MathematicsOfPK

# MathematicsOfPK - Inside PK Cryptography Math and...

This preview shows pages 1–11. Sign up to view the full content.

Inside PK Cryptography: Math and Implementation S rira m  S rinivas a n (“Ram ”)             [email protected]

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

View Full Document
Sriram Srinivasan 2/47 Agenda Introduction to PK Cryptography Essential Number Theory  Fundamental Number Theorem GCD, Euclid’s algorithm Linear combinations  Modular Arithmetic Euler’s Totient Function Java implementation of RSA
Sriram Srinivasan 3/47 Security Issues Authentication, Authorization, and Encryption, Non- repudiation Shared Secrets  (e.g passwords, Enigma) Something shared, something (else) secret Concept by Ellis, Cocks and Williams Popularly attributed to Diffie and Hellman  Algorithm by Rivest, Shamir and Adelman Used everywhere: https, SSL, email, certificates.

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

View Full Document
Sriram Srinivasan 4/47 Public Key Cryptography Consider a pair of magic pens.  Write with one, use the other to decode. Symmetric: either can be used to encode You want to send a message to me You borrow one of my pens and write with it.  I decode it with my other pen. Avoids problems of shared secrets Same tools for authentication, encryption and non- repudiation.
Mathematics

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

View Full Document
Sriram Srinivasan 6/47 Fundamental Theorem of Arithmetic All numbers are expressible as a unique product of  primes 10 = 2 * 5,    60 = 2 * 2 * 3  * 5 Proof in two parts 1. All numbers are expressible as products    of primes 2. There is only one such product sequence     per number
Sriram Srinivasan 7/47 Fundamental Theorem proof First part of proof All numbers are products of primes Let S = {x | x is not expressible as a product of primes} Let c = min{S}.     c cannot be prime Let c = c 1  . c 2 c 1 , c 2   < c   c 1 , c 2    S (because c is min{S})  c 1 , c 2  are products of primes   c is too  S is an empty set

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

View Full Document
Sriram Srinivasan 8/47 Fundamental Theorem proof Second part of proof The product of primes is unique Let n = p 1 p 2 p 3 p 4 …  = q 1 q 2 q 3 q 4 Cancel common primes. Now unique primes on both sides Now, p 1  | p 1 p 2 p 3 p 4    p 1  | q 1 q 2 q 3 q 4 p 1   | one of q 1 , q 2 , q 3 , q 4 p 1   = q i  which is a contradiction
Sriram Srinivasan 9/47 GCD (Greatest Common Divisor) gcd(a,b) = the greatest of the divisors of a,b Many ways to compute gcd Extract common prime factors Express a, b as products of primes Extract common prime factors gcd(18, 66) = gcd( 2*3 *3,  2*3 *11) = 2*3 = 6 Factoring is hard. Not practical Euclid’s algorithm

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

View Full Document
Sriram Srinivasan 10/47 r r 1 r r = a % b Euclid’s algorithm a b b r % r 1   = 0.      gcd (a,b) = r 1 r 1  = b % r 1 2 3
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

### Page1 / 49

MathematicsOfPK - Inside PK Cryptography Math and...

This preview shows document pages 1 - 11. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online