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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**Example text**

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.

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

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