1 / 32

An Introduction to Cryptology and Coding Theory

An Introduction to Cryptology and Coding Theory. Communication System. Cryptology. Cryptography Inventing cipher systems; protecting communications and storage Cryptanalysis Breaking cipher systems. Cryptography. Cryptanalysis. What is used in Cryptology?. Cryptography:

leora
Download Presentation

An Introduction to Cryptology and Coding Theory

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript


  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…

More Related