Read e-book online An Introduction to the General Number Field Sieve PDF

By Briggs M.

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

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Example text

3) for 0 ≤ i ≤ l − 1 and assume the condition that wi , T (wj ) = 0 for i = j. It can then be shown that any such set must be linearly independent. To see this, assume a0 w0 +· · · +al−1 wl−1 = 0 for with ai ∈ F for 0 ≤ i ≤ l − 1. Then T (a0w0 + · · · + al−1 wl−1 ) = 0 and hence for any j with 0 ≤ j ≤ l − 1 it is seen that 0 = wj , 0 = wj , T (a0w0 + · · · + al−1 wl−1 ) = wj , a0T (w0) + · · · + al−1 T (wl−1) = a0 wj , T (w0) + · · · + aj wj , T (wj ) + · · · + al−1 wj , T (wl−1) = aj wj , T (wj ) and hence aj = 0 since wj , T (wj ) = 0 and wi , T (wj ) = 0 for all i = j with 0 ≤ j ≤ l − 1.

Let α ∈ F q and β ∈ F q . Then σp (α · β) = (α · β)p = αp · β p = σp (α) · σp(β) so the multiplicative structure of F q is preserved by σp . Also, p p−1 p p−2 2 α β+ α β 1 2 p p−i i p + ··· + α β + ··· + αβ p−1 + β p i p−1 = αp + β p = σp(α) + σp(β) σp (α + β) = (α + β)p = αp + since the binomial coefficients occurring in the middle terms of the expansion of (α + β)p p i = p! (p − i)! i! are a multiples of p and hence equivalent to 0 in F q , since i < p and p−i < p for 1 ≤ i ≤ p−1. d d To show that σp is one-to-one, suppose σp (α) = αp = 0.

Then T (a0w0 + · · · + al−1 wl−1 ) = 0 and hence for any j with 0 ≤ j ≤ l − 1 it is seen that 0 = wj , 0 = wj , T (a0w0 + · · · + al−1 wl−1 ) = wj , a0T (w0) + · · · + al−1 T (wl−1) = a0 wj , T (w0) + · · · + aj wj , T (wj ) + · · · + al−1 wj , T (wl−1) = aj wj , T (wj ) and hence aj = 0 since wj , T (wj ) = 0 and wi , T (wj ) = 0 for all i = j with 0 ≤ j ≤ l − 1. But since j was arbitrary it follows that the set Wl must be linearly independent. As in the proof of Gram-Schmidt, induction will be used on subsets of W and W .

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


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