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The algorithm was introduced by Roland Silver, but first published by Stephen Pohlig and Martin Hellman, who credit Silver with its earlier independent but unpublished discovery. Pohlig and Hellman also list Richard Schroeppel and H. Block as having found the same algorithm, later than Silver, but again without publishing it. [2]
In computational number theory, the index calculus algorithm is a probabilistic algorithm for computing discrete logarithms. Dedicated to the discrete logarithm in ( Z / q Z ) ∗ {\displaystyle (\mathbb {Z} /q\mathbb {Z} )^{*}} where q {\displaystyle q} is a prime, index calculus leads to a family of algorithms adapted to finite fields and to ...
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.
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 ...
To find the needed , , , and the algorithm uses Floyd's cycle-finding algorithm to find a cycle in the sequence =, where the function: + is assumed to be random-looking and thus is likely to enter into a loop of approximate length after steps.
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]
As n increases, the performance of the algorithm or method in question degrades rapidly. For example, the Pohlig–Hellman algorithm for computing discrete logarithms has a running time of O ( n 1/2 )—for groups of n -smooth order .
An algorithm is fundamentally a set of rules or defined procedures that is typically designed and used to solve a specific problem or a broad set of problems.. Broadly, algorithms define process(es), sets of rules, or methodologies that are to be followed in calculations, data processing, data mining, pattern recognition, automated reasoning or other problem-solving operations.