Unformatted text preview: to n - 1 form what is called a complete set of residues modulo n. This means that, for every integer a, its residue modulo n is some number from 0 to n - 1. The operation a mod n denotes the residue of a, such that the residue is some integer from 0 to n - 1. This operation is modular reduction. For example, 5 mod 3 = 2. This definition of mod may be different from the definition used in some programming languages. For example, PASCAL’s modulo operator sometimes returns a negative number. It returns a number between -(n - 1) and n - 1. In C, the % operator returns the remainder from the division of the first expression by the second; this can be a negative number if either operand is negative. For all the algorithms in this book, make sure you add n to the result of the modulo operator if it returns a negative number. Modular arithmetic is just like normal arithmetic: It’s commutative, associative, and distributive. Also, reducing each intermediate result modulo n yields the same result as doing the whole calculation and then reducing the end result modulo n. a + b) mod n = ((a mod n) + (b mod n)) mod n (a - b) mod n = ((a mod n) - (b mod n)) mod n (a*b) mod n = ((a mod n)*(b mod n)) mod n (a*(b + c)) mod n = (((a*b) mod n) + ((a*c) mod n)) mod n Cryptography uses computation mod n a lot, because calculating discrete logarithms and square roots mod n can be hard problems. Modular arithmetic is also easier to work with on computers, because it restricts the range of all intermediate values and the result. For a k- bit modulus, n, the intermediate results of any addition, subtraction, or multiplication will not be more than 2kbits long. So we can perform exponentiation in modular arithmetic without generating huge intermediate results. Calculating the power of some number modulo some number, ax mod n, is just a series of multiplications and divisions, but there are speedups. One kind of speedup aims to minimize the number of modular multiplications; another kind aims to optimize...
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- Fall '10
- Cryptography, Bruce Schneier, Applied Cryptography, EarthWeb, Search Search Tips