The Chinese Remainder Theorem is often used as a way to speed up modular exponentiation. In this problem we go through the procedure of using CRT. Suppose we want to compute yd mod N, where y = 680, d 1325, and N 1739. = To use the CRT technique we must know the factorization of N. In the case of this problem, N where p = 37 and q = 47. Step 1) We first compute Up = y mod p and Yqy mod q. Ур Yq dp dq Step 2) - We compute the exponents dp = d mod p - 1 and dq= = d mod q - 1. Notice that this step uses Fermat's Little Theorem. - Xp xq = = Step 3) In this stage we do the exponentiation in the smaller groups. dp = yp = da Yq mod p = = pq mod q=

The Chinese Remainder Theorem is often used as a way to speed up modular exponentiation. In this problem we go through the procedure of using CRT. Suppose we want to compute yd mod N, where y = 680, d 1325, and N 1739. = To use the CRT technique we must know the factorization of N. In the case of this problem, N where p = 37 and q = 47. Step 1) We first compute Up = y mod p and Yqy mod q. Ур Yq dp dq Step 2) - We compute the exponents dp = d mod p - 1 and dq= = d mod q - 1. Notice that this step uses Fermat's Little Theorem. - Xp xq = = Step 3) In this stage we do the exponentiation in the smaller groups. dp = yp = da Yq mod p = = pq mod q=