Tutorial5_2010

Tutorial5_2010 - COMP 170 Spring 2010 Tutorial 5 Version...

Info iconThis preview shows pages 1–11. Sign up to view the full content.

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

View Full Document Right Arrow Icon

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

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

Unformatted text preview: COMP 170 Spring 2010 Tutorial 5 Version 1.1 Version of March 15, 2010 • Review of the Chinese Remainder Theorem • RSA & The Chinese Remainder Theorem Example: m = 13 , n = 11 , a = 3 , b = 7 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . Example: m = 13 , n = 11 , a = 3 , b = 7 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . • m = 6 and n = 6 since 13 · 6 mod 11 = 78 mod 11 = 1 11 · 6 mod 13 = 66 mod 13 = 1 Example: m = 13 , n = 11 , a = 3 , b = 7 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . • m = 6 and n = 6 since 13 · 6 mod 11 = 78 mod 11 = 1 11 · 6 mod 13 = 66 mod 13 = 1 • Set y = a nn + b mm = 3 · 6 · 11 + 7 · 6 · 13 = 744 Example: m = 13 , n = 11 , a = 3 , b = 7 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . • m = 6 and n = 6 since 13 · 6 mod 11 = 78 mod 11 = 1 11 · 6 mod 13 = 66 mod 13 = 1 • Set y = a nn + b mm = 3 · 6 · 11 + 7 · 6 · 13 = 744 • 744 = 5 * 143 + 29 so x = y mod ( nm ) = 29 Example: m = 13 , n = 11 , a = 3 , b = 7 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . • m = 6 and n = 6 since 13 · 6 mod 11 = 78 mod 11 = 1 11 · 6 mod 13 = 66 mod 13 = 1 • Set y = a nn + b mm = 3 · 6 · 11 + 7 · 6 · 13 = 744 • 744 = 5 * 143 + 29 so x = y mod ( nm ) = 29 • Reality Check: 29 mod 13 = 3 29 mod 11 = 7 Example: m = 13 , n = 11 , a = 4 , b = 8 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . Example: m = 13 , n = 11 , a = 4 , b = 8 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 . • m = 6 and n = 6 since 13 · 6 mod 11 = 78 mod 11 = 1 11 · 6 mod 13 = 66 mod 13 = 1 Example: m = 13 , n = 11 , a = 4 , b = 8 Theorem 2.24 (Chinese Remainder Theorem) If m and n are relatively prime integers, then the equations x mod m = a ∈ Z m and x mod n = b ∈ Z n have one and only one solution for an integer x between and mn- 1 ....
View Full Document

{[ snackBarMessage ]}

Page1 / 47

Tutorial5_2010 - COMP 170 Spring 2010 Tutorial 5 Version...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online