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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]
Pohlig–Hellman algorithm; Pollard's kangaroo algorithm; Pollard's rho algorithm for logarithms; R. Rational reconstruction (mathematics) T. Tonelli–Shanks algorithm
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.
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 .
He also relates his subsequent work in cryptography with Steve Pohlig (the Pohlig–Hellman algorithm) and others. Hellman addresses the National Security Agency's (NSA) early efforts to contain and discourage academic work in the field, the Department of Commerce's encryption export restrictions, and key escrow (the so-called Clipper chip).