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Unformatted text preview: Administration Midterms 1. Midterm 1. Wednesday. October 5. 810 PM. Dwinelle 155. 2. Midterm 2. Wednesday. November 9. 810 PM. Dwinelle 155. CS70: Satish Rao: Lecture 9. Outline. 1. Extended GCD. 2. Cryptography 3. Public Key Cryptography 4. RSA system 4.1 Efficiency: Repeated Squaring. 4.2 Correctness: Fermat’s Theorem. 4.3 Construction. Euclid’s GCD algorithm. (define (gcd x y) (if (= y 0) x (gcd y (mod x y)))) Euclid’s GCD algorithm. (define (gcd x y) (if (= y 0) x (gcd y (mod x y)))) Computes the gcd ( x , y ) in O ( n ) divisions. Euclid’s GCD algorithm. (define (gcd x y) (if (= y 0) x (gcd y (mod x y)))) Computes the gcd ( x , y ) in O ( n ) divisions. For x and m , if gcd ( x , m ) = 1 then x has an inverse modulo m . Multiplicative Inverse. GCD algorithm used to tell if there is a multiplicative inverse. Multiplicative Inverse. GCD algorithm used to tell if there is a multiplicative inverse. How do we find a multiplicative inverse? Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . “Make gcd ( x , y ) out of x and y .” Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . “Make gcd ( x , y ) out of x and y .” What is multiplicative inverse of x modulo m ? Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . “Make gcd ( x , y ) out of x and y .” What is multiplicative inverse of x modulo m ? By extended GCD theorem, wehen gcd ( x , m ) = 1. Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . “Make gcd ( x , y ) out of x and y .” What is multiplicative inverse of x modulo m ? By extended GCD theorem, wehen gcd ( x , m ) = 1. ax + bm = 1 Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . “Make gcd ( x , y ) out of x and y .” What is multiplicative inverse of x modulo m ? By extended GCD theorem, wehen gcd ( x , m ) = 1. ax + bm = 1 ax ≡ 1 bm ≡ 1 ( mod m ) . Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) . “Make gcd ( x , y ) out of x and y .” What is multiplicative inverse of x modulo m ? By extended GCD theorem, wehen gcd ( x , m ) = 1. ax + bm = 1 ax ≡ 1 bm ≡ 1 ( mod m ) . So is a is multiplicative inverse of x if gcd ( x , m ) = 1!! Extended GCD Euclid’s Extended GCD Theorem: For any x , y there are integers a , b such that ax + by = gcd ( x , y ) ....
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 Fall '11
 Rau
 Cryptography, Alice, gcd

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