Modular Arithmetic and Primality

For encryption we need to compute x y mod N for values of x,
y, and N which are several hundred bits long
38295034738204523523… 4645987345009439845234… mod 5345098234509345823…
Way too big (and slow) unless we keep all intermediate
numbers in the modulus range N.
Algorithm to solve x y mod N
How about multiplying by x mod N, y times, each time doing
modular reduction to keep the number less than N
– Multiplication is n 2 , where n (=log(N)) is length of {x, y, N} in bits.
Relatively fast since each intermediate result is less than N thus each
multiplication time is log 2 (N). Much better than the huge number you
would end up with for non-modular exponentiation.
– However, must still do y multiplies where y can be huge. Requires 2 |y|
multiplications, with y being potentially hundreds of bits long
CS 312 - Complexity Examples - Arithmetic and RSA 20