This booklet is principally dedicated to a few computational and algorithmic difficulties in finite fields corresponding to, for instance, polynomial factorization, discovering irreducible and primitive polynomials, the distribution of those primitive polynomials and of primitive issues on elliptic curves, developing bases of assorted forms and new functions of finite fields to different parts of arithmetic. For completeness we in clude certain chapters on a few contemporary advances and functions of the speculation of congruences (optimal coefficients, congruential pseudo-random quantity gener ators, modular mathematics, etc.) and computational quantity conception (primality checking out, factoring integers, computation in algebraic quantity thought, etc.). the issues thought of the following have many functions in machine technological know-how, Cod ing concept, Cryptography, Numerical equipment, etc. There are a couple of books dedicated to extra common questions, however the effects contained during this booklet haven't until eventually now been accrued lower than one conceal. within the current paintings the writer has tried to show new hyperlinks between varied components of the speculation of finite fields. It includes many extremely important effects which formerly will be discovered in simple terms in broadly scattered and rarely to be had convention complaints and journals. particularly, we broadly overview effects which initially seemed simply in Russian, and aren't popular to mathematicians outdoor the previous USSR.

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