Information Theory Rusty Nyffler Introduction n n Entropy

Information Theory Rusty Nyffler

Introduction n n Entropy Example of Use Perfect Cryptosystems Common Entropy

Entropy n Measure of uncertainty n Entropy of a fair coin toss = 1

Why is entropy important? n n n Measures the likelihood of the eavesdropper knowing something about the key, given the ciphertext Greater entropy usually means greater security Entropy of a cryptosystem (K = number of possible keys (if all keys are equally likely)

Example n n 3 possible plaintexts: a(. 5), b(. 3), c(. 2) Two keys k 1 and k 2, equally likely (. 5) Possible ciphertexts U, V, W If someone obtains sent ciphertext, then information is known about the plaintext. ¡ ¡ If U was received, then a was the text If V was received, then b or c was the text

Example Continued n Moreover, ¡ p(b|V) =. 6 ¡ p(c|V) =. 4 ¡ p(b|W) =. 6 ¡ p(c|W) =. 4 ¡ n Which means it’s more likely that b is the plaintext when V or W is caught Entropy shows that the eavesdropper knows more about the plaintext when the ciphertext is intercepted

Perfect Cryptosystems n n n In a perfect cryptosystem (i. e. unbreakable), the ciphertext should not give any more information about the plaintext or the key H(P) = H(P|C) H(K) = H(K|C)

Common Entropy n The English language ¡ ¡ n Somewhere between 1. 42 and. 72 English is around 75% redundant In many algorithms, the longer the key, the more entropy the system has ¡ Vigenere

Common Entropy n RSA ¡ ¡ Entropy = 0 All the info you need is in n, e, and c … except it’ll take a while to factor n

References 1. 2. 3. 4. Mac. Kay, David J. C. Information Theory, Inference, and Learning Algorithms. 18 April 2003. http: //www. inference. phy. cam. ac. uk/itprnn/book. l. pdf North Carolina State University. Cryptography FAQ. 21 March 2003. http: //isc. faqs. org/faqs/cryptography-faq/part 01/ Raynal, Frederick. Weak Algorithms. 2002. http: //www. owasp. org/asac/cryptographic/algorithms. shtml Trappe, Wade and Lawrence C. Washington. Introduction to Cryptography with Coding Theory. Upper Saddle River, New Jersey: Prentice-Hall, Inc. , 2002.