Search results
Results From The WOW.Com Content Network
The algorithm was introduced in 1978 by the number theorist John M. Pollard, in the same paper as his better-known Pollard's rho algorithm for solving the same problem. [1] [2] Although Pollard described the application of his algorithm to the discrete logarithm problem in the multiplicative group of units modulo a prime p, it is in fact a ...
For any number a in this list, one can compute log 10 a. For example, log 10 10000 = 4, and log 10 0.001 = −3. These are instances of the discrete logarithm problem. Other base-10 logarithms in the real numbers are not instances of the discrete logarithm problem, because they involve non-integer exponents.
In group theory, a branch of mathematics, the baby-step giant-step is a meet-in-the-middle algorithm for computing the discrete logarithm or order of an element in a finite abelian group by Daniel Shanks. [1] The discrete log problem is of fundamental importance to the area of public key cryptography.
The discrete logarithm problem in a finite field consists of solving the equation = for ,, a prime number and an integer. The function f : F p n → F p n , a ↦ a x {\displaystyle f:\mathbb {F} _{p^{n}}\to \mathbb {F} _{p^{n}},a\mapsto a^{x}} for a fixed x ∈ N {\displaystyle x\in \mathbb {N} } is a one-way function used in cryptography .
Let be a cyclic group of order , and given ,, and a partition =, let : be the map = {and define maps : and : by (,) = {() + (,) = {+ ()input: a: a generator of G b: an element of G output: An integer x such that a x = b, or failure Initialise i ← 0, a 0 ← 0, b 0 ← 0, x 0 ← 1 ∈ G loop i ← i + 1 x i ← f(x i−1), a i ← g(x i−1, a i−1), b i ← h(x i−1, b i−1) x 2i−1 ← ...
This was considered a minor step compared to the others for smaller discrete log computations. However, larger discrete logarithm records [1] [2] were made possible only by shifting the work away from the linear algebra and onto the sieve (i.e., increasing the number of equations while reducing the number of variables).
ECC2K-108, involving taking a discrete logarithm on a Koblitz curve over a field of 2 108 elements. The prize was awarded on 4 April 2000 to a group of about 1300 people represented by Robert Harley. They used a parallelized Pollard rho method with speedup. ECC2-109, involving taking a discrete logarithm on a curve over a field of 2 109 ...
The aforementioned process achieves a t-bit security level with 4t-bit signatures. For example, a 128-bit security level would require 512-bit (64-byte) signatures. The security is limited by discrete logarithm attacks on the group, which have a complexity of the square-root of the group size.