Midterm_2_2007_sol_Slides

# Midterm_2_2007_sol_Slides - COMP 170 Fall 2007 Midterm 2...

This preview shows pages 1–12. Sign up to view the full content.

COMP 170 – Fall 2007 Midterm 2 Solution

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Q1. Recall the RSA public key cryptography scheme. Bob posts a public key P = ( n,e ) and keeps a secret key S = ( n,d ) . When Alice wants to send a message 0 < M < n to Bob, she calculates M 0 = M e mod n and sends M 0 to Bob. Bob then decrypts this by calculating ( M 0 ) d mod n . In class we learnt that in order for this scheme to work, n,e,d must have special properties.
(Note: In real life, to ensure a high level of security, n, e, d have to be very large numbers. For simplicity, however, we do not consider that fact here and use small numbers in this question.) For each of the three Public/Secret ( P/S ) key pairs listed below: (i) say whether it is a valid set of RSA Public/Secret key pairs and (ii) justify your answer. (a) P = (91 , 25) , S = (91 , 51) (b) P = (91 , 25) , S = (91 , 49) (c) P = (84 , 25) , S = (84 , 37)

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Solution: Recall that the conditions for a pair to be correct is that (i) n = pq where p and q are prime numbers and (ii) e · d mod T = 1 where T = ( p - 1)( q - 1) .
Solution: Recall that the conditions for a pair to be correct is that (i) n = pq where p and q are prime numbers and (ii) e · d mod T = 1 where T = ( p - 1)( q - 1) . (a) P = (91 , 25) , S = (91 , 51) This is not a valid key pair. It is true that n = 7 · 13 so p, q are prime. But T = 72 and 25 · 51 mod 72 6 = 1 . It is true that 25 · 51 mod 91 = 1 but that is not the RSA condition.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Solution: Recall that the conditions for a pair to be correct is that (i) n = pq where p and q are prime numbers and (ii) e · d mod T = 1 where T = ( p - 1)( q - 1) . (b) P = (91 , 25) , S = (91 , 49) This is a valid key pair since n = 7 · 13 and 25 · 49 mod 72 = 1 .
Solution: Recall that the conditions for a pair to be correct is that (i) n = pq where p and q are prime numbers and (ii) e · d mod T = 1 where T = ( p - 1)( q - 1) . (c) P = (84 , 25) , S = (84 , 37) This is not a valid key pair since n = 7 · 12 and 12 is not prime .

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Solution: Recall that the conditions for a pair to be correct is that (i) n = pq where p and q are prime numbers and (ii) e · d mod T = 1 where T = ( p - 1)( q - 1) . (c) P = (84 , 25) , S = (84 , 37) This is not a valid key pair since n = 7 · 12 and 12 is not prime . Note. It is true that e · d mod n = 1 and e · d mod (6 · 11) = 1 but this doesn’t mean anything.
Q2. Calculate the value of 3 1032 mod 50 . Show the steps to obtain the result.

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
Solution: Use repeated squaring to calculate 3 1 mod 50 = 3 3 2 mod 50 = 9 3 4 mod 50 = 9 2 mod 50 = 31 3 8 mod 50 = 31 2 mod 50 = 11 3 16 mod 50 = 11 2 mod 50 = 21 3 32 mod 50 = 21 2 mod 50 = 41 3 64 mod 50 = 41 2 mod 50 = 31 3 128 mod 50 = 31 2 mod 50 = 11 3 256 mod 50 = 11 2 mod 50 = 21 3 512 mod 50 = 21 2 mod 50 = 41 3 1024 mod 50 = 41 2 mod 50 = 31
Solution: Use repeated squaring to calculate 3 1 mod 50 = 3 3 2 mod 50 = 9 3 4 mod 50 = 9 2 mod 50 = 31 3 8 mod 50 = 31 2 mod 50 = 11 3 16 mod 50 = 11 2 mod 50 = 21 3 32 mod 50 = 21 2 mod 50 = 41 3 64 mod 50 = 41 2 mod 50 = 31 3

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

## This note was uploaded on 08/25/2010 for the course COMP COMP170 taught by Professor M.j.golin during the Spring '10 term at HKUST.

### Page1 / 44

Midterm_2_2007_sol_Slides - COMP 170 Fall 2007 Midterm 2...

This preview shows document pages 1 - 12. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online