By Nigel Smart

In this introductory textbook the writer explains the foremost issues in cryptography. he's taking a contemporary process, the place defining what's intended via "secure" is as very important as growing anything that achieves that aim, and protection definitions are relevant to the dialogue throughout.

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Biggs. Discrete Mathematics. Oxford University Press, 1989. H. Rosen. Discrete Mathematics and Its Applications. McGraw-Hill, 1999. CHAPTER 2 Primality Testing and Factoring Chapter Goals • • • • • To explain the basics of primality testing. To describe the most used primality-testing algorithm, namely Miller–Rabin. To examine the relationship between various mathematical problems based on factoring. To explain various factoring algorithms. To sketch how the most successful factoring algorithm works, namely the Number Field Sieve.

In the elliptic curve variant of ElGamal we require an elliptic curve over a ﬁnite ﬁeld, such that the order of the elliptic curve is divisible by a large prime q. Luckily we shall see that testing a number for primality can be done very fast using very simple code, but with an algorithm that has a probability of error. By repeating this algorithm we can reduce the error probability to any value that we require. Some of the more advanced primality-testing techniques will produce a certiﬁcate which can be checked by a third party to prove that the number is indeed prime.

Then we compute √ si ← a (mod pi ) for 1 ≤ i ≤ k. 3 from Chapter 1). Then we compute the value of x using the Chinese Remainder Theorem on the data (s1 , p1 ), . . , (sk , pk ). We have to be a little careful if powers of pi greater than one divide N . However, this is easy to deal with and will not concern us here, since we are mainly interested in integers N which are the product of two primes. Hence, ﬁnding square roots modulo N is no harder than factoring. 5; where we have specialized the game to one of integers N which are the product of two prime factors.