Cellular Encryption CREU Project Team: Alburn Brown Orkun Kaya Isaac Rieksts Eric Thorpe

2004Kutztown University2 Overview Construct software simulation Construct software simulation Compare 3 cellular encryption algorithms Compare 3 cellular encryption algorithms Choose the best one Choose the best one Implement it in hardware Implement it in hardware

2004Kutztown University3 Background :: Cellular Computation Cellular Automata Cellular Automata Cellular Automata with Shadow Cells Cellular Automata with Shadow Cells Hybrid Cellular Automata Hybrid Cellular Automata Integer Functions Integer Functions Cellular Encryption Cellular Encryption

2004Kutztown University4 Cellular Automata John von Neumann John von Neumann Uniform cells Uniform cells “Peer Pressure Automata” “Peer Pressure Automata” Uniform transition function Uniform transition function Turing Machine computing power Turing Machine computing power

2004Kutztown University5 Cellular Automata with Shadow Cells Need to compute integer functions Need to compute integer functions Limitation of classic CAs Limitation of classic CAs Shadow cells enable integer computation Shadow cells enable integer computation Wasted memory Wasted memory

2004Kutztown University6 Hybrid Cellular Automata Relaxing the uniform transition rule Relaxing the uniform transition rule Memory savings Memory savings All integer functions computable All integer functions computable

2004Kutztown University7 Integer Functions Block cipher = integer function Block cipher = integer function Example: RSA Example: RSA How many integer functions? How many integer functions? How many identity avoiding functions? How many identity avoiding functions?

2004Kutztown University8 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

2004Kutztown University9 RSA is an integer function  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)  Clearly an integer function

2004Kutztown University10 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!

2004Kutztown University11 How Many Are Identity Avoiding? 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?  1-1, onto, identity avoiding functions  Over the integers: 0 to N-1 N! / e N! / e

2004Kutztown University12 Identity Avoiding Integer Functions Over the integers 0 to M-1 Over the integers 0 to M-1 There are :: N! / e There are :: N! / e e is a constant e is a constant Order of magnitude :: O (N!) Order of magnitude :: O (N!)

2004Kutztown University13 Cellular Encryption Integers 0 to 2 k – 1 represent: Integers 0 to 2 k – 1 represent:  Plain text  Cipher text Choose at random from 2 k !/e possibilities Choose at random from 2 k !/e possibilities k cells of hybrid CA k cells of hybrid CA  Will encrypt  By computing chosen integer function Each cell computes one Boolean function Each cell computes one Boolean function

2004Kutztown University14 Encryption Cell Details Represent integer in binary Represent integer in binary 3 bit example 3 bit example  Base 10:: f(3) = 5  Binary:: f(011) = 101 Work is spread among 3 cells Work is spread among 3 cells  Cell 0 :: f 0 (011) = 1  Cell 1 :: f 1 (011) = 0  Cell 2 :: f 2 (011) = 1

2004Kutztown University15 Integer ► Boolean An integer function – base 10 & binary: An integer function – base 10 & binary:  f(0) = 3f(000) = 011  f(1) = 6f(001) = 110  f(2) = 7f(010) = 111  f(3) = 5f(011) = 101  f(4) = 2f(100) = 010  f(5) = 1f(101) = 001  f(6) = 0f(110) = 000  f(7) = 4f(111) = 100

2004Kutztown University16 Integer ► Boolean Cell by cell Cell by cell f(000) = 011 ► f 0 (000) = 0 f 1 (000) = 1 f 2 (000) = 1f(000) = 011 ► f 0 (000) = 0 f 1 (000) = 1 f 2 (000) = 1 f(001) = 110 ► f 0 (001) = 1 f 1 (001) = 1 f 2 (001) = 0f(001) = 110 ► f 0 (001) = 1 f 1 (001) = 1 f 2 (001) = 0 f(010) = 111 ► f 0 (010) = 1 f 1 (010) = 1 f 2 (010) = 1f(010) = 111 ► f 0 (010) = 1 f 1 (010) = 1 f 2 (010) = 1 f(011) = 101 ► f 0 (011) = 1 f 1 (011) = 0 f 2 (011) = 1f(011) = 101 ► f 0 (011) = 1 f 1 (011) = 0 f 2 (011) = 1 f(100) = 010 ► f 0 (100) = 0 f 1 (100) = 1 f 2 (100) = 0f(100) = 010 ► f 0 (100) = 0 f 1 (100) = 1 f 2 (100) = 0 f(101) = 001 ► f 0 (101) = 0 f 1 (101) = 0 f 2 (101) = 1f(101) = 001 ► f 0 (101) = 0 f 1 (101) = 0 f 2 (101) = 1 f(110) = 000 ► f 0 (110) = 0 f 1 (110) = 0 f 2 (110) = 0f(110) = 000 ► f 0 (110) = 0 f 1 (110) = 0 f 2 (110) = 0 f(111) = 100 ► f 0 (111) = 1 f 1 (111) = 0 f 2 (111) = 0f(111) = 100 ► f 0 (111) = 1 f 1 (111) = 0 f 2 (111) = 0

2004Kutztown University17 Hardware Implementation Plain text is k bits long Plain text is k bits long There are k cells There are k cells Each cell gets k bits of input Each cell gets k bits of input Each cell computes its own Boolean function Each cell computes its own Boolean function Together cells compute k-bit integer function Together cells compute k-bit integer function

THE END