I once ran into laypeople, who were able to understand what a modulus was – so that the output of a series of computations would never equal or exceed that modulus – and who were able to understand what the exponent function is, but who were incredulous, when I told them that it was possible to compute the result of raising a 2048-bit number, to a 2048-bit exponent, on the basis that the result only needs to fit inside a 2048-bit modulus.
I believe that the most common way in which this is done, is based on the assumption that two 2048-bit numbers can be multiplied, and the result brought back down to a 2048-bit modulus. This notion can be extended, to mean that a 2048-bit number can also be squared, and the result written in the 2048-bit modulus…
Well to achieve the exponent-function, one needs a base-register, an accumulator-register, and the exponent. The accumulator is initialized to the value (1).
If the exponent has 2048 bits, then the operation can be repeated 2048 times:
- Square the value in the accumulator.
- Left-Shift the Most-Significant Bit of the Exponent out, into a bit-register that can be examined.
- If that bit-register is equal to (1) and not (0), multiply the base-register into the accumulator an extra time.
Because the Most-Significant Bit of the Exponent was shifted out first, its being (1) would mean that the value in the Accumulator was multiplied by the Base earlier, so that this exponent of the Base will also be squared, as a factor of the Accumulator, by the highest number of iterations, thus effectively raising the Base to the power of 1, 2, 4, 8, 16, 32, … 2^2047 , in combinations.