An elliptic curve processing apparatus that performs operations on elliptic curves specified over binary polynomial fields includes a functional unit that has a digit serial multiplier with a digit size of at least two bits. The elliptic curve processing apparatus performs reduction for respective generic curves using arbitrary irreducible polynomials, which correspond to respective ones of the generic curves. The elliptic curve processing apparatus may include hardwired reduction circuits in the functional unit for use with respective named curves. A storage location in the elliptic curve processing apparatus may be used to specify whether an operation is for one of the named curves or for one of the generic curves. The elliptic curve processing apparatus responds to an arithmetic instruction to utilize a respective one of the hardwired reduction circuits for reduction for respective named curves and a multiplier circuit for reduction for a plurality of generic curves, the multiplier coupled to perform reduction for respective generic curves using arbitrary irreducible polynomials, the arbitrary irreducible polynomials corresponding to respective ones of the generic curves. The elliptic curve processing apparatus operable on elliptic curves specified over binary polynomial fields performs a conditional branch according to whether a curve being processed is a generic curve or a named curve.