Abstract
Probability plays a fundamental role in complexity theory, which in turn is one of the pillars of modern cryptology. However, security practitioners are not always familiar with probability theory, and thus fail to foresee the impact of (seemingly small) deviations from the theoretical description of a scheme at the implementation level. On the other hand, many cryptographic scenarios involve mutually distrusting parties, which need however to cooperate towards a joint goal. In order to attain assurance of the good behavior of one party, interactive validation methods (also known as interactive proof systems) are employed. Randomness is at the core of such methods, which most often will only provide relative assurance, in the sense that they will establish correctness in a probabilistic way. In this paper we will briefly discuss the role of probability theory within modern cryptology, reviewing probabilistic proof systems as a powerful tool towards efficient protocol design, and provable security, as an invaluable framework for deriving formal security proofs.
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Notes
- 1.
The provable security paradigm has been questioned by different authors, see [15].
- 2.
We follow standard notation and denote by \(\mathbb {Z}_n^*\) the group of units in \(\{1,\dots , n-1\},\) where product is defined modulo n. Also, as standard, throughout the paper, by “u.a.r.” we mean uniformly at random.
- 3.
Actually, the terms interactive and probabilistic are often used as synonyms in this setting.
- 4.
For soundness: if \(G_0\) and \(G_1\) were isomorphic, we take that \(\alpha _i\) will equal 1 with probability \(\frac{1}{2};\) as a result, the probability that the verifier does not reject in this case is at most \(\frac{1}{2^m}\).
- 5.
Here \(\Pr (P=m\,|\, C=c)\) denotes conditional probability, i.e., the probability of \(P=m\) once we know the ciphertext is c.
- 6.
A formal discussion on entropy and information can be found in [12].
- 7.
Informally, a negligible function has domain in \(\mathbb {N}\), range in \(\mathbb {R}^+\) and goes to zero faster than the inverse of any polynomial.
References
Barak B (2016) Lecture notes: zero knowledge proofs. http://www.boazbarak.org
Bellare M, Rogaway P (1993) Random oracles are practical: a paradigm for designing efficient protocols. In: Denning D, Pyle R, Ganesan R, Sandhu R, Ashby V (eds) Proceedings of 1st ACM conference on computer and communications security. ACM, New York
Bellare M, Rogaway P (1996) The exact security of digital signatures - how to sign with RSA and Rabin. In: Maurer U (ed) Advances in cryptology EUROCRYPT’96, vol 1070. Lecture notes in computer science. Springer, Berlin
Dent A (2006) A note on game-hopping proofs. IACR Cryptology ePrint Archive: Report 2006/260
Fiat A, Shamir A (1987) How to prove yourself: practical solutions to identification and signature problems. In: Odlyzko AM (ed) Advances in cryptology CRYPTO’86, vol 263. Lecture notes in computer science. Springer, Berlin
Gil P (2007) Por qué teoría de la información? Bol Soc Estad Investig Oper 23(3):8–9
Goldreich O (2004) The foundations of cryptography - volume 1, basic tools. Cambridge University Press, Cambridge
Goldreich O (2008) Computational complexity, a conceptual perspective. Cambridge Univeristy Press, Cambridge
Goldwasser S, Micali S (1984) Probabilistic encryption. J Comput Syst Sci 28(2):270–299
Goldwasser S, Micali S, Rackoff C (1985) The knowledge complexity of interactive proof-systems. In: Sedgewick R (ed) STOC’85 Proceedings of 17th annual ACM symposium on theory of computing. ACM, New York
Goldreich O, Micali S, Wigderson A (1991) Proofs that yield nothing but their validity or all languages in NP vave zero-knowledge proof systems. J ACM 38(3):691–729
Gray RM (2013) Entropy and information theory. Springer, New York
Jaikin A (2013) Grafos, grupos y variedades: un punto de encuentro. Gaceta Real Soc Matem Española 16(4):761–776
Katz J (2010) Digital signatures. Springer, New York
Koblitz N, Menezes A (2007) Another look at “provable security”. J Cryptol 20(1):3–37
Maurer UM (1993) The role of information theory in cryptography. In: Farrell PG (ed) Codes and ciphers: cryptography and coding IV, proceedings of 4th IMA conference on cryptography and coding. IMA Press, Berlin
Shannon C (1948) A mathematical theory of communication. Bell Syst Tech J 27(3):379–423, 623–656
Shor P (1994) Algorithms for quantum computation: Discrete logarithms and factoring. In: SFCS’94 proceedings of the 35th annual symposium on foundations of computer science. IEEE Computer Society, Washington
Shoup V (2004) Sequences of games: a tool for taming complexity in security proofs. IACR cryptology ePrint archive: report 2004/332
Smart N (2003) Cryptography: an introduction, 3rd edn. McGraw-Hill College, New York
Stebila D (2014) An introduction to provable security. Lecture notes from AMSI winter school on cryptography. https://www.douglas.stebila.ca/teaching/amsi-winter-school/
Stinson D (1997) Cryptography: theory and practice. CRC Press, Boca Raton
Vernam GS (1926) Cipher printing telegraph systems for secret wire and radio telegraphic communications. J Am Inst Electron Eng 55:109–115
Wolf S (1998) Unconditional security in cryptography. In: Damgard I (ed) Lectures on data security, modern cryptology in theory and practice. Springer, Berlin
Acknowledgements
This paper is affectionately dedicated to Pedro, who enthusiastically lead the first steps of so many students in the information theory pathways. Authors 2,3 and 4 have been partially supported by project MTM2013-45588-C3-1-P. Authors 2 and 3 have been partially supported by project GRUPIN 14-142, Principado de Asturias.
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González Vasco, M.I., González, S., Martínez, C., Suárez Corona, A. (2018). The Roll of Dices in Cryptology. In: Gil, E., Gil, E., Gil, J., Gil, M. (eds) The Mathematics of the Uncertain. Studies in Systems, Decision and Control, vol 142. Springer, Cham. https://doi.org/10.1007/978-3-319-73848-2_46
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