By Neal Koblitz

ISBN-10: 0387942939

ISBN-13: 9780387942933

This can be a considerably revised and up to date advent to mathematics subject matters, either historical and sleek, which have been on the centre of curiosity in functions of quantity conception, really in cryptography. As such, no history in algebra or quantity conception is thought, and the publication starts with a dialogue of the elemental quantity conception that's wanted. The technique taken is algorithmic, emphasising estimates of the potency of the ideas that come up from the idea, and one targeted characteristic is the inclusion of contemporary functions of the idea of elliptic curves. broad workouts and cautious solutions are an essential component the entire chapters.

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Additional resources for A Course in Number Theory and Cryptography (2nd Edition) (Graduate Texts in Mathematics, Volume 114)

Example text

J, n) = 1. Proof. Any element of F; can be written as a power gj of the generator g. A power of 9 is 1 if and only if the power is divisible by q - 1. Thus, an element gj is an n-th root of unity if and only if nj == 0 mod q - 1. (n, q -1). 1, the equation nj == 0 mod q - 1 (with j the unknown) is equivalent to the equation Jj == 0 mod ~. Since n / d is prime to (q - 1) / d, the latter congruence is equivalent to requiring j to be a multiple of (q - 1) / d. In other words, the d distinct powers of g(q-l)/d are precisely the n-th roots of unity.

II k is a positive integer, then an element 01 F q can be raised to the k-th power in O(log k log3 q) bit operations. Proof. An element of F q is a polynomial with coefficients in F v = Z/pZ regarded modulo F(X). To multiply two such elements, we multiply the polynomials - this requires O(P) multiplications of integers modulo p (and some additions of integers modulo p, which take much less time) - and then divide the polynomial F(X) into the product, taking the remainder polynomial as our answer.

This concludes the proof. Definition. A generator g of a finite field F q is an element of order q -1; equivalently, the powers of g run through all of the elements of F~. The next proposition is one of the very basic facts about finite fields. , they are all powers of a single element. 2. Every finite field has a generator. (j, q - 1) = 1. In particular, there are a total of ip( q - 1) different generators of F~. Proof. , ad = 1 and no lower power of a gives 1. 1, d divides q - 1. Since ad is the smallest power which equals 1, it follows that the elements a, a2 , ••• , ad = 1 are distinct.

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