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square root: four different dependencies were run in parallel on four 250 MHZ SGI Origin 2000 processors at CWI; three of them found the factors of RSA-140 after 14.2, 19.0 and 19.0 CPU-hours eleven weeks (including four weeks for polynomial selection, one month for sieving, one week for data filtering and matrix construction, five days for the ...
The RSA Factoring Challenge was a challenge put forward by RSA Laboratories on March 18, ... 140 463 US$17,226 February 2, 1999 Herman te Riele et al. RSA150: 150
Multiply the corresponding a i together and give the result mod n the name a; similarly, multiply the b i together which yields a B-smooth square b 2. We are now left with the equality a 2 = b 2 mod n from which we get two square roots of (a 2 mod n), one by taking the square root in the integers of b 2 namely b, and the other the a computed in ...
The RSA problem is defined as the task of taking e th roots modulo a composite n: recovering a value m such that c ≡ m e (mod n), where (n, e) is an RSA public key, and c is an RSA ciphertext. Currently the most promising approach to solving the RSA problem is to factor the modulus n .
More specifically, the RSA problem is to efficiently compute P given an RSA public key (N, e) and a ciphertext C ≡ P e (mod N). The structure of the RSA public key requires that N be a large semiprime (i.e., a product of two large prime numbers ), that 2 < e < N , that e be coprime to φ ( N ), and that 0 ≤ C < N .
As far as is known, this is not possible using classical (non-quantum) computers; no classical algorithm is known that can factor integers in polynomial time. However, Shor's algorithm shows that factoring integers is efficient on an ideal quantum computer, so it may be feasible to defeat RSA by constructing a large quantum computer.
As for why this happens: "The previous challenge, RSA-140, was factored earlier this year. RSA-155 was significant because it matched a common benchmark, 512 bits. RSA-150 hasn't been done yet, since RSA-155 was the more interesting milestone, even though RSA-150 would have been a little easier (since it's shorter than RSA-155)."
The Gaussian integers are complex numbers of the form α = u + vi, where u and v are ordinary integers [note 2] and i is the square root of negative one. [142] By defining an analog of the Euclidean algorithm, Gaussian integers can be shown to be uniquely factorizable, by the argument above . [ 43 ]