Lecture

Discrete Log Problem: Pollard's Rho Method

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In course

MATH-489: Number theory II.d - Cryptography

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Description

This lecture covers the discrete log problem and introduces Pollard's rho method, which aims to find collisions in a cyclic group of order N generated by g. The method involves partitioning the group, fixing initial values, and iteratively computing solutions. By adapting Pollard's idea, the lecture explores how to factor integers and find collisions efficiently, with a focus on constant memory usage and recursive computation.

Instructor

Dimitar Petkov Jetchev

Official source

Related concepts (32)

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In number theory, integer factorization is the decomposition, when possible, of a positive integer into a product of smaller integers. If the factors are further restricted to be prime numbers, the process is called prime factorization, and includes the test whether the given integer is prime (in this case, one has a "product" of a single factor). When the numbers are sufficiently large, no efficient non-quantum integer factorization algorithm is known. However, it has not been proven that such an algorithm does not exist.

Quadratic integer

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Algebraic integer

In algebraic number theory, an algebraic integer is a complex number which is integral over the integers. That is, an algebraic integer is a complex root of some monic polynomial (a polynomial whose leading coefficient is 1) whose coefficients are integers. The set of all algebraic integers A is closed under addition, subtraction and multiplication and therefore is a commutative subring of the complex numbers. The ring of integers of a number field K, denoted by OK, is the intersection of K and A: it can also be characterised as the maximal order of the field K.

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