# quadratic sieve in a sentence

###### SentencesMobile
• This example will demonstrate standard quadratic sieve without logarithm optimizations or prime powers.
• It starts at an undergraduate level and gets up through the quadratic sieve.
• Atkins et al . used the quadratic sieve algorithm invented by Carl Pomerance in 1981.
• In such cases other methods are used such as the quadratic sieve and the general number field sieve ( GNFS ).
• This demonstration should also serve to show that the quadratic sieve is only appropriate when " n " is large.
• The quadratic sieve searches for smooth numbers using a technique called sieving, discussed later, from which the algorithm takes its name.
• As a consequence of the Chinese remainder theorem, the number 1 has at least four distinct square roots factoring algorithms like the quadratic sieve.
• The principle of the number field sieve ( both special and general ) can be understood as an improvement to the simpler rational sieve or quadratic sieve.
• The quadratic sieve speeds up the process of finding relations by taking " x " close to the square root of " n ".
• The sieve methods discussed in this article are not closely related to the integer factorization sieve methods such as the quadratic sieve and the general number field sieve.
• While the asymptotically faster number field sieve had just been invented, it was not clear at the time that it would be better than the quadratic sieve for 129-digit numbers.
• Congruences of squares are extremely useful in integer factorization algorithms and are extensively used in, for example, the quadratic sieve, general number field sieve, continued fraction factorization, and Dixon's factorization.
• The quadratic sieve consists of computing " a " 2 mod " n " for several " a ", then finding a subset of these whose product is a square.
• The block Lanczos algorithm is amongst the most efficient methods known for finding nullspaces, which is the final stage in integer factorization algorithms such as the quadratic sieve and number field sieve, and its development has been entirely driven by this application.
• Note the set "'A "'does not have to be a set of prime factors, but it is typically a proper subset of the primes as seen in the factor base of Dixon's factorization method and the Quadratic sieve.
• He also invented the block Lanczos algorithm for finding nullspace of a matrix over a finite field, which is very widely used for the quadratic sieve and number field sieve methods of factorization; he has been involved in the computations which set a number of integer factorization records.
• This approach ( called MPQS, Multiple Polynomial Quadratic Sieve ) is ideally suited for processor involved in the factorization can be given " n ", the factor base and a collection of polynomials, and it will have no need to communicate with the central processor until it is finished with its polynomials.
• This is finding a square root modulo a prime, for which there exist efficient algorithms, such as the Shanks & ndash; Tonelli algorithm . ( This is where the quadratic sieve gets its name : " y " is a quadratic polynomial in " x ", and the sieving process works like the Sieve of Eratosthenes .)
• His algorithms include : Baby-step giant-step algorithm for computing the discrete logarithm, which is useful in public-key cryptography; Shanks'square forms factorization, an integer factorization method that generalizes Fermat's factorization method; and the Tonelli Shanks algorithm that finds square roots moduli a prime, which is useful for the quadratic sieve method of integer factorization.
• Due to Schroeppel's apparent prejudice against publishing ( though he freely circulated his ideas within the research community ), and in spite of Pomerance noting that his quadratic sieve factoring algorithm owed a debt to Schroeppel's earlier work, the latter's contribution is often overlooked . ( See the section on " Smooth Numbers " on pages 1476-1477 of Pomerance's " A Tale of Two Sieves, " " Notices of the AMS ", Vol . 43, No . 12, December 1996 .)
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