Advanced Encryption Techniques: Transpositions in Cryptography

1. Chapter 2Basic Encryption and Decryption (part B)

2. Transpositions (Permutations) • P.52 • Transposition: an encryption in which the letters of the message are rearranged • Also known as permutations • Compare the goals: • Substitution  confusion • Transposition  diffusion

3. Confusion vs Diffusion • Confusion: making it difficult to determine how a message and key were transformed into ciphertext. • Diffusion: spreading the information from the message or the key out widely across the ciphertext • See p.62 for more discussions.

4. Columnar Transpositions • A rearrangement of the plaintext characters into columns. • The ciphertext is generated from the columns. • Example: p.53

5. Complexity of Columnar Transpositions • Time: proportional to the length of the message, that is, O(n) or at the order of function n. • Space: depends on the length of the message. • Output cannot be produced until all characters of the message have been read. • Initial delay varies, depending on the length of the message. C.f., constant initial delay in the previous (substitution) algorithms.

6. Digrams, Trigrams, & Other Patterns • Digrams: groups of two letters • Trigrams: groups of three letters • Table 2-2 (p.54): Most Frequent digrams an Trigrams Note: not counting digrams that consist of the last letter of one word and the first letter of the next word.

7. Cryptanalysis by Digram Analysis • To compute the letter frequencies • Clue: The fact that all letters appear with their normal frequencies implies that a transposition has been performed. • (Same IC as monoalphabrtic 0.068 same distribution graph as plaintext English) • To find where in the ciphertext a pair of adjacent columns lies (that is, to determine the width of a row in the original table used for encryption) • The ‘moving window’ method: Fig. 2-5 (p.56)

8. The ‘moving window’ method

9. The ‘moving window’ method • Pick a window size, say n. • Compare every Ci, 1  i  n,in the window to Ci+n and determine if the two form a common digram by checking their frequency. • Do most of the digrams look reasonable? Compute their mean and std. deviation

10. Double transposition (P.51) • Involves two columnar transpositions • An example of product ciphers, in which one encryption is applied to the result of another C = E2 (E1 (P) )

11. Fractionated Morse • A keyed monoalphabetic cipher • Uses Morse code as its base • Double encodings (P  Morse code  P’) • 3 steps: • Convert P to Morse code, using separator(s) between letters and words. • Divide the Morse code messages into blocks of 3 symbols. • Each block of symbols is encoded as the letter corresponding to that 3-symbol pattern

12. Stream versus Block Ciphers • Stream ciphers: The plaintext characters are encoded by the sender letter-by-letter as sent to the receiver. • Example: substitution ciphers • Block ciphers: Blocks of plaintext are encoded into ciphertext before being sent. • Example: columnar transposition

13. Stream Ciphers + Fast + Little storage space + Low error propagation, meaning that encoding errors affect just one character in the ciphertext • low diffusion, meaning that individual characters in the ciphertext can be analyzed using frequency distribution, digram analysis, IC and the Kasiski method • Susceptibility to malicious insertions and modifications

14. Block Ciphers - Slow - Require more storage space - Error propagation + High diffusion + High immunity to insertions

15. 4 cryptanalysis cases & 5 approaches • Ciphertext only • Ciphertext-only attack • Full or partial plaintext • Known plaintext attack • Probable plaintext analysis • Ciphertext of any plaintext • Chosen plaintext attack • Algorithm + Ciphertext • Chosen ciphertext attack

16. Brute Force Known Plaintext Chosen Plaintext Adaptive Chosen Plaintext Ciphertext only Chosen Ciphertext Adaptive Chosen Ciphertext Meet in the Middle Differential Cryptanalysis Linear Cryptanalysis Differential Linear Cryptanalysis Factoring Statistical Complete listing of examples of cryptanalysis methods

17. Data Encryption Standard (DES) • 1976: officially adopted as a U.S. federal standard • Was authorized for use on all public and private sector unclassified communication • Later became an international standard by the ISO • The strength of DES comes from repeated application of the two encryption techniques, substitution and permutation, one on top of the other, for a total of 16 cycles. • Product cipher: C = E2 ( E1(P) ) • A secret key method

18. Chronology • 1976 DES is approved as a standard • 1977 DES is published as a FIPS standard FIPS PUB 46 • 1992 Biham and Shamir report the first theoretical attack with less complexity than brute force (differential cryptanalysis). However, it requires an unrealistic 247 chosen plaintexts. • 1993 DES is reaffirmed for the third time as FIPS 46-2 • June1997 The DESCHALL Project breaks a message encrypted with DES for the first time in public. • July1998 The EFF's DES cracker (Deep Crack) breaks a DES key in 56 hours. • January1999 DES key in 22 hours and 15 minutes.25 October • 1999 DES is reaffirmed for the fourth time as FIPS 46-3, which specifies the preferred use of Triple DES, with single DES permitted only in legacy systems. • 2001 The Advanced Encryption Standard is published in FIPS 197 • 2002 The AES standard becomes effective • 2004 The withdrawal of FIPS 46-3 (and a couple of related standards) is proposed in the Federal Register[2]19 May2005NIST withdraws FIPS 46-315 March • 2007 The FPGA based parallel machine of the University of Bochum and Kiel, Germany, breaks DES in 6.4 days at $10,000 hardware cost

19. Data Encryption Standard (DES) • 16 cycles of substitutions and permutations: Fig.10-7 (p.647) • Steps in a single cycle: Fig. 647, Fig. 10-9 Operates on blocks of data: 64 bits per block Splits a data block in half: left half, right half (32 bits) Scrambles each half independently: shifted left, permuted Combines the key with the right half: the result is then combined with the left half Swaps the two halves: T = right half; right half += key; right half += left half; left half = T.

20. Data Encryption Standard (DES) (1 CYCLE)

21. Key Transformation • 64-bit key, but only 56 of the bits are used (bits 8, 16, 24, 32, 40, 48, 56, 64 are parity bits) • At each step, the key is split into two 28-bit halves. • The halves are shifted by a specified number of digits (Table 10-2, p.651), pasted together again, and then 48 of these 56 bits are permuted as a key during this step (choice permutation: Table 10-3 P. 651).

22. Data Encryption Standard (DES) • How to combine a 32-bit right half with a 56-bit key? p.649: Fig. 10-11 Expansion permutation: The 32-bit half is expanded to 48 bits by repeating certain bits See Table 10-1 p.649 Permuted choices: The 56-bit key is reduced to 48 bits by choosing only certain bits • Fig. 10-11: a detailed single cycle

23. Substitutions and S-boxes • An S-box is a table in which 6 bits of data are replaced by 4 bits. • The 48-bit input is divided into eight 6-bit blocks  8 S-boxes for substitutions (Table 10-4, p.652) • Substitutions by S-boxes

24. S-box substitutions • Use Table 10-4 • Input: Bj = 6 bits (b1 b2 b3 b4 b5 b6) • Output: a 4-bit binary number • Process: • r = b1 b6 • c = b2 b3 b4 b5 • result = the value at (r, c) of the S-box j (Sj) of Table 3-6. • Convert the result to a binary number.

25. S-box substitutions • Example: p.650 Input = Bj =B7 = 010011b r = 01b = 1 c = 1001b = 9 result = (r, c) = (1, 9) in S7 = 3 = 0011b • Suppose the Input = B4 = 101010b. What’s the output of the substitution?

26. P-box permutations • Permutations: Initial permutation Permutation in each cycle Final inverse permutation • inverse & final permutations: Table 3-8, 3-9, p.110 • Per-cycle permutation: p.653: Table 10-5 Examples Bit 1  bit 9 Bit 23 bit 11 Bit 19  ? Bit 27  ?

27. Summary of DES Encryption • Input = (64-bit data block, 64-bit key) • process 64-bit key  56 bits Initial permutation 16 cycles: Key transformation (split, shift, permuted) Data blocks substitution & permutation Lj = Rj-1 ; Rj = Lj-1 XOR f(Rj-1, kj) S-boxes, P-boxes Final permutation • Output = 64-bit ciphertext

28. DES Decryption • Encryption: (Lj-1 ,Rj-1)  (Lj ,Rj) Lj = Rj-1 ; Rj = Lj-1 XOR f(Rj-1, kj) • Decryption: (Lj ,Rj)  (Lj-1 ,Rj-1) Rj-1 =Lj Lj-1 = Rj XOR f(Lj, kj) • Reverse key transformation