Cryptography Made Simple by Nigel Smart

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.

The chapters partially 1 supply a quick creation to the mathematical foundations: modular mathematics, teams, finite fields, and likelihood; primality checking out and factoring; discrete logarithms; elliptic curves; and lattices. half 2 of the ebook indicates how ancient ciphers have been damaged, hence motivating the layout of contemporary cryptosystems because the Nineteen Sixties; this half additionally encompasses a bankruptcy on information-theoretic protection. half three covers the center elements of recent cryptography: the definition of safety; sleek flow ciphers; block ciphers and modes of operation; hash services, message authentication codes, and key derivation services; the "naive" RSA set of rules; public key encryption and signature algorithms; cryptography in keeping with computational complexity; and certificate, key delivery and key contract. ultimately, half four addresses complicated prot ocols, the place the events could have diversified or maybe conflicting safeguard pursuits: mystery sharing schemes; commitments and oblivious move; zero-knowledge proofs; and safe multi-party computation.

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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.

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