 114–116Google Scholar
  13. 13.
    Overbeck R, Sendrier N (2009) Code-based cryptographyGoogle Scholar
  14. 14.
    Merkle R (1979) Secrecy, authentication and public key systems/a certified digital signature. PhD thesis, Stanford UniversityGoogle Scholar
  15. 15.
    Peikert C (2016) A decade of lattice cryptography 10:283–424Google Scholar
  16. 16.
    Ajtai M, Dwork C (1997) A public-key cryptosystem with worst-case/average-case equivalence. In: Proceedings of the twenty-ninth annual ACM symposium on theory of computing. ACM, pp 284–293Google Scholar
  17. 17.
    Goldreich O, Goldwasser S, Halevi S (1997) Public-key cryptosystems from lattice reduction problems. In: Annual international cryptology conference. Springer, pp 112–131Google Scholar
  18. 18.
    Regev O (2004) New lattice-based cryptographic constructions. J ACM (JACM) 51(6):899–942MathSciNetCrossRefGoogle Scholar
  19. 19.
    Micciancio D, Regev O (2004) Worst-case to average-case reductions based on gaussian measures. SIAM J Comput (SICOMP) 37(1):267–302. Extended abstract in FOCS 2004Google Scholar
  20. 20.
    Gentry C, Peikert C, Vaikuntanathan V (2008) Trapdoors for hard lattices and new cryptographic constructions. In: STOC, pp 197–206Google Scholar
  21. 21.
    Micciancio D, Peikert C (2013) Hardness of sis and lwe with small parameters. In: CryptoGoogle Scholar
  22. 22.
    Regev O (2009) On lattices, learning with errors, random linear codes, and cryptography. J ACM 56(6). Extended abstract in STOC’05Google Scholar
  23. 23.
    Lyubashevsky V, Peikert C, Regev O (2010) On ideal lattices and learning with errors over rings. In: EUROCRYPT, vol 6110Google Scholar
  24. 24.
    Peikert C (2009) Public-key cryptosystems from the worst-case shortest vector problem. In: STOC, pp 333–342Google Scholar
  25. 25.
    Brakerski Z, Langlois A, Peikert C, Regev O, Stehlé D (2013) Classical hardness of learning with errors. In: Proceedings of the forty-fifth annual ACM symposium on theory of computing, STOC ’13. ACMGoogle Scholar
  26. 26.
    Hoffstein J, Pipher J, Silverman JH (1998) Ntru: a ring-based public key cryptosystem. In: Buhler JP (ed) Algorithmic number theory: third international symposium, ANTS-III Portland, Oregon, USA, 21–25 June 998, ProceedingsGoogle Scholar
  27. 27.
    Stehlé D, Steinfeld R (2011) Making ntru as secure as worst-case problems over ideal lattices. In: Proceedings of the 30th annual international conference on theory and applications of cryptographic techniques: advances in cryptology, EUROCRYPT’11Google Scholar
  28. 28.
    López-Alt A, Tromer E, Vaikuntanathan V (2012) On-the-fly multiparty computation on the cloud via multikey fully homomorphic encryption. In: Proceedings of the forty-fourth annual ACM symposium on theory of computing, STOC ’12Google Scholar
  29. 29.
    Barak B, Dodis Y, Krawczyk H, Pereira O, Pietrzak K, Standaert FX, Yu Y (2011) Leftover hash lemma, revisited. In: Annual cryptology conference. Springer, pp 1–20Google Scholar

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© Springer Nature Singapore Pte Ltd. 2020

Authors and Affiliations

  1. 1.Indian Institute of Technology MadrasChennaiIndia

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