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Number Theory Algorithms and Cryptography Algorithms Prepared by John Reif, Ph.D. Analysis of Algorithms

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Number Theory Algorithms a)GCD b)Multiplicative Inverse c)Fermat & Euler ’ s Theorems d)Public Key Cryptographic Systems e)Primality Testing

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Number Theory Algorithms (cont ’ d) Main Reading Selections: CLR, Chapter 33

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Euclid ’ s Algorithm Greatest Common Divisor Euclid ’ s Algorithm

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Euclid ’ s Algorithm (cont ’ d) Inductive proof of correctness:

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Euclid ’ s Algorithm (cont ’ d) Time Analysis of Euclid ’ s Algorithm for n bit numbers u,v

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Euclid ’ s Algorithm (cont ’ d) Fibonacci worst case:

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Euclid ’ s Algorithm (cont ’ d) Improved Algorithm

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Extended GCD Algorithm

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Extended GCD Algorithm (cont ’ d) Theorem Proof

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Extended GCD Algorithm (cont ’ d) Corollary If gcd(x,y) = 1 then x' is the modular inverse of x modulo y Proof

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Modular Laws Gives Algorithm for Modular Laws

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Modular Laws (cont ’ d)

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Fermat ’ s Little Theorem If n prime then a n = a mod n Proof by Euler

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Fermat ’ s Little Theorem (cont ’ d)

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Euler ’ s Theorem Φ(n) = number of integers in {1,…, n-1} relatively prime to n Euler ’ s Theorem Proof

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Euler ’ s Theorem (cont ’ d) Lemma Proof

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Euler ’ s Theorem (cont ’ d) By Law A and Lemma By Law B

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Taking Powers mod n by “ Repeated Squaring ” Problem: Compute a e mod b

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Taking Powers mod n by “ Repeated Squaring ” (cont ’ d) Time Cost

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Rivest, Sharmir, Adelman (RSA) Encryption Algorithm M = integer message e = “ encryption integer ” for user A Cryptogram

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Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont ’ d) Method

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Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont ’ d) Theorem

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Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont ’ d) Proof

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Rivest, Sharmir, Adelman (RSA) Encryption Algorithm (cont ’ d) By Euler ’ s Theorem

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Security of RSA Cryptosystem Theorem If can compute d in polynomial time, then can factor n in polynomial time Proof e· d-1 is a multiple of φ(n) But Miller has shown can factor n from any multiple of φ(n)

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Security of RSA Cryptosystem (cont ’ d)

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Rabin ’ s Public Key Crypto System Use private large primes p, q public key n=q p message M cryptogram M 2 mod n Theorem If cryptosystem can be broken, then can factor key n

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Rabin ’ s Public Key Crypto System (cont ’ d) Proof In either case, two independent solutions for M give factorization of n, i.e., a factor of n is gcd (n, γ -β).

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Rabin ’ s Public Key Crypto System (cont ’ d) Rabin ’ s Algorithm for factoring n, given a way to break his cryptosystem.

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Quadratic Residues

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Jacobi Function

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Jacobi Function (cont ’ d) Gauss ’ s Quadratic Reciprocity Law Rivest Algorithm

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Jacobi Function (cont ’ d) Theorem (Fermat)

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Theorem: Primes are in NP Proof

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Theorem & Primes NP (cont ’ d) Note

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Primality Testing Testing Goal of Randomized Primality Testing

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Primality Testing (cont ’ d) Solovey & Strassen Primality Test quadratic reciprocal law

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Definitions

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Theorem of Solovey & Strassen Theorem Proof

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Theorem of Solovey & Strassen (cont ’ d)

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Then by Chinese Remainder Theorem, Since a is relatively prime to n,

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Theorem of Solovey & Strassen (cont ’ d)

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Miller Miller ’ s Primality Test

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Theorem (Miller) Assuming the extended RH, if n is composite, then W n (a) holds for some a ∈ {1,2,…, c log 2 n} Miller ’ s Test assumes extended RH (not proved) Miller (cont ’ d)

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Miller – Rabin Randomized Primality Test Theorem

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Number Theory Algorithms and Cryptography Algorithms Prepared by John Reif, Ph.D. Analysis of Algorithms

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