Search results
Results From The WOW.Com Content Network
The general number field sieve, on the other hand, manages to search for smooth numbers that are subexponential in the size of n. Since these numbers are smaller, they are more likely to be smooth than the numbers inspected in previous algorithms. This is the key to the efficiency of the number field sieve.
Number field sieve (NFS) is an integer factorization method, it can be: General number field sieve (GNFS): Number field sieve for any integer Special number field sieve (SNFS): Number field sieve for integers of a certain special form
For example, 15 is a composite number because 15 = 3 · 5, but 7 is a prime number because it cannot be decomposed in this way. If one of the factors is composite, it can in turn be written as a product of smaller factors, for example 60 = 3 · 20 = 3 · (5 · 4) .
The second-fastest is the multiple polynomial quadratic sieve, and the fastest is the general number field sieve. The Lenstra elliptic-curve factorization is named after Hendrik Lenstra . Practically speaking, ECM is considered a special-purpose factoring algorithm, as it is most suitable for finding small factors.
In number theory, a branch of mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from it. The special number field sieve is efficient for integers of the form r e ± s, where r and s are small (for instance Mersenne numbers).
The quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve). It is still the fastest for integers under 100 decimal digits or so, and is considerably simpler than the number field sieve. It is a general-purpose factorization algorithm, meaning ...
If a quantum computer with a sufficient number of qubits could operate without succumbing to quantum noise and other quantum-decoherence phenomena, then Shor's algorithm could be used to break public-key cryptography schemes, such as The RSA scheme; The Finite Field Diffie-Hellman key exchange; The Elliptic Curve Diffie-Hellman key exchange [10]
Smooth numbers have a number of applications to cryptography. [11] While most applications center around cryptanalysis (e.g. the fastest known integer factorization algorithms, for example: General number field sieve algorithm), the VSH hash function is another example of a constructive use of smoothness to obtain a provably secure design.