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The algorithm is performed in three stages. The first two stages depend only on the generator g and prime modulus q, and find the discrete logarithms of a factor base of r small primes. The third stage finds the discrete log of the desired number h in terms of the discrete logs of the factor base.
The implementation used 2000 CPU cores and took about 6 months to solve the problem. [ 40 ] On 16 June 2020, Aleksander Zieniewicz (zielar) and Jean Luc Pons ( JeanLucPons ) announced the solution of a 114-bit interval elliptic curve discrete logarithm problem on the secp256k1 curve by solving a 114-bit private key in Bitcoin Puzzle ...
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. For example, the equation log 10 53 = 1.724276… means that 10 1.724276… = 53.
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.
A quantum algorithm for solving this problem exists. This algorithm is, like the factor-finding algorithm, due to Peter Shor and both are implemented by creating a superposition through using Hadamard gates, followed by implementing f {\displaystyle f} as a quantum transform, followed finally by a quantum Fourier transform. [ 3 ]
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.
Computing the discrete logarithm is the only known method for solving the CDH problem. But there is no proof that it is, in fact, the only method. It is an open problem to determine whether the discrete log assumption is equivalent to the CDH assumption, though in certain special cases this can be shown to be the case. [3] [4]
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 .