(1 point) The Miller-Rabin primality test is based around the following observation. 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 = is maximal, and r = 12 where Now we randomly select some a Є Zn, such as a = 1441. Initially we compute ar = 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). 2r 4r Otherwise we compute the list of values a²™, a², ..., a²², ..., a²r. Note a²ur 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. 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)? What conclusion should we draw from what we have done? 1) n is prime. 2) n is composite. 3) The test is inconclusive.
(1 point) The Miller-Rabin primality test is based around the following observation. 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 = is maximal, and r = 12 where Now we randomly select some a Є Zn, such as a = 1441. Initially we compute ar = 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). 2r 4r Otherwise we compute the list of values a²™, a², ..., a²², ..., a²r. Note a²ur 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. 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)? What conclusion should we draw from what we have done? 1) n is prime. 2) n is composite. 3) The test is inconclusive.
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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give the correct answer show all steps, please make sure all answers are correct.
Transcribed Image Text:(1 point) The Miller-Rabin primality test is based around the following observation.
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 =
is maximal, and
r =
12 where
Now we randomly select some a Є Zn, such as a = 1441.
Initially we compute
ar =
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).
2r 4r
Otherwise we compute the list of values a²™, a², ..., a²², ..., a²r. Note a²ur
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.
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)?
What conclusion should we draw from what we have done?
1) n is prime.
2) n is composite.
3) The test is inconclusive.
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