By Henri Cohen

A description of 148 algorithms primary to number-theoretic computations, particularly for computations relating to algebraic quantity concept, elliptic curves, primality trying out and factoring. the 1st seven chapters advisor readers to the guts of present study in computational algebraic quantity concept, together with contemporary algorithms for computing type teams and devices, in addition to elliptic curve computations, whereas the final 3 chapters survey factoring and primality checking out tools, together with an in depth description of the quantity box sieve set of rules. the complete is rounded off with an outline of accessible machine applications and a few valuable tables, sponsored through a variety of routines. Written through an expert within the box, and one with nice useful and instructing adventure, this can be bound to turn into the normal and integral reference at the topic.

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Extra resources for A Course in Computational Algebraic Number Theory (Graduate Texts in Mathematics, Volume 138)

Sample text

We can do this by slightly changing step 4 so as to always end up with an odd value of b. This however may have disastrous effects on the running time, which may become exponential instead of polynomial time (see [Bac-Sha] and Exercise 24). 10 can be slightly improved (by a small constant factor) by adding the following statement at the end of the assignments of step 4, before going back to step 3: If a > r/2, then a = a - r. e. between -r/2 and r/2. This modification could also be used in Euclid's algorithms if desired, if tests suggest that it is faster in practice.

Yare encountered. 3 Euclid's Algorithms We now consider the problem of computing the GCD of two integers a and b. The naIve answer to this problem would be to factor a and b, and then multiply together the common prime factors raised to suitable powers. Indeed, this method works well when a and b are very small, say less than 100, or when a or b is known to be prime (then a single division is sufficient). In general this is not feasible, because one of the important facts of life in number theory is that factorization is difficult and slow.

That a is a quadratic non-residue mod p). 1. [Find generator] Choose numbers n at random until (~) = -1. Then set z f - n q (mod p). 2. [Initialize] Set y f x f - ax (mod p). Z, r f- e, x f- a(q-l)/2 (mod p), b f- ax 2 (mod p), 3. [Find exponent] If b == 1 (mod p), output x and terminate the algorithm. Otherwise, find the smallest m ~ 1 such that b2m == 1 (mod p). If m = r, output a message saying that a is not a quadratic residue mod p. 4. [Reduce exponent] Set t f - y2 r - m - 1 , y f - t 2, r operations done modulo p), and go to step 3.