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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 ....
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 Spring '10
 M.J.Golin
 Computer Science, Prime number, Z1176, Z29

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