By Nigel Smart

In this introductory textbook the writer explains the most important themes in cryptography. he is taking a latest method, the place defining what's intended through "secure" is as very important as developing anything that achieves that aim, and protection definitions are crucial to the dialogue throughout.

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If A is an algorithm which can factor integers of size v, then there is an eﬃcient algorithm B which can solve SQRROOT for integers of size v. In particular (A) = AdvSQRROOT (B). AdvFACTOR v v 36 2. PRIMALITY TESTING AND FACTORING Proof. Assume we are given a factoring algorithm A; we wish to show how to use this to extract square roots modulo a composite number N . Namely, given a = x2 (mod N ) we wish to compute x. First we factor N into its prime factors p1 , p2 , . . , pk , using the factoring oracle A.

Such a witness is called a proof of primality. In practice such programs are only used when we are morally certain that the number we are testing for primality is actually prime. In other words the number has already passed the Miller–Rabin Test for a number of bases and all we now require is a proof of the primality. The most successful of these primality-proving algorithms is one based on elliptic curves called ECPP (for Elliptic Curve Primality Prover). This itself is based on an older primality-proving algorithm based on ﬁnite ﬁelds due to Pocklington and Lehmer; the elliptic curve variant is due to Goldwasser and Kilian.

If a (mod 4) = 3 and b (mod 4) = 3 then j ← −j. a ← a (mod b). if b = 1 then return j. return 0. • n is the product of two primes, n = p · q: • Qn ⊂ Jn . • #Qn = #(Jn \ Qn ) = (p − 1)(q − 1)/4. The sets Qn and Jn will be seen to be important in a number of algorithms and protocols, especially in the case where n is a product of two primes. 9. Square Roots Modulo n = p · q: We now look at how to compute a square root modulo a composite number n = p·q. Suppose we wish to compute the square root of a modulo n.