March 15, 2018

# An Introduction to the General Number Field Sieve - download pdf or read online By Briggs M.

The overall quantity box Sieve (GNFS) is the quickest recognized technique 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 procedure, probably the most usual tools for transmitting and receiving mystery information. in truth, GNFS used to be used lately to issue a 130-digit "challenge" quantity released by means of RSA, the most important variety of cryptographic value ever factored.

Read Online or Download An Introduction to the General Number Field Sieve PDF

Similar introduction books

Christophe Bobda (auth.), Christophe Bobda (eds.)'s Introduction to Reconfigurable Computing: Architectures, PDF

“Introduction to Reconfigurable Computing” offers a entire learn of the sphere Reconfigurable Computing. It offers an access aspect to the beginner keen to maneuver within the examine box reconfigurable computing, FPGA and procedure on programmable chip layout. The publication is usually used as educating reference for a graduate path in desktop engineering, or as connection with boost electric and computing device engineers.

Extra info for An Introduction to the General Number Field Sieve

Example text

Fortunately this problem is addressed by a solution to another problem that crops up when adapting the Lanczos method for use in GNFS, outlined below. 3, the goal is to find a dependency among the columns of the matrix B, which amounts to finding a non-trivial vector x such that B · x = 0. 4 is the zero vector in this case. 4 is the trivial vector x = 0. 4 can fail with binary vectors. 4 is adapted to a “block” scheme that works with subspaces of vectors instead of individual vectors. First, the matrix A is formed as A = B T B as alluded to earlier.

Experimentation and experience  have dictated that for factoring an integer with more than 110 digits, the degree d be set to 5. For integers between 50 and 80 digits a value of 3 for d is used. 6, early implementations of GNFS restricted d to an odd integer. In this case, d = 5 is usually substituted for d = 4. Having selected a value for d, the choice of f(x) and m is usually made simultaneously. First m is chosen with m ≈ n1/d and such that the quotient of n divided by md is exactly one. A “base-m” expansion [5, Section 3] of n then gives n = md + ad−1md−1 + · · · + a1m + a0 with coefficients 0 ≤ ai < m for 0 ≤ i < d.

1) i=0 Proof. 3] from abstract algebra that the non-zero elements of a field form a group under multiplication. In this case, that means the p − 1 non-zero elements of Z /pZ form a finite group of order p − 1 under multiplication. Then for any 0 < a < p it follows that ap−1 ≡ 1 (mod p) and therefore ap ≡ a (mod p) for all a with 0 ≤ a < p. Rearranging the last congruence yields ap − a ≡ 0 (mod p) and therefore a is seen to be a root of xp − x (mod p) for 0 ≤ a < p. This determines p roots for xp − x (mod p).