A cryptographic accelerator performs various modular arithmetic operations producing unreduced results bounded by the double of the modulus (i.e.: 2*M). In doing so, various processing elements of an ALU of the cryptographic accelerator can begin to process respective data word portions of a modular arithmetic operations before the entirety of one or more operands are loaded. Similarly, various processing elements may begin to store their respective data word portions of a modular arithmetic result before the entirety of the result is calculated.