By Henri Cohen

A description of 148 algorithms primary to number-theoretic computations, particularly for computations with regards to algebraic quantity conception, elliptic curves, primality trying out and factoring. the 1st seven chapters advisor readers to the guts of present study in computational algebraic quantity thought, together with contemporary algorithms for computing classification 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 desktop applications and a few precious tables, subsidized via various workouts. Written through an expert within the box, and one with nice useful and instructing event, this is often guaranteed to develop into the normal and critical reference at the topic.

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Example text

First, it is not well suited to handling large numbers (in our sense, say numbers with 50 or 100 decimal digits). This is because each Euclidean step requires a long division, which takes time O(ln2 N). When carelessly programmed, the algorithm takes time O(ln3 N). If, however, at each step the precision is decreased as a function of a and b, and if one also notices that the time to compute a Euclidean step a = bq + r is O((ln a) (In q + 1)), then the total time is bounded by O((lnN)((2:lnq) + O(lnN))).

Then it is easy to see that for a given integer a, the congruence (mod p) can have either no solution (we say in this case that a is a quadratic nonresidue mod p), one solution if a == 0 (mod p), or two solutions (we then say that a is a quadratic residue mod p). Define the Legendre symbol (~) as being -1 if a is a quadratic non-residue, 0 if a = 0, and 1 if a is a quadratic residue. Then the number of solutions modulo p of the above congruence is 1 + (~). g. 6. e. (~) G) = (~) In particular, the product of two quadratic non-residues is a quadratic residue.

This leads to the following algorithm. Here we assume that we have an algorithm digit(k, N, 1) which gives digit number f of N expressed in base 2k. 4 (Left-Right Base 2k). Given 9 E G and nEZ, this algorithm computes gn in G. If n i= 0, we assume also given the unique integer e such that 2ke :::; Inl < 2k(e+l). We write 1 for the unit element of G. 1. [Initialize] If n = 0, output 1 and terminate. If n < 0 set N z +--- g-l. Otherwise, set N +--- nand z +--- 9 and f +--- e. 2. [Precomputations] Compute and store +--- -n and z3, z5, ...