An Introduction to Cryptology and Coding Theory Communication
- Slides: 32
An Introduction to Cryptology and Coding Theory
Communication System Digital Source Digital Sink Source Encoding Encryption Source Decoding Decryption Error Control Encoding Error Control Decoding Modulation Channel Demodulation
Cryptology n Cryptography ¡ n Inventing cipher systems; protecting communications and storage Cryptanalysis ¡ Breaking cipher systems
Cryptography
Cryptanalysis
What is used in Cryptology? n Cryptography: ¡ n Linear algebra, abstract algebra, number theory Cryptanalysis: ¡ Probability, statistics, combinatorics, computing
Caesar Cipher n ABCDEFGHIJKLMNOPQRSTUVWXYZ Key = 3 DEFGHIJKLMNOPQRSTUVWXYZABC n Example n n ¡ ¡ Plaintext: Encryption: Ciphertext: Decryption: OLINCOLLEGE Shift by KEY = 3 ROLQFROOHJH Shift backwards by KEY = 3
Cryptanalysis of Caesar n Try all 26 possible shifts n Frequency analysis
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… n Substitute H for A, Q for B, etc. n Example n ¡ ¡ Plaintext: OLINCOLLEGE Key: PSEOAPSSIFI
Cryptanalysis of Substitution Ciphers n n Try all 26! permutations – TOO MANY! Bigger than Avogadro's Number! Frequency analysis
One-Time Pads n n n n 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
Modular Arithmetic
One-Time Pads n Unconditionally secure n Problem: Exchanging the key n There are some clever ways to exchange the key – we will study some of them!
Public-Key Cryptography n n n Diffie & Hellman (1976) Known at GCHQ years before Uses one-way (asymmetric) functions, public keys, and private keys
Public Key Algorithms n Based on two hard problems ¡ ¡ Factoring large integers The discrete logarithm problem
WWII Folly: The Weather. Beaten Enigma
Need more than secrecy…. n Need reliability! n Enter coding theory…. .
What is Coding Theory? n n Coding theory is the study of errorcontrol codes Error control codes are used to detect and correct errors that occur when data are transferred or stored
What IS Coding Theory? n 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
General Problem n We want to send data from one place to another… ¡ n or we want to write and later retrieve data… ¡ n channels: telephone lines, internet cables, fiber-optic lines, microwave radio channels, cell phone channels, etc. 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.
General Solution n Add controlled redundancy to the message to improve the chances of being able to recover the original message n Trivial example: The telephone game
The ISBN Code x 1 x 2… x 10 n x 10 is a check digit chosen so that S = x 1 + 2 x 2 + … + 9 x 9 + 10 x 10 = 0 mod 11 n Can detect all single and all transposition errors n
ISBN Example n n n 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!!
UPC (Universal Product Code) n x 1 x 2… x 12 is a check digit chosen so that S = 3 x 1 + 1 x 2 + … + 3 x 11 + 1 x 12 = 0 mod 10 n Can detect all single and most transposition errors n What transposition errors go undetected? n
The Repetition Code n Send 0 and 1 n Noise may change 0 to 1 or change 1 to 0 n Instead, send codewords 00000 and 11111 n If noise corrupts up to 2 bits, decoder can use majority vote and decode received word as 00000
The Repetition Code n The distance between the two codewords is 5, because they differ in 5 spots ¡ n 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!
When is a Code “Good”? n 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
How Good Does It Get? n What are the ideal trade-offs between rate, error-correcting capability, and number of codewords? n What is the biggest distance you can get given a fixed rate or fixed number of codewords? n What is the best rate you can get given a fixed distance or fixed number of codewords?
1969 Mariner Mission n We’ll learn how Hadamard matrices were used on the 1969 Mariner Mission to build a rate 6/32 code that is approximately 100, 000 x better at correcting errors than the binary repetition code of length 5
1980 -90’s Voyager Missions n Better pictures need better codes need more sophisticated mathematics… n Picture transmitted via Reed-Solomon codes
Summary n From Caesar to Public-Key…. from Repetition Codes to Reed-Solomon Codes…. ¡ n 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!
Who Cares? n You and me! ¡ ¡ n 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, i. Tunes…
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