Search results
Results From The WOW.Com Content Network
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 Tonelli–Shanks algorithm (referred to by Shanks as the RESSOL algorithm) is used in modular arithmetic to solve for r in a congruence of the form r 2 ≡ n (mod p), where p is a prime: that is, to find a square root of n modulo p.
A method analogous to piece-wise linear approximation but using only arithmetic instead of algebraic equations, uses the multiplication tables in reverse: the square root of a number between 1 and 100 is between 1 and 10, so if we know 25 is a perfect square (5 × 5), and 36 is a perfect square (6 × 6), then the square root of a number greater than or equal to 25 but less than 36, begins with ...
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.
For encryption, a square modulo n must be calculated. This is more efficient than RSA, which requires the calculation of at least a cube. For decryption, the Chinese remainder theorem is applied, along with two modular exponentiations. Here the efficiency is comparable to RSA.
It is useful in computer science, especially in the field of public-key cryptography, where it is used in both Diffie–Hellman key exchange and RSA public/private keys. Modular exponentiation is the remainder when an integer b (the base) is raised to the power e (the exponent), and divided by a positive integer m (the modulus); that is, c = b ...
RSA Laboratories stated: "Now that the industry has a considerably more advanced understanding of the cryptanalytic strength of common symmetric-key and public-key algorithms, these challenges are no longer active." [6] When the challenge ended in 2007, only RSA-576 and RSA-640 had been factored from the 2001 challenge numbers. [7]