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An Introduction to Cryptology and Coding Theory

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  1. An Introduction to Cryptology and Coding Theory

  2. Communication System

  3. Cryptology • Cryptography • Inventing cipher systems; protecting communications and storage • Cryptanalysis • Breaking cipher systems

  4. Cryptography

  5. Cryptanalysis

  6. What is used in Cryptology? • Cryptography: • Linear algebra, abstract algebra, number theory • Cryptanalysis: • Probability, statistics, combinatorics, computing

  7. Caesar Cipher • ABCDEFGHIJKLMNOPQRSTUVWXYZ • Key = 3 • DEFGHIJKLMNOPQRSTUVWXYZABC • Example • Plaintext: OLINCOLLEGE • Encryption: Shift by KEY = 3 • Ciphertext: ROLQFROOHJH • Decryption: Shift backwards by KEY = 3

  8. Cryptanalysis of Caesar • Try all 26 possible shifts • Frequency analysis

  9. Substitution Cipher • Permute A-Z randomly: A B C D E F G H I J K L M N O P… becomes H Q A W I N F T E B X S F O P C… • Substitute H for A, Q for B, etc. • Example • Plaintext: OLINCOLLEGE • Key: PSEOAPSSIFI

  10. Cryptanalysis of Substitution Ciphers • Try all 26! permutations – TOO MANY! Bigger than Avogadro's Number! • Frequency analysis

  11. One-Time Pads • Map A, B, C, … Z to 0, 1, 2, …25 • A B … M N … T U • 0 1 … 13 14 … 20 21 • Plaintext: MATHISUSEFULANDFUN • Key: NGUJKAMOCTLNYBCIAZ • Encryption: “Add” key to message mod 26 • Ciphertext: BGO….. • Decryption: “Subtract” key from ciphertext mod 26

  12. Modular Arithmetic

  13. One-Time Pads • Unconditionally secure • Problem: Exchanging the key • There are some clever ways to exchange the key – we will study some of them!

  14. Public-Key Cryptography • Diffie & Hellman (1976) • Known at GCHQ years before • Uses one-way (asymmetric) functions, public keys, and private keys

  15. Public Key Algorithms • Based on two hard problems • Factoring large integers • The discrete logarithm problem

  16. WWII Folly: The Weather-Beaten Enigma

  17. Need more than secrecy…. • Need reliability! • Enter coding theory…..

  18. What is Coding Theory? • Coding theory is the study of error-control codes • Error control codes are used to detect and correct errors that occur when data are transferred or stored

  19. What IS Coding Theory? • A mix of mathematics, computer science, electrical engineering, telecommunications • Linear algebra • Abstract algebra (groups, rings, fields) • Probability&Statistics • Signals&Systems • Implementation issues • Optimization issues • Performance issues

  20. General Problem • We want to send data from one place to another… • channels: telephone lines, internet cables, fiber-optic lines, microwave radio channels, cell phone channels, etc. • or we want to write and later retrieve data… • channels: hard drives, disks, CD-ROMs, DVDs, solid state memory, etc. • BUT! the data, or signals, may be corrupted • additive noise, attenuation, interference, jamming, hardware malfunction, etc.

  21. General Solution • Add controlled redundancy to the message to improve the chances of being able to recover the original message • Trivial example: The telephone game

  22. The ISBN Code • x1 x2…x10 • x10 is a check digit chosen so that S=x1 + 2x2 + … + 9x9 + 10x10 =0 mod 11 • Can detect all single and all transposition errors

  23. ISBN Example • Cryptology by Thomas Barr: 0-13-088976-? • Want 1(0) + 2(1) + 3(3) + 4(0) + 5(8) + 6(8) + 7(9) + 8(7) + 9(6) + 10(?) = multiple of 11 • Compute 1(0) + 2(1) + 3(3) + 4(0) + 5(8) + 6(8) + 7(9) + 8(7) + 9(6) = 272 • Ponder 272 + 10(?) = multiple of 11 • Modular arithmetic shows that the check digit is 8!!

  24. UPC (Universal Product Code) • x1 x2…x12 • x12 is a check digit chosen so that S = 3x1 + 1x2 + … + 3x11 + 1x12 =0 mod 10 • Can detect all single and most transposition errors • What transposition errors go undetected?

  25. The Repetition Code • Send 0 and 1 • Noise may change 0 to 1 or change 1 to 0 • Instead, send codewords 00000 and 11111 • If noise corrupts up to 2 bits, decoder can use majority vote and decode received word as 00000

  26. The Repetition Code • The distance between the two codewords is 5, because they differ in 5 spots • Large distance between codewords is good! • The “rate” of the code is 1/5, since for every bit of information, we need to send 5 coded bits • High rate is good!

  27. When is a Code “Good”? • Important Code Parameters (n, M, d) • Length (n) • Number of codewords (M) • Minimum Hamming distance (d): Directly related to probability of decoding correctly • Code rate: Ratio of information bits to codeword bits

  28. How Good Does It Get? • What are the ideal trade-offs between rate, error-correcting capability, and number of codewords? • What is the biggest distance you can get given a fixed rate or fixed number of codewords? • What is the best rate you can get given a fixed distance or fixed number of codewords?

  29. 1969 Mariner Mission • We’ll learn how Hadamard matrices were used on the 1969 Mariner Mission to build a rate 6/32 code that is approximately 100,000x better at correcting errors than the binary repetition code of length 5

  30. 1980-90’s Voyager Missions • Better pictures need better codes need more sophisticated mathematics… • Picture transmitted via Reed-Solomon codes

  31. Summary • From Caesar to Public-Key…. from Repetition Codes to Reed-Solomon Codes…. • More sophisticated mathematics  better ciphers/codes • Cryptology and coding theory involve abstract algebra, finite fields, rings, groups, probability, linear algebra, number theory, and additional exciting mathematics!

  32. Who Cares? • You and me! • Shopping and e-commerce • ATMs and online banking • Satellite TV & Radio, Cable TV, CD players • Corporate/government espionage • Who else? • NSA, IDA, RSA, Aerospace, Bell Labs, AT&T, NASA, Lucent, Amazon, iTunes…