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Steps of the Pohlig–Hellman algorithm. In group theory, the Pohlig–Hellman algorithm, sometimes credited as the Silver–Pohlig–Hellman algorithm, [1] is a special-purpose algorithm for computing discrete logarithms in a finite abelian group whose order is a smooth integer.
Stephen C. Pohlig (1952/1953 in Washington, D.C. – April 14, 2017) was an American electrical engineer who worked in the MIT Lincoln Laboratory.As a graduate student of Martin Hellman's at Stanford University in the mid-1970s, he helped develop the underlying concepts of Diffie-Hellman key exchange, [1] including the Pohlig–Hellman exponentiation cipher and the Pohlig–Hellman algorithm ...
The extended Euclidean algorithm finds k quickly. With Diffie–Hellman, a cyclic group modulo a prime p is used, allowing an efficient computation of the discrete logarithm with Pohlig–Hellman if the order of the group (being p−1) is sufficiently smooth, i.e. has no large prime factors.
The baby-step giant-step algorithm could be used by an eavesdropper to derive the private key generated in the Diffie Hellman key exchange, when the modulus is a prime number that is not too large. If the modulus is not prime, the Pohlig–Hellman algorithm has a smaller algorithmic complexity, and potentially solves the same problem. [2]
The algorithm computes integers ... If used together with the Pohlig–Hellman algorithm, the running time of the combined algorithm is () , where is ...
All generic attacks on the discrete logarithm problem in finite abelian groups such as the Pohlig–Hellman algorithm and Pollard's rho method can be used to attack the DLP in the Jacobian of hyperelliptic curves. The Pohlig-Hellman attack reduces the difficulty of the DLP by looking at the order of the group we are working with.
Martin Edward Hellman (born October 2, 1945) is an American cryptologist and mathematician, best known for his invention of public-key cryptography in cooperation with Whitfield Diffie and Ralph Merkle. [2] [3] Hellman is a longtime contributor to the computer privacy debate, and has applied risk analysis to a potential failure of nuclear ...
The basic idea of the algorithm is due to Western and Miller (1968), [4] which ultimately relies on ideas from Kraitchik (1922). [5] The first practical implementations followed the 1976 introduction of the Diffie-Hellman cryptosystem which relies on the discrete logarithm. Merkle's Stanford University dissertation (1979) was credited by Pohlig ...