MTH314: Discrete Mathematics for Engineers Lecture 8: Public-Key - PowerPoint PPT Presentation

MTH314: Discrete Mathematics for Engineers Lecture 8: Public-Key Cryptography Dr Ewa Infeld Ryerson University Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem Think of a number x such that x ≡ 5 ( mod 7) and x ≡ 2 ( mod 4) . Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem Think of a number x such that x ≡ 5 ( mod 7) and x ≡ 2 ( mod 4) You had to go all the way up to 26. Or 54. Or 82. Every 28 numbers, there’s only one of those. Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem Think of a number x such that x ≡ 5 ( mod 7) and x ≡ 2 ( mod 4) You had to go all the way up to 26. Or 54. Or 82. Every 28 numbers, there’s only one of those. These two congruence equations have a common solution x ≡ 26 ( mod 28) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem Theorem Suppose that m , n are coprime. Then: 1. For all integers a , b the linear congruences x ≡ a ( mod m ) , x ≡ b ( mod n ) have a common solution. 2. If we have two solutions x 0 and x 1 such that: x 0 ≡ a ( mod m ) , x 0 ≡ b ( mod n ) x 1 ≡ a ( mod m ) , x 1 ≡ b ( mod n ) Then: x 0 ≡ x 1 ( mod m · n ) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) where GCD ( m , n ) = 1. GCD (7 , 4) = 1 Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) where GCD ( m , n ) = 1. GCD (7 , 4) = 1 We can write x = q 1 · m + a , x = q 2 · n + b for some q 1 , q 2 ∈ Z . x = q 1 · 7 + 5 = q 2 · 4 + 2 Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) where GCD ( m , n ) = 1. GCD (7 , 4) = 1 We can write x = q 1 · m + a , x = q 2 · n + b for some q 1 , q 2 ∈ Z . x = q 1 · 7 + 5 = q 2 · 4 + 2 We can mod both sides of this equation by either m or n . Suppose it’s n . q 1 · m + a ≡ b ( mod n ) q 1 · 7 + 1 ≡ 2 ( mod 4) q 1 · m ≡ b − a ( mod n ) q 1 · 7 ≡ 1 ( mod 4) Now we can find a possible value for q 1 with an LDE. Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) where GCD ( m , n ) = 1. GCD (7 , 4) = 1 We can write x = q 1 · m + a , x = q 2 · n + b for some q 1 , q 2 ∈ Z . x = q 1 · 7 + 5 = q 2 · 4 + 2 We can mod both sides of this equation by either m or n . Suppose it’s n . q 1 · m + a ≡ b ( mod n ) q 1 · 7 + 1 ≡ 2 ( mod 4) q 1 · m ≡ b − a ( mod n ) q 1 · 7 ≡ 1 ( mod 4) Now we can find a possible value for q 1 with an LDE. q 1 = 3 works. Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) where GCD ( m , n ) = 1. GCD (7 , 4) = 1 q 1 · m + a ≡ b ( mod n ) q 1 · 7 + 1 ≡ 2 ( mod 4) q 1 · m ≡ b − a ( mod n ) q 1 · 7 ≡ 1 ( mod 4) Now we can find a possible value for q 1 with an LDE. q 1 = 3 works. So x = q 1 · 7 + 5 = 26 is one solution. Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) where GCD ( m , n ) = 1. GCD (7 , 4) = 1 q 1 · m + a ≡ b ( mod n ) q 1 · 7 + 1 ≡ 2 ( mod 4) q 1 · m ≡ b − a ( mod n ) q 1 · 7 ≡ 1 ( mod 4) Now we can find a possible value for q 1 with an LDE. q 1 = 3 works. The congruence class of x = q 1 · m + a mod m · n is one solution. So x = q 1 · 7 + 5 = 26 ( mod 28) is one solution. Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Chinese Remainder Theorem So how do we use that to solve systems of linear congruences? Suppose we have two congruence equations: x ≡ a ( mon m ) x ≡ 5 ( mod 7) x ≡ b ( mon n ) x ≡ 2 ( mod 4) Check that GCD ( m , n ) = 1. Write the equations as q 1 · m + a = q 2 · n + b . Mod both sides by either m to get an equation for q 2 , or by n to get an equation q 1 . Solve the resulting equation with an LDE. Once you hav one value for q , we can find a possible x and its congruence class mod m · n is the common solution of the system of congruences. Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1a x ≡ − 4 ( mod 13) x ≡ 5002 ( mod 5) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1a x ≡ − 4 ( mod 13) x ≡ 5002 ( mod 5) x ≡ 9 ( mod 13) x ≡ 2 ( mod 5) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1a x ≡ − 4 ( mod 13) x ≡ 5002 ( mod 5) x ≡ 9 ( mod 13) x = 13 · q 1 + 9 x ≡ 2 ( mod 5) x = 5 · q 2 + 2 13 · q 1 + 9 = 5 · q 2 + 2 Mod both sides by 5: 13 · q 1 + 4 ≡ 2 ( mod 5) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1a x ≡ − 4 ( mod 13) x ≡ 5002 ( mod 5) x ≡ 9 ( mod 13) x = 13 · q 1 + 9 x ≡ 2 ( mod 5) x = 5 · q 2 + 2 13 · q 1 + 9 = 5 · q 2 + 2 Mod both sides by 5: 13 · q 1 + 4 ≡ 2 ( mod 5) 13 · q 1 ≡ 3 ( mod 5) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1a x = 13 · q 1 + 9 x = 5 · q 2 + 2 13 · q 1 + 9 = 5 · q 2 + 2 Mod both sides by 5: 13 · q 1 + 4 ≡ 2 ( mod 5) 13 · q 1 ≡ 3 ( mod 5) You should solve to LDE... except this time we can see that q 1 = 1 works. So 13 · 1 + 9 = 22 works. The solution is: x ≡ 22 ( mod 13 · 5) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1c 4 x ≡ 2 ( mod 6) 3 x ≡ 5 ( mod 8) We have to solve the linear congruences first (see last lecture). Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1c 4 x ≡ 2 ( mod 6) 3 x ≡ 5 ( mod 8) We can to solve the linear congruences first (see last lecture). 4 x ≡ 2 ( mod 6) is equivalent to x ≡ 2 ( mod 3) Or we can just write the equations as: 4 x = 6 · q 1 + 2 3 x = 8 · q 2 + 5 Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1c 4 x ≡ 2 ( mod 6) 3 x ≡ 5 ( mod 8) We can to solve the linear congruences first (see last lecture). 4 x ≡ 2 ( mod 6) is equivalent to x ≡ 2 ( mod 3) Or we can just write the equations as: x = 3 · q 1 + 2 3 x = 8 · q 2 + 5 So: 3(3 · q 1 + 2) = 8 · q 2 + 5 Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Exercise 1c 4 x ≡ 2 ( mod 6) 3 x ≡ 5 ( mod 8) We can to solve the linear congruences first (see last lecture). 4 x ≡ 2 ( mod 6) is equivalent to x ≡ 2 ( mod 3) Or we can just write the equations as: x = 3 · q 1 + 2 3 x = 8 · q 2 + 5 So: 3(3 · q 1 + 2) = 8 · q 2 + 5 Mod both sides by 8: 9 · q 1 ≡ 7 ( mod 8) q 1 = 7 works, and gives x = 23. The solution is x ≡ 23 ( mod 24). Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Fermat’s Little Theorem Theorem If p is prime and a , p are coprime, then: a p − 1 ≡ 1 ( mod p ) . Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Fermat’s Little Theorem Theorem If p is prime and a , p are coprime, then: a p − 1 ≡ 1 ( mod p ) . Example: If p = 7, then for any integer a that is not a multiple of 7: a 6 ≡ 1 ( mod 7) . Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Fermat’s Little Theorem Theorem If p is prime and a , p are coprime, then: a p − 1 ≡ 1 ( mod p ) . Example: If p = 7, then for any integer a that is not a multiple of 7: a 6 ≡ 1 ( mod 7) . Example: What is the congruence class of n 154 ( mod 23)? Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Fermat’s Little Theorem Theorem If p is prime and a , p are coprime, then: a p − 1 ≡ 1 ( mod p ) . Example: If p = 7, then for any integer a that is not a multiple of 7: a 6 ≡ 1 ( mod 7) . Example: What is the congruence class of n 154 ( mod 23)? 154 = 11 · 2 · 7 = 22 · 7 Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers

Fermat’s Little Theorem Theorem If p is prime and a , p are coprime, then: a p − 1 ≡ 1 ( mod p ) . Example: If p = 7, then for any integer a that is not a multiple of 7: a 6 ≡ 1 ( mod 7) . Example: What is the congruence class of n 154 ( mod 23)? 154 = 11 · 2 · 7 = 22 · 7 n 154 = ( n 22 ) 7 ≡ 1 7 ≡ 1 ( mod 23) Dr Ewa Infeld Ryerson University MTH314: Discrete Mathematics for Engineers