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(1 point) The Miller-Rabin primality test is based around the following observation. 2 If p is prime and x² = 1 mod p then x = ±1 mod p. Note that x = -1 mod p and x = p-1 mod p mean the same thing. We will use the Miller-Rabin test to test n = 2353 for primality. To do this we will closely examine an-1 mod n for various integers a. First we divide out all of the 2's from n - 1. We can write n u = 4 is maximal, and r = 147 12 where Now we randomly select some a Є Zn, such as a = 1441. Initially we compute = | mod n. If this number is 1 then the test is inconclusive and another a is selected (up until the number of a's determined by the security level). 4r Otherwise we compute the list of values a²r, a¹², ..., a²²r, ..., a²r. Note a²"r Please enter this list for n = 2353 and a = 1441 in comma separated format. When the list is computed one of a few things could happen. 1) The value 1 never occurs on the list. = =an-1 2) The value -1 = p-1 mod p occurs on the list, immediately followed by 1. 3) The value 1 occurs in the list, but *not* preceded by -1. Which one of these possibilities holds in our case (enter the number)? 2 What conclusion should we draw from what we have done? 1) n is prime. 2) n is composite. 3) The test is inconclusive. 3

(1 point) The Miller-Rabin primality test is based around the following observation. 2 If p is prime and x² = 1 mod p then x = ±1 mod p. Note that x = -1 mod p and x = p-1 mod p mean the same thing. We will use the Miller-Rabin test to test n = 2353 for primality. To do this we will closely examine an-1 mod n for various integers a. First we divide out all of the 2's from n - 1. We can write n u = 4 is maximal, and r = 147 12 where Now we randomly select some a Є Zn, such as a = 1441. Initially we compute = | mod n. If this number is 1 then the test is inconclusive and another a is selected (up until the number of a's determined by the security level). 4r Otherwise we compute the list of values a²r, a¹², ..., a²²r, ..., a²r. Note a²"r Please enter this list for n = 2353 and a = 1441 in comma separated format. When the list is computed one of a few things could happen. 1) The value 1 never occurs on the list. = =an-1 2) The value -1 = p-1 mod p occurs on the list, immediately followed by 1. 3) The value 1 occurs in the list, but *not* preceded by -1. Which one of these possibilities holds in our case (enter the number)? 2 What conclusion should we draw from what we have done? 1) n is prime. 2) n is composite. 3) The test is inconclusive. 3

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
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(1 point) The Miller-Rabin primality test is based around the following observation. 2 If p is prime and x² = 1 mod p then x = ±1 mod p. Note that x = -1 mod p and x = p-1 mod p mean the same thing. We will use the Miller-Rabin test to test n = 2353 for primality. To do this we will closely examine an-1 mod n for various integers a. First we divide out all of the 2's from n - 1. We can write n u = 4 is maximal, and r = 147 12 where Now we randomly select some a Є Zn, such as a = 1441. Initially we compute = | mod n. If this number is 1 then the test is inconclusive and another a is selected (up until the number of a's determined by the security level). 4r Otherwise we compute the list of values a²r, a¹², ..., a²²r, ..., a²r. Note a²"r Please enter this list for n = 2353 and a = 1441 in comma separated format. When the list is computed one of a few things could happen. 1) The value 1 never occurs on the list. = =an-1 2) The value -1 = p-1 mod p occurs on the list, immediately followed by 1. 3) The value 1 occurs in the list, but *not* preceded by -1. Which one of these possibilities holds in our case (enter the number)? 2 What conclusion should we draw from what we have done? 1) n is prime. 2) n is composite. 3) The test is inconclusive. 3
Transcribed Image Text:(1 point) The Miller-Rabin primality test is based around the following observation. 2 If p is prime and x² = 1 mod p then x = ±1 mod p. Note that x = -1 mod p and x = p-1 mod p mean the same thing. We will use the Miller-Rabin test to test n = 2353 for primality. To do this we will closely examine an-1 mod n for various integers a. First we divide out all of the 2's from n - 1. We can write n u = 4 is maximal, and r = 147 12 where Now we randomly select some a Є Zn, such as a = 1441. Initially we compute = | mod n. If this number is 1 then the test is inconclusive and another a is selected (up until the number of a's determined by the security level). 4r Otherwise we compute the list of values a²r, a¹², ..., a²²r, ..., a²r. Note a²"r Please enter this list for n = 2353 and a = 1441 in comma separated format. When the list is computed one of a few things could happen. 1) The value 1 never occurs on the list. = =an-1 2) The value -1 = p-1 mod p occurs on the list, immediately followed by 1. 3) The value 1 occurs in the list, but *not* preceded by -1. Which one of these possibilities holds in our case (enter the number)? 2 What conclusion should we draw from what we have done? 1) n is prime. 2) n is composite. 3) The test is inconclusive. 3
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