Fermat’s theorem Euler’s generalization Application to cryptography Section 20 – Fermat’s and Euler’s theorems Instructor: Yifan Yang Spring 2007 Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography The multiplicative group of nonzero elements in a field Theorem The nonzero elements of a field form a group under the field multiplication. Proof. Straightforward. See Exercise 37 of Section 18. Notation The mutliplicative group of nonzero elements in a field F will be denoted by F × . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography The multiplicative group of nonzero elements in a field Theorem The nonzero elements of a field form a group under the field multiplication. Proof. Straightforward. See Exercise 37 of Section 18. Notation The mutliplicative group of nonzero elements in a field F will be denoted by F × . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography The multiplicative group of nonzero elements in a field Theorem The nonzero elements of a field form a group under the field multiplication. Proof. Straightforward. See Exercise 37 of Section 18. Notation The mutliplicative group of nonzero elements in a field F will be denoted by F × . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Fermat’s theorem Theorem (20.1, Little theorem of Fermat) Let p be a prime. Then for all integers a not divisible by p, we have a p − 1 ≡ 1 mod p . Proof. The group Z × p has p − 1 elements. Then by the Lagrange p , a p − 1 ≡ 1 mod p . theorem (Theorem 10.10), for all a ∈ Z × Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Fermat’s theorem Theorem (20.1, Little theorem of Fermat) Let p be a prime. Then for all integers a not divisible by p, we have a p − 1 ≡ 1 mod p . Proof. The group Z × p has p − 1 elements. Then by the Lagrange p , a p − 1 ≡ 1 mod p . theorem (Theorem 10.10), for all a ∈ Z × Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Corollary and examples Corollary (20.2) Let p be a prime. Then a p ≡ a mod p for all a ∈ Z . Example 1. Let us compute the remainder of 7 103 when divided by 17. Solution. By Fermat’s theorem, we have 7 16 ≡ 1 mod 17. Thus, 7 103 = 7 6 × 16 + 7 = ( 7 16 ) 6 ( 7 7 ) ≡ 7 7 = 7 ( 7 3 ) 2 = 7 ( 343 ) 2 ≡ 7 · 9 ≡ 12 mod 17 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Corollary and examples Corollary (20.2) Let p be a prime. Then a p ≡ a mod p for all a ∈ Z . Example 1. Let us compute the remainder of 7 103 when divided by 17. Solution. By Fermat’s theorem, we have 7 16 ≡ 1 mod 17. Thus, 7 103 = 7 6 × 16 + 7 = ( 7 16 ) 6 ( 7 7 ) ≡ 7 7 = 7 ( 7 3 ) 2 = 7 ( 343 ) 2 ≡ 7 · 9 ≡ 12 mod 17 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Corollary and examples Corollary (20.2) Let p be a prime. Then a p ≡ a mod p for all a ∈ Z . Example 1. Let us compute the remainder of 7 103 when divided by 17. Solution. By Fermat’s theorem, we have 7 16 ≡ 1 mod 17. Thus, 7 103 = 7 6 × 16 + 7 = ( 7 16 ) 6 ( 7 7 ) ≡ 7 7 = 7 ( 7 3 ) 2 = 7 ( 343 ) 2 ≡ 7 · 9 ≡ 12 mod 17 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Examples Example 2. Prove that n 33 − n is divisible by 15 for all n . Solution. We need to show that n 33 − n is divisible by both 3 and 5. Here we demonstrate n 33 − n ≡ 0 mod 5, and leave n 33 − n ≡ 0 mod 3 as an exercise. If 5 | n , then n 33 is clearly congruent to n modulo 5. If 5 ∤ n , then n 33 − n = n ( n 32 − 1 ) = n (( n 4 ) 8 − 1 ) ≡ n ( 1 − 1 ) = 0 mod 5 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Examples Example 2. Prove that n 33 − n is divisible by 15 for all n . Solution. We need to show that n 33 − n is divisible by both 3 and 5. Here we demonstrate n 33 − n ≡ 0 mod 5, and leave n 33 − n ≡ 0 mod 3 as an exercise. If 5 | n , then n 33 is clearly congruent to n modulo 5. If 5 ∤ n , then n 33 − n = n ( n 32 − 1 ) = n (( n 4 ) 8 − 1 ) ≡ n ( 1 − 1 ) = 0 mod 5 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Examples Example 2. Prove that n 33 − n is divisible by 15 for all n . Solution. We need to show that n 33 − n is divisible by both 3 and 5. Here we demonstrate n 33 − n ≡ 0 mod 5, and leave n 33 − n ≡ 0 mod 3 as an exercise. If 5 | n , then n 33 is clearly congruent to n modulo 5. If 5 ∤ n , then n 33 − n = n ( n 32 − 1 ) = n (( n 4 ) 8 − 1 ) ≡ n ( 1 − 1 ) = 0 mod 5 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Examples Example 2. Prove that n 33 − n is divisible by 15 for all n . Solution. We need to show that n 33 − n is divisible by both 3 and 5. Here we demonstrate n 33 − n ≡ 0 mod 5, and leave n 33 − n ≡ 0 mod 3 as an exercise. If 5 | n , then n 33 is clearly congruent to n modulo 5. If 5 ∤ n , then n 33 − n = n ( n 32 − 1 ) = n (( n 4 ) 8 − 1 ) ≡ n ( 1 − 1 ) = 0 mod 5 . Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Euler’s generalization Theorem (20.6) The set Z × n of nonzero elements of Z n that are not zero divisors forms a group. Proof. • closed: • Suppose that a and b are not 0 nor zero divisors. We need to show that ab is neither 0 nor a zero divisor. • Since a and b are not 0 nor zero divisors, ab � = 0. • Now suppose that ( ab ) c = 0. • Then a ( bc ) = 0. Since a is not 0 nor a zero divisors, bc = 0. • By the same token bc = 0 implies c = 0. Thus ab is not a zero divisor. Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Euler’s generalization Theorem (20.6) The set Z × n of nonzero elements of Z n that are not zero divisors forms a group. Proof. • closed: • Suppose that a and b are not 0 nor zero divisors. We need to show that ab is neither 0 nor a zero divisor. • Since a and b are not 0 nor zero divisors, ab � = 0. • Now suppose that ( ab ) c = 0. • Then a ( bc ) = 0. Since a is not 0 nor a zero divisors, bc = 0. • By the same token bc = 0 implies c = 0. Thus ab is not a zero divisor. Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Euler’s generalization Theorem (20.6) The set Z × n of nonzero elements of Z n that are not zero divisors forms a group. Proof. • closed: • Suppose that a and b are not 0 nor zero divisors. We need to show that ab is neither 0 nor a zero divisor. • Since a and b are not 0 nor zero divisors, ab � = 0. • Now suppose that ( ab ) c = 0. • Then a ( bc ) = 0. Since a is not 0 nor a zero divisors, bc = 0. • By the same token bc = 0 implies c = 0. Thus ab is not a zero divisor. Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Euler’s generalization Theorem (20.6) The set Z × n of nonzero elements of Z n that are not zero divisors forms a group. Proof. • closed: • Suppose that a and b are not 0 nor zero divisors. We need to show that ab is neither 0 nor a zero divisor. • Since a and b are not 0 nor zero divisors, ab � = 0. • Now suppose that ( ab ) c = 0. • Then a ( bc ) = 0. Since a is not 0 nor a zero divisors, bc = 0. • By the same token bc = 0 implies c = 0. Thus ab is not a zero divisor. Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Fermat’s theorem Euler’s generalization Application to cryptography Euler’s generalization Theorem (20.6) The set Z × n of nonzero elements of Z n that are not zero divisors forms a group. Proof. • closed: • Suppose that a and b are not 0 nor zero divisors. We need to show that ab is neither 0 nor a zero divisor. • Since a and b are not 0 nor zero divisors, ab � = 0. • Now suppose that ( ab ) c = 0. • Then a ( bc ) = 0. Since a is not 0 nor a zero divisors, bc = 0. • By the same token bc = 0 implies c = 0. Thus ab is not a zero divisor. Instructor: Yifan Yang Section 20 – Fermat’s and Euler’s theorems
Recommend
More recommend
