Public Key Cryptography with a Group of Unknown Order

We present algorithms for digital signature generation and verification using public key cryptography over an arbitrary group G. The algorithms are similar to those of ElGamal, but do not require a knowledge of the group order. Forging signatures or determining the secret key requires the solution of a discrete logarithm problem over G, or the solution of other problems which appear at least as difficult. The algorithms can be modified to work over arbitrary semigroups.