{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

# lec-9 - Administration Midterms 1 Midterm 1 Wednesday...

This preview shows pages 1–17. Sign up to view the full content.

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

View Full Document

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

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

Unformatted text preview: Administration Midterms 1. Midterm 1. Wednesday. October 5. 8-10 PM. Dwinelle 155. 2. Midterm 2. Wednesday. November 9. 8-10 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 ) ....
View Full Document

{[ snackBarMessage ]}

### Page1 / 94

lec-9 - Administration Midterms 1 Midterm 1 Wednesday...

This preview shows document pages 1 - 17. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online