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.
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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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