This is because such a b is a square root of 1 modulo N, other than 1 and-1, whose existence is guaranteed by the Chinese remainder theorem, since N is not a prime power.
42.
They are threshold secret sharing schemes, in which the shares are generated by reduction modulo the integers m _ i, and the secret is recovered by essentially solving the system of congruences using the Chinese Remainder Theorem.
43.
By the Chinese Remainder Theorem, each R / I can further be decomposed into a direct sum of submodules of the form R / P ^ i, where P ^ i is a power of a prime ideal.
44.
Moreover, as long as the polynomial factors at each stage are relatively prime ( which for polynomials means that they have no common roots ), one can construct a dual algorithm by reversing the process with the Chinese Remainder Theorem.
45.
For example, number theoretic ideas; using the fundamental theorem of arithmetic is a straightforward way, but there are also more economic approaches, such as using the pairing function combined with the Chinese remainder theorem in a sophisticated way.
46.
:Oddly enough, this sequence doesn't appear in Sloane's, but it seems easy enough to keep generating new terms with the Chinese remainder theorem . Keenan Pepper 01 : 58, 13 June 2006 ( UTC)
47.
The Chinese remainder theorem asserts that if the are pairwise coprime, and if are integers such that for every, then there is one and only one integer, such that and the remainder of the Euclidean division of by is for every.
48.
Using reconstruction techniques ( Chinese remainder theorem, rational reconstruction, etc . ) one can recover the GCD of " f " and " g " from its image modulo a number of ideals " I ".
49.
It's faster ( except in coding time, perhaps ) to just find the numbers ( the algorithm at Chinese remainder theorem is O ( N \ ln M ) or so when Tardis 19 : 17, 27 July 2006 ( UTC)
50.
By an ingenious use of the Chinese remainder theorem, we can constructively define such a recursive function \ beta ( using simple number-theoretical functions, all of which can be defined in a total recursive way ) fulfilling the specifications given above.
