Cryptographic key exchange using pre-computation

A public-key method of cryptographic key exchange using modular exponentiation in which memory, for storing pre-computed results, can be flexibly traded off against the computational complexity of key-exchange. In a typical embodiment, the invention performs key exchange by the method of Diffie-Hellman but with exponents having a constrained form such that by use of a small table of pre-computed powers of a user's public key, any possible shared secret key within the allowed set can be computed with many fewer modular multiplications than the number of bits of effective key-length thereby obtained. The table of pre-computed powers is transmitted as part of the key exchange protocol. The party in receipt of the table uses the pre-computed powers of the sender's public key to replace calculations that would otherwise need to be done at key-exchange time. The method allows a flexible trade-off between computation and table size. The method of accelerating modular exponentiation can also be applied to other cryptographic operations in which the number to be raised to a power is fixed or known in advance and where the exponent is allowed to be of the specified form.