MathematicsOfPK

MathematicsOfPK - Inside PK Cryptography: Math and...

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Inside PK Cryptography: Math and Implementation S rira m  S rinivas a n (“Ram ”)             sriram@malhar.net
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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
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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.
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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.
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Mathematics
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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
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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
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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 
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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
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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
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MathematicsOfPK - Inside PK Cryptography: Math and...

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