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Pollard's rho algorithm for logarithms is an algorithm introduced by John Pollard in 1978 to solve the discrete logarithm problem, analogous to Pollard's rho algorithm to solve the integer factorization problem.
In cryptography, the computational complexity of the discrete logarithm problem, along with its application, was first proposed in the Diffie–Hellman problem. Several important algorithms in public-key cryptography , such as ElGamal , base their security on the hardness assumption that the discrete logarithm problem (DLP) over carefully ...
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms.Dedicated to the discrete logarithm in (/) where is a prime, index calculus leads to a family of algorithms adapted to finite fields and to some families of elliptic curves.
Symbolab is an answer engine [1] that provides step-by-step solutions to mathematical problems in a range of subjects. [2] It was originally developed by Israeli start-up company EqsQuest Ltd., under whom it was released for public use in 2011. In 2020, the company was acquired by American educational technology website Course Hero. [3] [4]
The computation solve DLP in the 1551-bit field GF(3 6 · 163), taking 1201 CPU hours. [ 21 ] [ 22 ] in 2012 by a joint Fujitsu, NICT, and Kyushu University team, that computed a discrete logarithm in the field of 3 6 · 97 elements and a size of 923 bits, [ 23 ] using a variation on the function field sieve and beating the previous record in a ...
There are two other well known algorithms that solve the discrete logarithm problem in sub-exponential time: the index calculus algorithm and a version of the Number Field Sieve. [5] In their easiest forms both solve the DLP in a finite field of prime order but they can be expanded to solve the DLP in as well.
The DLP, as it has come to be known, is a widely used approach to public key cryptography, and the difficulty in solving this problem determines the level of security of the cryptosystem. This article covers algorithms to count points on elliptic curves over fields of large characteristic, in particular p > 3.
The primary difference between a computer algebra system and a traditional calculator is the ability to deal with equations symbolically rather than numerically. The precise uses and capabilities of these systems differ greatly from one system to another, yet their purpose remains the same: manipulation of symbolic equations.