Briggs M.'s An Introduction to the General Number Field Sieve PDF

By Briggs M.

The overall quantity box Sieve (GNFS) is the quickest identified process for factoring "large" integers, the place huge is usually taken to intend over a hundred and ten digits. This makes it the simplest set of rules for trying to unscramble keys within the RSA [2, bankruptcy four] public-key cryptography method, essentially the most time-honored tools for transmitting and receiving mystery info. in truth, GNFS was once used lately to issue a 130-digit "challenge" quantity released through RSA, the most important variety of cryptographic value ever factored.

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2, if the polynomial f(x) is irreducible over Z /pZ then f(x) divides any polynomial g(x), modulo p, which shares a root with f(x). 2 the finite field F q with q = pd elements can be represented as elements of F p (θp) where θp is a root of f(x) in the splitting field of f(x) over Z /pZ . 1. 1 it follows that f(x) does as well. Hence f(x) has d distinct roots in the splitting field of f(x) over Z /pZ . Furthermore, the splitting field F q of xq − x contains all these roots of f(x). 2 carry over to the finite F q , in that there are exactly d automorphisms defined on F q , each of which sends θp to a distinct root of f(x) in F q .

1. 1 it follows that f(x) does as well. Hence f(x) has d distinct roots in the splitting field of f(x) over Z /pZ . Furthermore, the splitting field F q of xq − x contains all these roots of f(x). 2 carry over to the finite F q , in that there are exactly d automorphisms defined on F q , each of which sends θp to a distinct root of f(x) in F q . 1. Let f(x) be a monic polynomial of degree d with integer coefficients that is irreducible modulo p for some prime integer p. 6) where the σi are the distinct automorphisms of F q which map θp to the d distinct roots of f(x) in F q .

Thus, values xi = x (mod pi ) may be computed easily assuming that βp may be calculated from δp in F p (θp), which turns the problem into one of extracting square roots in a finite field efficiently. 8. It could so happen that for primes pi and pj for which f(x) is irreducible that different square roots of δ are computed in the fields F pi (θpi ) and F pi (θpj ). More explicitly, it could be that βpi is computed in one field and −βpj and the other, both of which are valid square roots of δ in the respective fields.

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An Introduction to the General Number Field Sieve by Briggs M.


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