Abstract
There are many situations where one wants to know if a large number n is prime. For example, in the RSA public key cryptosystem and in various cryptosystems based on the discrete log problem in finite fields, we need to find a large “random” prime. One interpretation of what this means is to choose a large odd integer \(n_{0}\) using a generator of random digits and then test \(n_{0}, n_{0}+2, \dotsc\) for primality until we obtain the first prime which is \(\ge n_{0}\). A second type of use of primality testing is to determine whether an integer of a certain very special type is a prime. For example, for some large prime f we might want to know whether \(2^{f}-1\) is a Mersenne prime. If we’re working in the field of \(2^{f}\) elements, we saw that every element ≠ 0, 1 is a generator of \({\pmb{\text{F}}}^{*}_{2^{f}}\) if (and only if) \(2^{f}-1\) is prime (see Exercise 13(a) of §11.1).
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References for § V.1
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© 1987 Springer-Verlag New York Inc.
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Koblitz, N. (1987). Primality and Factoring. In: A Course in Number Theory and Cryptography. Graduate Texts in Mathematics, vol 114. Springer, New York, NY. https://doi.org/10.1007/978-1-4684-0310-7_5
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DOI: https://doi.org/10.1007/978-1-4684-0310-7_5
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