A processor includes a hash register and a hash generating circuit. The hash generating circuit includes a novel programmable nonlinearizing function circuit as well as a modulo-2 multiplier, a first modulo-2 summer, a modulor-2 divider, and a second modulo-2 summer. The nonlinearizing function circuit receives a hash value from the hash register and performs a programmable nonlinearizing function, thereby generating a modified version of the hash value. In one example, the nonlinearizing function circuit includes a plurality of separately enableable S-box circuits. The multiplier multiplies the input data by a programmable multiplier value, thereby generating a product value. The first summer sums a first portion of the product value with the modified hash value. The divider divides the resulting sum by a fixed divisor value, thereby generating a remainder value. The second summer sums the remainder value and the second portion of the input data, thereby generating a hash result.