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# Cryptology

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1. Cryptology • The study of secret messages • Caesar cipher • f (p) = (p + 3) mod 26 • “MEET YOU IN THE PARK” becomes • “PHHW BRX LQ WKH SDUN”

2. Security Now Podcast Leo LaPorte, twit.tv, leoville.com Steve Gibson, grc.com

3. Decoder Ring (SN-31)

4. Basic Terminology plaintext - the original message ciphertext - the coded message cipher - algorithm for transforming plaintext to ciphertext key - info used in cipher known only to sender/receiver encipher (encrypt) - converting plaintext to ciphertext decipher (decrypt) - recovering ciphertext from plaintext cryptography - study of encryption principles/methods cryptanalysis (codebreaking) - the study of principles/ methods of deciphering ciphertext without knowing key cryptology - the field of both cryptography and cryptanalysis

5. Cryptology • The study of secret messages • Caesar cipher • f (p) = (p + 3) mod 26 • “MEET YOU IN THE PARK” becomes • “PHHW BRX LQ WKH SDUN”

6. English Letter Frequencies

7. Letter Pair Frequencies • For instance, given a section of English language, E tends to be very common, while X is very rare. Likewise, ST, NG, TH, and QU are common pairs of letters (termed bigrams or digraphs), while NZ and QJ are rare. The mnemonics phrase "ETAOIN SHRDLU" encodes the 12 most frequent letters in typical English language text.

8. EXCLUSIVE OR The EXCLUSIVE OR is true when exactly one of its operands is true. p q p Å q ____________________ F F F F T T T F T T T F

9. Reversible XOR 1100 plaintext Å 1010 encrypt ----- 0110 ciphertext Å 1010 decrypt ----- 1100 plaintext

10. Random Numbers • Generating a sequence of random numbers is often useful • In a game, it ensures that a player does not seethe same behavior each time • In a simulation of a complex system,random numbers can be used tohelp generate random events • Car crash in a simulationof a highway system • Likelihood of a gene in cell mutation • Weather simulation

11. Uniform Random Numbers • Uniform random number sequence • A sequence of random numbers where • Each value in the sequence is drawn from the same range of numbers • In each position of the sequence, any value in the number range is equally likely to occur

12. Random Numbers • Examples • Generate a uniform randomnumber sequence in the range1 to 6 • Use a fair six-sided dice • Each roll represents a new random number • Do two die produce uniform random numbers in the range 1 ... 12? • Generate a uniform random numbersequence in the range 1 to 2 • Use a fair coin • Heads: 1, Tails: 2

13. Random Numbers • We can write an algorithmfor generating what lookslike random numbers • Because it’s an algorithm,we know the rules for generating the next number • The generated numbers are not really random • They are properly called pseudorandom numbers 30 21 9 28 29 ...

14. Stdlib Library • Provides in part an interface to functions that generate pseudorandom numbers • rand() -- returns a uniform pseudorandom number (unsigned int) from the inclusive interval 0 to RAND_MAX • Consider rand.cpp #include <iostream> #include <string> using namespace std; #include <stdlib.h> int main() { for (int i = 1; i <= 5; ++i) cout << rand() << endl; return 0; }

15. Different Sequences • To produce a different sequence, invokevoid srand(unsigned int); • Consider seed.cpp int main() { cout << "Enter a seed: "; unsigned int Seed; cin >> Seed; srand(Seed); for (int i = 1; i <= 5; ++i) cout << rand() << endl; return 0; }

16. Different Sequences • To get a different sequence each time • Need a method of setting the seed to a random value • The standard method is to use the computer's clock as the value of the seed • The function invocation time() can be used • Returns an integral value of type time_t • Invocation time(0) returns a suitable value for generating a random sequence

17. Randseed.cpp #include <iostream> #include <string> using namespace std; #include <stdlib.h> #include <time.h> int main() { srand((unsigned int) time(0)); for (int i = 1; i <= 5; ++i) cout << rand() << endl; return 0; }

18. Symmetric Key • Symmetric-key algorithms are a class of algorithms for cryptography that use trivially related, often identical, cryptographic keys for both decryption and encryption.

19. Diffie-Hellman Key Exchange • We both agree on a public key a • I raise ab for some large integer b • You raise ac for integer c • We exchange values • You raise ab to the c power • I raise ac to the b power • We both have abc for our session key (RNG seed) No one eavesdropping can perform log ab or log ac for large values of a, b, c Security Now 34, 7:30mins-11:30

20. RSA Encryption • C = Me mod n • C = encrypted message • M = plaintext message (converted to #s via ASCII) • n is product of two primes pq • e is exponent relatively prime to (p-1)(q-1) (public key) • The formula is invertible if p, q are known • So, huge prime values are chosen for p,q (200 digits) • n cannot be factored in a reasonable amount of time • (2 billion years of computer time) • Receiver of message is given decryption key: p & q • RSA Demo

21. RSA Encryption • The RSA algorithm, invented in 1977 by Ron Rivest, Adi Shamir, and Leonard Adleman, is used frequently for public-key encryption and decryption. • It works as follows: Take two large prime numbers, p and q, and find their product n = pq; n is called the modulus. Choose a number, e, less than n and relatively prime to (p1)(q1), and find d, an inverse of e modulo (p1)(q1), which means that ed 1 mod (p1)(q1); • e and d are called the public and private exponents, respectively. The public key is the pair (n,e); the private key is d. The factors p and q must be kept secret, or destroyed. It is difficult (presumably) to obtain the private key d from the public key (n,e). If one could factor n into p and q (which can be difficult for large values of p and q) however, then one could obtain the private key d. • RSA Demo

22. Outline • Overview of Cryptography • Classical Symmetric Cipher • Modern Symmetric Ciphers (DES)

23. Basic Terminology • plaintext - the original message • ciphertext - the coded message • cipher - algorithm for transforming plaintext to ciphertext • key - info used in cipher known only to sender/receiver • encipher (encrypt) - converting plaintext to ciphertext • decipher (decrypt) - recovering ciphertext from plaintext • cryptography - study of encryption principles/methods • cryptanalysis (codebreaking) - the study of principles/ methods of deciphering ciphertext without knowing key • cryptology - the field of both cryptography and cryptanalysis

24. Classification of Cryptography • Number of keys used • Hash functions: no key • Secret key cryptography: one key • Public key cryptography: two keys - public, private • Type of encryption operations used • substitution / transposition / product • Way in which plaintext is processed • block / stream

25. Secret Key vs. Secret Algorithm • Secret algorithm: additional hurdle • Hard to keep secret if used widely: • Reverse engineering, social engineering • Commercial: published • Wide review, trust • Military: avoid giving enemy good ideas

26. Cryptanalysis Scheme • Ciphertext only: • Exhaustive search until “recognizable plaintext” • Need enough ciphertext • Known plaintext: • Secret may be revealed (by spy, time), thus <ciphertext, plaintext> pair is obtained • Great for monoalphabetic ciphers • Chosen plaintext: • Choose text, get encrypted • Useful if limited set of messages

27. Unconditional vs. Computational Security • Unconditional security • No matter how much computer power is available, the cipher cannot be broken • The ciphertext provides insufficient information to uniquely determine the corresponding plaintext • Only one-time pad scheme qualifies • Computational security • The cost of breaking the cipher exceeds the value of the encrypted info • The time required to break the cipher exceeds the useful lifetime of the info

28. Brute Force Search • Always possible to simply try every key • Most basic attack, proportional to key size • Assume either know / recognise plaintext

29. Outline • Overview of Cryptography • Classical Symmetric Cipher • Substitution Cipher • Transposition Cipher • Modern Symmetric Ciphers (DES)

30. Symmetric Cipher Model

31. Requirements • Two requirements for secure use of symmetric encryption: • a strong encryption algorithm • a secret key known only to sender / receiver Y = EK(X) X = DK(Y) • Assume encryption algorithm is known • Implies a secure channel to distribute key

32. Classical Substitution Ciphers • Letters of plaintext are replaced by other letters or by numbers or symbols • Plaintext is viewed as a sequence of bits, then substitution replaces plaintext bit patterns with ciphertext bit patterns

33. Caesar Cipher • Earliest known substitution cipher • Replaces each letter by 3rd letter on • Example: meet me after the toga party PHHW PH DIWHU WKH WRJD SDUWB

34. Caesar Cipher • Define transformation as: a b c d e f g h i j k l m n o p q r s t u v w x y z D E F G H I J K L M N O P Q R S T U V W X Y Z A B C • Mathematically give each letter a number a b c d e f g h i j k l m 0 1 2 3 4 5 6 7 8 9 10 11 12 n o p q r s t u v w x y Z 13 14 15 16 17 18 19 20 21 22 23 24 25 • Then have Caesar cipher as: C = E(p) = (p + k) mod (26) p = D(C) = (C – k) mod (26)

35. Cryptanalysis of Caesar Cipher • Only have 25 possible ciphers • A maps to B,..Z • Given ciphertext, just try all shifts of letters • Do need to recognize when have plaintext • E.g., break ciphertext "GCUA VQ DTGCM"

36. Monoalphabetic Cipher • Rather than just shifting the alphabet • Could shuffle (jumble) the letters arbitrarily • Each plaintext letter maps to a different random ciphertext letter • Key is 26 letters long Plain: abcdefghijklmnopqrstuvwxyz Cipher: DKVQFIBJWPESCXHTMYAUOLRGZN Plaintext: ifwewishtoreplaceletters Ciphertext: WIRFRWAJUHYFTSDVFSFUUFYA

37. Monoalphabetic Cipher Security • Now have a total of 26! = 4 x 1026 keys • Is that secure? • Problem is language characteristics • Human languages are redundant • Letters are not equally commonly used

38. English Letter Frequencies

39. Example Cryptanalysis • Given ciphertext: UZQSOVUOHXMOPVGPOZPEVSGZWSZOPFPESXUDBMETSXAIZ VUEPHZHMDZSHZOWSFPAPPDTSVPQUZWYMXUZUHSX EPYEPOPDZSZUFPOMBZWPFUPZHMDJUDTMOHMQ • Count relative letter frequencies (see text) • Guess P & Z are e and t • Guess ZW is th and hence ZWP is the • Proceeding with trial and error finally get: it was disclosed yesterday that several informal but direct contacts have been made with political representatives of the viet cong in moscow

40. One-Time Pad • If a truly random key as long as the message is used, the cipher will be secure - One-Time pad • E.g., a random sequence of 0’s and 1’s XORed to plaintext, no repetition of keys • Unbreakable since ciphertext bears no statistical relationship to the plaintext • For any plaintext, it needs a random key of the same length • Hard to generate large amount of keys • Have problem of safe distribution of key

41. Transposition Ciphers • Now consider classical transposition or permutation ciphers • These hide the message by rearranging the letter order, without altering the actual letters used • Can recognise these since have the same frequency distribution as the original text

42. Rail Fence cipher • Write message letters out diagonally over a number of rows • Then read off cipher row by row • E.g., write message out as: m e m a t r h t g p r y e t e f e t e o a a t • Giving ciphertext MEMATRHTGPRYETEFETEOAAT

43. Product Ciphers • Ciphers using substitutions or transpositions are not secure because of language characteristics • Hence consider using several ciphers in succession to make harder, but: • Two substitutions make a more complex substitution • Two transpositions make more complex transposition • But a substitution followed by a transposition makes a new much harder cipher • This is bridge from classical to modern ciphers

44. Outline • Overview of Cryptography • Classical Symmetric Cipher • Modern Symmetric Ciphers (DES)

45. Block vs Stream Ciphers • Block ciphers process messages in into blocks, each of which is then en/decrypted • Like a substitution on very big characters • 64-bits or more • Stream ciphers process messages a bit or byte at a time when en/decrypting • Many current ciphers are block ciphers, one of the most widely used types of cryptographic algorithms

46. Block Cipher Principles • Most symmetric block ciphers are based on a Feistel Cipher Structure • Block ciphers look like an extremely large substitution • Would need table of 264 entries for a 64-bit block • Instead create from smaller building blocks • Using idea of a product cipher

47. Substitution-Permutation Ciphers • Substitution-permutation (S-P) networks [Shannon, 1949] • modern substitution-transposition product cipher • These form the basis of modern block ciphers • S-P networks are based on the two primitive cryptographic operations • substitution (S-box) • permutation (P-box) • provide confusion and diffusion of message

48. Confusion and Diffusion • Cipher needs to completely obscure statistical properties of original message • A one-time pad does this • More practically Shannon suggested S-P networks to obtain: • Diffusion – dissipates statistical structure of plaintext over bulk of ciphertext • Confusion – makes relationship between ciphertext and key as complex as possible

49. Feistel Cipher Structure • Feistel cipher implements Shannon’s S-P network concept • based on invertible product cipher • Process through multiple rounds which • partitions input block into two halves • perform a substitution on left data half • based on round function of right half & subkey • then have permutation swapping halves

50. Feistel Cipher Structure