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(1 point) 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 xd mod N, where x = 2956, d=3323, and N = 5191. To use the CRT technique we must know the factorization of N. In the case of this problem, N p = 179 and q = 29. = pq where Step 1) We first compute xp = x mod Ρ and = x Ꮖ q mod q. xp x q = Step 2) We compute the exponents dp = d mod p 1 and da: = d mod q - 1. Notice that this step uses Fermat's Little Theorem. dp da = = Step 3) In this stage we do the exponentiation in the smaller groups. = Ур xp dp mod p = = Уд xq da mod q= 29 Step 4) We now return to the big group using the formula y = qCpYpPcqyq mod N. In this formula, Cp = q-1 mod Ρ 1 and = Са Ρ mod q. Ср = Са = And finally, y = =

(1 point) 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 xd mod N, where x = 2956, d=3323, and N = 5191. To use the CRT technique we must know the factorization of N. In the case of this problem, N p = 179 and q = 29. = pq where Step 1) We first compute xp = x mod Ρ and = x Ꮖ q mod q. xp x q = Step 2) We compute the exponents dp = d mod p 1 and da: = d mod q - 1. Notice that this step uses Fermat's Little Theorem. dp da = = Step 3) In this stage we do the exponentiation in the smaller groups. = Ур xp dp mod p = = Уд xq da mod q= 29 Step 4) We now return to the big group using the formula y = qCpYpPcqyq mod N. In this formula, Cp = q-1 mod Ρ 1 and = Са Ρ mod q. Ср = Са = And finally, y = =

Database System Concepts
Database System Concepts
7th Edition
ISBN:9780078022159
Author:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Publisher:Abraham Silberschatz Professor, Henry F. Korth, S. Sudarshan
Chapter1: Introduction
Section: Chapter Questions
Problem 1PE
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(1 point) 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 xd mod N, where x = 2956, d=3323, and N = 5191. To use the CRT technique we must know the factorization of N. In the case of this problem, N p = 179 and q = 29. = pq where Step 1) We first compute xp = x mod Ρ and = x Ꮖ q mod q. xp x q = Step 2) We compute the exponents dp = d mod p 1 and da: = d mod q - 1. Notice that this step uses Fermat's Little Theorem. dp da = = Step 3) In this stage we do the exponentiation in the smaller groups. = Ур xp dp mod p = = Уд xq da mod q= 29 Step 4) We now return to the big group using the formula y = qCpYpPcqyq mod N. In this formula, Cp = q-1 mod Ρ 1 and = Са Ρ mod q. Ср = Са = And finally, y = =
Transcribed Image Text:(1 point) 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 xd mod N, where x = 2956, d=3323, and N = 5191. To use the CRT technique we must know the factorization of N. In the case of this problem, N p = 179 and q = 29. = pq where Step 1) We first compute xp = x mod Ρ and = x Ꮖ q mod q. xp x q = Step 2) We compute the exponents dp = d mod p 1 and da: = d mod q - 1. Notice that this step uses Fermat's Little Theorem. dp da = = Step 3) In this stage we do the exponentiation in the smaller groups. = Ур xp dp mod p = = Уд xq da mod q= 29 Step 4) We now return to the big group using the formula y = qCpYpPcqyq mod N. In this formula, Cp = q-1 mod Ρ 1 and = Са Ρ mod q. Ср = Са = And finally, y = =
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