mathematics the Function Field Sieve is one of the most efficient algorithms to solve the Discrete Logarithm Problem (DLP) in a finite field. It has heuristic...

In number theory, the general number field sieve (GNFS) is the most efficient classical algorithm known for factoring integers larger than 10100. Heuristically...

quadratic sieve algorithm (QS) is an integer factorization algorithm and, in practice, the second-fastest method known (after the general number field sieve)....

mathematics, the special number field sieve (SNFS) is a special-purpose integer factorization algorithm. The general number field sieve (GNFS) was derived from...

In mathematics, the sieve of Eratosthenes is an ancient algorithm for finding all prime numbers up to any given limit. It does so by iteratively marking...

mathematics, the sieve of Atkin is a modern algorithm for finding all prime numbers up to a specified integer. Compared with the ancient sieve of Eratosthenes...

{\displaystyle p} is large compared to q {\displaystyle q} , the function field sieve, L q [ 1 / 3 , 32 / 9 3 ] {\textstyle L_{q}\left[1/3,{\sqrt[{3}]{32/9}}\...

Function field may refer to: Function field of an algebraic variety Function field (scheme theory) Algebraic function field Function field sieve Function...

sophisticated sieves also do not work directly with sets per se, but instead count them according to carefully chosen weight functions on these sets (options...

prime. A prime sieve or prime number sieve is a fast type of algorithm for finding primes. There are many prime sieves. The simple sieve of Eratosthenes...

theorem Brun sieve Function field sieve General number field sieve Large sieve Larger sieve Quadratic sieve Selberg sieve Sieve of Atkin Sieve of Eratosthenes...

In mathematics, the sieve of Sundaram is a variant of the sieve of Eratosthenes, a simple deterministic algorithm for finding all the prime numbers up...

variant of the medium-sized base field function field sieve, for binary fields, to compute a discrete logarithm in a field of 21971 elements. In order to...

the size of the group). Baby-step giant-step Function field sieve Index calculus algorithm Number field sieve Pohlig–Hellman algorithm Pollard's rho algorithm...

Fermat's factorization method are the basis of the quadratic sieve and general number field sieve, the best-known algorithms for factoring large semiprimes...

exponent e when given b, c, and m – is believed to be difficult. This one-way function behavior makes modular exponentiation a candidate for use in cryptographic...

completed with a highly optimized implementation of the general number field sieve run on hundreds of machines. No algorithm has been published that can...

the rational sieve is a general algorithm for factoring integers into prime factors. It is a special case of the general number field sieve. While it is...

such cases other methods are used such as the quadratic sieve and the general number field sieve (GNFS). Because these methods also have superpolynomial...

suffices to replace everywhere 10 by 2. The second argument of the split_at function specifies the number of digits to extract from the right: for example,...

Observations analogous to the preceding can be applied recursively, giving the Sieve of Eratosthenes. One way to speed up these methods (and all the others mentioned...

{\tilde {O}}(\log(n)^{10.5})} , later reduced using additional results from sieve theory to O ~ ( log ⁡ ( n ) 7.5 ) {\displaystyle {\tilde {O}}(\log(n)^{7...

 0) = rN−1. function gcd(a, b) if b = 0 return a else return gcd(b, a mod b) (As above, if negative inputs are allowed, or if the mod function may return...

giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve Greatest common divisor Binary Euclidean Extended Euclidean Lehmer's...

bound for the number of Carmichael numbers is lower than the prime number function n/log(n)) there are enough of them that Fermat's primality test is not...

Mathematica as the function LatticeReduce Number Theory Library (NTL) as the function LLL PARI/GP as the function qflll Pymatgen as the function analysis...

giant-step Pollard rho Pollard kangaroo Pohlig–Hellman Index calculus Function field sieve Greatest common divisor Binary Euclidean Extended Euclidean Lehmer's...

most efficient known classical factoring algorithm, the general number field sieve, which works in sub-exponential time: O ( e 1.9 ( log ⁡ N ) 1 / 3 ( log...

compute the multiplicative inverse in algebraic field extensions and, in particular in finite fields of non prime order. It follows that both extended...

\end{aligned}}} where the first equality uses the Binomial Theorem in a finite field, which is ( x + y ) M p ≡ x M p + y M p ( mod M p ) {\displaystyle (x+y)^{M_{p}}\equiv...