Abstract
In the previous chapters we have introduced the most important aspects of private-key cryptography and we have noticed that prime numbers underlie many of the constructions and algorithms discussed. Also, computational number-theoretic problems which are presumed to be hard made their appearance and we mentioned, in particular, the integer factorization problem and the discrete logarithm problem. In the coming chapters we will study publickey cryptography and we will see that all these aspects play a relevant role in this setting. In fact, number theory and, in particular, presumedly hard number-theoretic problems such as the ones just mentioned, are of central importance for public-key cryptography because the security of most public-key schemes relies on the hardness of some of these problems. The study of the known algorithms to solve these hard problems can thus be seen as a form of cryptanalysis and, as such, is an indispensable complement to cryptography and a prerequisite for the practical evaluation of the security of public-key schemes in order to establish, for example, the key sizes that should be used. Thus we devote this chapter to the aspects of algorithmic number theory which are most relevant to public-key cryptographic schemes.
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Notes
- 1.
This is the so-called ‘American’ version of the logarithmic integral, while the ‘European’ version \({\text{ li}}(x) = \int _2^x \frac{dt}{\ln t}\) is related to it by the formula \({\text{ li}}(x) = {\text{ Li}}(x) - {\text{ Li}}(2) = {\text{ Li}}(x) - 1.045163780\) (the notations \({\text{ Li}}\) and \({\text{ li}}\) are usually interchanged but we use \({\text{ Li}}\) this way to conform to Maple’s usage).
- 2.
The term prime gap refers to the difference between two consecutive primes.
- 3.
This conjecture postulates that Sophie Germain primes are fairly frequent, see [180, Conjecture 5.24] and the discussions in [180, 21.2, 21.3] and [60, 4.5.1, 4.5.2]. There is some heuristic evidence in favor of this hypothesis, however it is not even known whether or not there are infinitely many of these primes.
- 4.
Regarding this probability we can mention the following quote from Borel cited in [118, p. 126]: Un phénomène dont la probabilité est \(10^{-50}\) ne se produira donc jamais, ou du moins ne sera jamais observé.
- 5.
Let us assume here that the mathematical proof gives absolute certainty, something that is by no means obvious and is subject to much philosophical debate.
- 6.
This problem highlights again the difference between primality testing and finding random primes.
- 7.
To overcome this difficulty a more efficient encoding of the binary matrix—such as storing eight entries per byte—should be used or, better still, sparse encoding in the form of a listing where the positions of the 1s appear, since most of the coefficients are 0. This also requires the use of a sparse-matrix method to compute the null space.
- 8.
Maple’s built-in function numtheory:-mlog computes modular discrete logarithms but we give explicit implementations of the different algorithms for this purpose. In Chap. 11, some of these implementations are adapted to deal with elliptic curve discrete logarithms.
- 9.
As mentioned, the current discrete logarithm record in an elliptic curve group, where only generic methods are available, is in a group whose order is a \(112\)-bit prime but, in the case of \(\mathbb Z _p^*\), the index calculus methods such as NFS—see Sect. 6.5.5—are able to deal with much larger primes.
- 10.
We will not explain these methods here but note that they are not necessary if \(p\) is a safe prime because then \(p-1\) is square-free.
- 11.
In fact, both rings share an even stronger property, namely both are ‘Euclidean domains’ since they have a ‘Euclidean function’ which provides a division algorithm and a Euclidean algorithm.
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© 2013 Springer-Verlag Berlin Heidelberg
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Gómez Pardo, J.L. (2013). Algorithmic Number Theory for Cryptography and Cryptanalysis: Primality, Factoring and Discrete Logarithms. In: Introduction to Cryptography with Maple. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-32166-5_6
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DOI: https://doi.org/10.1007/978-3-642-32166-5_6
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