Abstract: An optimized approach for machine computation of exponential values or functions is disclosed. The determination of the exponential values is performed using a "Big Multiply" approach and a "Big Mod" approach which involve dynamically determining the maximum size of an intermediate value required to determine an encryption key and allocating memory sufficiently large to store the maximum size intermediate value so that no additional memory allocation operations are required during the determination of the encryption keys. In addition, iterative multiplication and shift operations are performed on portions of the intermediate value in a cascade fashion to prevent spilling of the intermediate value. In one context, the computation of the exponential values is used in generating a key for exchange in a public key cryptosystem, such as the Diffie-Hellman public key cryptosystem.