Download e-book for kindle: An Introduction to Number Theory 2 DVD Set with Guidebook by Edward B. Burger

By Edward B. Burger

ISBN-10: 1598034200

ISBN-13: 9781598034202

2 DVD set with 24 lectures half-hour every one for a complete of 720 minutes...

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Example text

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.

3 (Lehmer). Let a and b be non-negative multi-precision integers, and assume that a 2 b. This algorithm computes (a, b), using the following auxiliary variables. e. less than M), and t and r are multi-precision variables. e. 1 and terminate. Otherwise, let a (resp. b) be the single precision number formed by the highest non-zero base M digit of a (resp. b). Set A f-- I, B f-- 0, C f-- 0, D f-- 1. 1. [Initialize) If b 2. [Test quotient) If b + C = 0 or b + D = 0 go to step 4. Otherwise, set q f-- l(a + A)/(b + C)J.

In fact the number of steps can easily be seen to be comparable to that of the Euclidean algorithm, hence this algorithm is very D efficient. 6 that the LLL algorithm allows us to determine efficiently small Z-linear combinations for more than two linearly independent vectors in a Euclidean space. It does not always give an optimal solution, but, in most situations, the result are sufficiently good to be very useful. e. of invertible elements of A. It is clear that A * is a group, and also that A* = A \ {O} if and only if A is a field.

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An Introduction to Number Theory 2 DVD Set with Guidebook by Edward B. Burger

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