Search results

...partial repeats: the search pattern is split into two parts (its critical factorization), represented only by the position of that split. Being a number less than ...libc/string/str-two-way.h.html}}</ref> As with most advanced string-search algorithms, the naïve implementation may be more efficient on small-enough instances;< ...

...ion|scalar multiple]]s of {{Mvar|P}}, say {{Math|[''n'']''P''}} for some [[integer]] {{Mvar|n}}, and they are based on the [[Exponentiation by squaring|double ==Algorithms and examples== ...

The algorithm does not require the factorization of the modulus, and uses modular operations that are often easy when the gi ...es in his original paper solve this equation by having <math>C</math> be a integer square and thus setting <math>z</math> to zero. ...

...innerton-Dyer]], that serve as examples where [[polynomial factorization]] algorithms have worst-case runtime. They have the property of being [[reducible polyn ...m. The polynomial <math>f_P(x)</math> has degree <math>2^{|P|}</math> and integer coefficients, which alternate in sign. If <math>|P|>1</math>, then <math>f ...

...r their small zeroes [[Modular arithmetic#Integers modulo m|modulo a given integer]]. The method uses the [[Lenstra–Lenstra–Lovász lattice basis reduction alg integer <math>|x_0|< M^{1/n}</math>. ...

...mbers (VSN).<ref name="main" /> This is assumed to be as hard as [[Integer factorization|factoring integers]]. For fixed constants {{Mvar|c}} and {{Mvar|n}}, an integer {{Mvar|m}} is a ''Very Smooth Number'' (VSN) if the largest prime factor of ...

{{broader|Root-finding algorithms}} {{section too long|date=January 2023|discuss=Talk:Polynomial root-finding algorithms#Section "Principles"}} ...

The test uses a partial factorization of <math>N - 1</math> to prove that an integer <math>N</math> is [[Prime number|prime]]. ...ith less effort than the [[Lucas primality test]], which requires the full factorization of <math>N - 1</math>. ...

There are many polynomial-time algorithms for list decoding. In this article, we first present an algorithm for [[RS Here is a plot of the rate R and distance <math> \delta</math> for different algorithms. ...

...l base for [[public-key cryptography]] in the future just as the [[integer factorization]] and [[discrete logarithm]] problem have served as the base for public key ...of two 384-bit primes but not the product of two 512-bit primes. [[Integer factorization]] forms the basis of the widely used [[RSA (cryptosystem)|RSA]] cryptograph ...

...function]] that gives the number of representations for a given positive [[integer]] {{math|''n''}} as the sum of {{math|''k''}} [[Square number|squares]], wh ...g the sum of two squares theorem are squares of possible distances between integer lattice points; values up to 100 are shown, with ...

...91 | pages = 268–275 | publisher = Springer | location = Berlin | series = Algorithms and Combinatorics | title = The mathematics of Paul Erdős, I | volume = 13 ...

...family of inequalities concerning a [[univariate polynomial|univariate]] [[integer polynomial]] ''f''(''x'') and one of its factors ''h''(''x''). A basic vers ...univariate [[complex polynomial]]s which later will be restricted to be [[integer polynomial]]s, i.e. in <math>\mathbb{Z}[x]</math>. Explicitly ...

...that there is a [[Modular multiplicative inverse|modular inverse]]. The [[factorization]] of ''N'' and the private key ''d'' are kept secret, so that only Bob can ...thm]], one can efficiently recover the secret key ''d'' if one knows the [[factorization]] of ''N''. By having the secret key ''d'', one can efficiently factor the ...

...concept of using [[Lenstra elliptic curve factorization|elliptic curves in factorization]] had been developed by [[H. W. Lenstra]] in 1985, and the implications for ...r factorization#Time complexity|become unwieldy with large input]]; modern algorithms treat the problems of determining whether a number is prime and what its fa ...

Linear integer equations. 69 rules and 36 examples.<ref name=mii27/> Contains factorization method,<ref name=mii27/> 11 rules and 7 examples.<ref name=mii27/> ...

...first3=Chiara |last4=Mosca |first4=Michele |date=1998-01-08 |title=Quantum Algorithms Revisited |url=http://arxiv.org/abs/quant-ph/9708016 |journal=Proceedings o ...l part of many quantum algorithms, including Shor’s algorithm, for integer factorization. ...

...root, one does not know whether it is because there is no real root. Some algorithms compute all complex roots, but, as there are generally much fewer real root ...ation algorithms began to be implemented on [[computer]]s it appeared that algorithms derived from Sturm's theorem are less efficient than those derived from [[D ...

...attack by a quantum cryptography. Several post quantum digital signature algorithms based on hard problems in lattices are being created replace the commonly u ...all of the widely used [[Public-key cryptography|public key]] cryptography algorithms used to protect privacy and digitally sign information on the internet.<ref ...

{{short description|Algorithms for polynomial evaluation}} ...lies <math>T(n)=O(n(\log n)^2)</math> by the [[Master theorem (analysis of algorithms)|master theorem]]. ...