Tractability & Cryptography Strange Symbiosis Points of Reference: Oskars Rieksts

2005Kutztown University2 Overview  Keyspace & exhaustive search  One way functions  Hidden clues  Construction/Search Ratio

2005Kutztown University3 Keyspace & Exhaustive Search  What is a Key?  What is Key Space?  What is Exhaustive Search  Cost of Exhaustive Search

2005Kutztown University4 What Is a Key?  Simple Key – Caesar Cipher  More Complex – Vigenere Cipher  Public/Private Key Pair – RSA  Integer function encryption

2005Kutztown University5 Simple Key – Caesar Cipher  Example: shift forward by 3  Plaintext: “I have a dream”  Cipher text: “LKDYHDGUHDP”  Symmetric decryption: shift back by 3  Decoded: “IHAVEADREAM”

2005Kutztown University6 Vigenere Cipher  Le Chiffre Indechiffrable  Multiple Shifts  Example: [5,17,9]  Plaintext: “I have a dream I have”  Cipher text: “NYJAVJIIFDRMREJ”

2005Kutztown University7 Public/Private Key Pair – RSA  Diffie, Hellman, Merkle  Rivest, Shamir, Adelman  Ellis, Cocks, Williamson

2005Kutztown University8 RSA Basics  Encryption: f(p) = p e mod n  e is encryption key  n = p*q  p & q, two very large primes  Decryption: f -1 (c) = c d mod n  d is decryption key  e*d mod  (n) = 1   (n) is Euler totient function   (n) = (p-1)*(q-1)

2005Kutztown University9 Integer Function Encryption  1-1 and onto integer function  Example: f(0) = 2 f(0) = 2 f(1) = 0 f(1) = 0 f(2) = 3 f(2) = 3 f(3) = 1 f(3) = 1  Encryption: f(x) = y  Decryption: f -1 (y) = x

2005Kutztown University10 What Is Key Space?  Caesar – 26 shifts  keyspace = 26  Vigenere – [26, 26,.., 26] shifts  keyspace = 26 k, for k different shifts for k different shifts  RSA –  p relp(p,q) where p,q are prime where p,q are prime and relp is number of pairs relatively and relp is number of pairs relatively prime to  (p*q) prime to  (p*q)

2005Kutztown University11 What Is Key Space?  Integer function over [0, 1,.., n-1]  Key space = n!  Note: Every block cipher incorporates one of the integer functions in its range.

2005Kutztown University12 What Is Exhaustive Search?  Try every key in the keyspace  Apply deciphering function  Check result to see if it could be a plain text  Works for Caesar cipher  Does not work for others

2005Kutztown University13 Cost of Exhaustive Search  Keyspace of Vigenere with 10 shifts = 1.4 x = 1.4 x  Seconds in one year = 60 x 60 x 24 x 365 = 3.15 x 10 7 = 3.15 x 10 7  If we could test one key per second it would take 4.44 x 10 6 years or 4,444 millenia!  INTRACTABLE!

2005Kutztown University14 More on Integer Functions  Block Cipher = Integer Function  RSA misses many integer functions  How many integer functions?  Identity avoiding functions  How many identity avoiding integer functions?

2005Kutztown University15 Block Cipher = Integer Function An integer can represent any text string An integer can represent any text string Let p represent a given plaintext Let p represent a given plaintext Let c represent the corresponding ciphertext Let c represent the corresponding ciphertext Then f(p) = c represents the encryption Then f(p) = c represents the encryption And f -1 (c) = p represents the decryption And f -1 (c) = p represents the decryption Both f and f -1 are integer functions Both f and f -1 are integer functions

2005Kutztown University16 RSA Misses Many Integer Functions Cycle Length Cycle Length –m = cycle length if f m (x) = x –Example: »f 1 (0) = 2 »f 2 (0) = 3 »f 3 (0) = 1 »f 4 (0) = 0 So m = 4 = n So m = 4 = n For RSA m << n For RSA m << n

2005Kutztown University17 How Many Integer Functions? Consider all 1-1 and onto functions Consider all 1-1 and onto functions Let range be 0 to N-1. Let range be 0 to N-1. Each function is one arrangement of integers 0 to N-1, i.e., a permutation. Each function is one arrangement of integers 0 to N-1, i.e., a permutation. Number of functions = number of permutations Number of functions = number of permutations How many? N! How many? N!

2005Kutztown University18 Identity Avoiding Integer Functions Want to avoid: Want to avoid:  f(p) = p, for all p within the range We call such functions “identity avoiding” We call such functions “identity avoiding” How many such functions? How many such functions?

2005Kutztown University19 How Many Identity Avoiding? How many such functions? How many such functions?  1-1, onto, identity avoiding functions  Over the integers: 0 to N-1 N! / e N! / e e is a constant e is a constant Order of magnitude :: O (N!) Order of magnitude :: O (N!)

More to Come