A Generic Algorithm for Small Weight Discrete Logarithms in Composite Groups

Let (G, center dot) be an arbitrary cyclic group of composite order N with G similar or equal to G(1) x G(2). We present a generic algorithm for solving the discrete logarithm problem in G with Hamming weight delta log N, delta is an element of (0, 1), in time (O) over tilde(root p + root vertical bar G(2)vertical bar(H(delta))), where p is the largest prime divisor in G(1) and H(center dot) is the binary entropy function. Our algorithm improves on the running time of Silver-Pohlig-Hellman's algorithm whenever delta not equal 1/2. Moreover, it improves on the Meet-in-the-Middle type algorithms of Heiman, Odlyzko and Coppersmith with running time (O) over tilde(root p + root vertical bar G(2)vertical bar(H(delta))) whenever p < vertical bar G vertical bar(H(delta)).