Search results

although there are many different ways of finding a factorization using an integer factorization algorithm, they all must produce the same result. Primes...

every unique factorization domain, every Gaussian integer may be factored as a product of a unit and Gaussian primes, and this factorization is unique up...

in several integer factorization algorithms that have applications in cryptography, such as Lenstra elliptic-curve factorization. The use of elliptic...

proven that none exists; see integer factorization for a discussion of this problem. The first RSA-512 factorization in 1999 used hundreds of computers...

the covering about the branch point. This leads to a transitive factorization. The moduli space M ¯ g , n {\displaystyle {\overline {\mathcal {M}}}_{g,n}}...

essential step in several integer factorization algorithms, such as Pollard's rho algorithm, Shor's algorithm, Dixon's factorization method and the Lenstra elliptic...

used for composite moduli: finding square roots modulo composite numbers is a computational problem equivalent to integer factorization. An equivalent, but...

splitting circle method is a numerical algorithm for the numerical factorization of a polynomial and, ultimately, for finding its complex roots. It was...

arbitrary order (hence its Frobenius norm, squared, is the sum of the squared moduli of the eigenvalues of A, while the Frobenius norm of A, squared, is the...

injective and that its image is a hypersurface, and finally a polynomial factorization to prove the irreducibility of the image. The set of n-by-n matrices...

residues (modulo the number being factorized) in an attempt to find a congruence of squares which will yield a factorization. The number field sieve is the...

\mathbf {Z} \left[{\frac {1+{\sqrt {-163}}}{2}}\right]} is a unique factorization domain. Here ( 1 + − 163 ) / 2 {\displaystyle (1+{\sqrt {-163}})/2}...

called the moduli. This representation is allowed by the Chinese remainder theorem, which asserts that, if M is the product of the moduli, there is, in...

example, if Vect n {\displaystyle \operatorname {Vect} _{n}} denotes the moduli stack of rank-n vector bundles, then there is a presentation Spec ⁡ ( k...

birational invariant for smooth complex projective varieties. The "Weak factorization theorem", proved by Abramovich, Karu, Matsuki, and Włodarczyk (2002)...

coefficients in intermediate calculations and data. It is used in polynomial factorization, a problem for which all known efficient algorithms use modular arithmetic...

composite numbers. Factorization is a method of expressing a number as a product. Specifically in number theory, integer factorization is the decomposition...

method that allows to "lift" the factorization modulo p of a polynomial with integer coefficients to a factorization modulo p n {\textstyle p^{n}} for...

Cipolla, an Italian mathematician who discovered it in 1907. Apart from prime moduli, Cipolla's algorithm is also able to take square roots modulo prime powers...

October 2024. The Return of Coppersmith’s Attack: Practical Factorization of Widely Used RSA Moduli Archived 2017-11-12 at the Wayback Machine, Matus Nemec...