Description of Algorithm
The Montgomery reduction algorithm calculates as follows:
- if return else return .
Note that only additions, subtractions, multiplications, and integer divides and modulos by R are used – all of which are 'cheap' operations.
To understand why this gives the right answer, consider the following:
- . But by the definition of and, is a multiple of, so . Therefore, ; in other words, is exactly divisible by, so is an integer.
- Furthermore, ; therefore, as required.
- Assuming, (as ). Therefore the return value is always less than .
Therefore, we can say that
Using this method to calculate is generally less efficient than a naive multiplication and reduction, as the cost of conversions to and from residue representation (multiplications by and modulo ) outweigh the savings from the reduction step. The advantage of this method becomes apparent when dealing with a sequence of multiplications, as required for modular exponentiation (e.g. exponentiation by squaring).
Read more about this topic: Montgomery Reduction
Famous quotes containing the words description of and/or description:
“A sound mind in a sound body, is a short, but full description of a happy state in this World: he that has these two, has little more to wish for; and he that wants either of them, will be little the better for anything else.”
—John Locke (1632–1704)
“Everything to which we concede existence is a posit from the standpoint of a description of the theory-building process, and simultaneously real from the standpoint of the theory that is being built. Nor let us look down on the standpoint of the theory as make-believe; for we can never do better than occupy the standpoint of some theory or other, the best we can muster at the time.”
—Willard Van Orman Quine (b. 1908)