# CS 392/681 Computer Security. Module 1 Private Key Cryptography

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1 CS 392/681 Computer Security Module 1 Private Key Cryptography

2 Logistics Office hours - Thursday 5 to 6 (tentative). Welcome to walk in anytime. s until midnight. AIM: evilproffy. Lab 0 due date postponed to Monday (13 th ). Lab 1 assigned. Due on 22 nd!! Undergrad/Grad issues. Cooperation on homework don t cross the line. ISIS issues? Schedule, lectures, announcements posted on web site Your resp onsibility to monitor this. Module 1 - Private Key Crypto 2

3 Data Encryption Encryption is the process of encoding a message such that its meaning is not obvious. Decryption is the reverse process, ie, transforming an encrypted message to its original form. Plaintext Encryption Ciphertext Decryption Plaintext We denote plaintext by P and ciphertext by C. C = E(P), P = D(C) and P = D(E(P)), where E() is the encryption function (algorithm) and D() the decryption function. Module 1 - Private Key Crypto 3

4 Kerckhoff s Principle How do you prevent and eavesdropper from computing P, given C? Keep the encryption algorithm E() secret. BAD IDEA!! Choose E() (and corresponding D()) from a large collection, based on secret key. GOOD IDEA!! Kerckhoff s principle. Secret Key Plaintext Encryption Ciphertext Decryption Plaintext C = E(K, P) and P = D(K, C) Module 1 - Private Key Crypto 4

5 Symmetric and Asymmetric Cryptosystems Just by changing key we have different encryptions of one plaintext. If the encryption key and the decryption key are the same then we have a symmetric encryption scheme (also private key, one-key). If the encryption key and the decryption key are different then we have an asymmetric encryption scheme (also public key, two-key). A cryptosystem then a five-tuple consisting of 1) The set of all plaintexts 2) The set of all ciphertexts 3) The set of all keys 4) A family of encryption functions 5) A family of decryption functions. Module 1 - Private Key Crypto 5

6 Example Caesar Cipher Let messages be all lower case from a through z (no spaces or punctuation). itsnotthathardtoread Represent letters by numbers from 0 to 25. Encryption function C i = E(P i ) = P i + K. where K is secret key and addition done modulo 26. Decryption is P i = D(C i ) = C i UNIX ROT13 uses K as 13. -K. Module 1 - Private Key Crypto 6

7 Cryptanalysis A cryptosystem had to be secure against the following kinds of attacks: Ciphertext only attack. Known plaintext attack. Chosen plaintext attack. Adaptive chosen plaintext attack. Chosen ciphertext attack. Chosen key attack. Of course there is one attack against which no cryptosystem can offer protection rubber hose attack. Module 1 - Private Key Crypto 7

8 Brute Force Attacks. Since the key space is finite, given a ciphertext a cryptanalyst can try and check all possible keys. For above to be not feasible, key space should be large!! How large? How about 2 56? Large enough to make it impractical for an adversary. But what is impractical today, may not be so tomorrow. In practice, for a good cryptosystem, the only possible attack should be the brute force attack, which should be impractical into the foreseeable future, as slong as message may have value. Module 1 - Private Key Crypto 8

9 Substitution Ciphers Basic idea substitute each block of plaintext by a different block. If plaintext is English then Mona-alphabetic substitution. Poly-alphabetic substitution. If plaintext is binary string then map one block of bits to another. Plaintext: Ciphertext: This is called block encryption. Very common. Module 1 - Private Key Crypto 9

10 Encryption by Mono-alphabetic Substitution. P O K E M O N M A S T E R 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 P O L Y T E C H N I U V R S B K W A D F G J M Q X Z K B U T R B S R P D F T A Key space is large 26! (How do you remember a key? See example). However, mono-alphabetic substitution is easy to break as it preserves source first order statistics. Large key space is necessary but not sufficient condition for security! Module 1 - Private Key Crypto 10

11 Encryption by Poly-alphabetic Substitution. P O L Y T E C H N I/ U V R S BJ K W A D F G M Q X Z Playfair cipher. Used by British Army in WW1 and WW2. Can be broken easily today with only a ciphertext of length about 100. Encrypt plaintext a pair at a time. Two letters specify a rectangle. Substitute by opposite corner pair. Eg: VX -> SM. If they fall in same row or column, then using next pair in circular manner. Eg: LY -> TP. Repeated letters are broken by filler letter. I/J chosen randomly. Module 1 - Private Key Crypto 11

12 Poly-alphabetic Cipher Vigenere. Use K mono-alphabetic ciphers E 1, E 2, E k. In position i, of plaintext, use cipher E i. Example using Caesar ciphers Plaintext: helloiloveyouwontyoutellmeyourname Key: polytechnicpolytechnicpolytechnicpoly Ciphertext: wswjhmnv coxc A little harder to break but trivial once you know key length! Some well known techniques for determining key length See text. Module 1 - Private Key Crypto 12

13 Vernam The Perfect Substitution Cipher. If we use Vigenere with key length as long as plaintext, then cryptanalysis will be difficult! If we change key every time we encrypt then cryptanalyst s job becomes even more difficult. Onetime pad or Vernam Cipher. How do we get such long keys? A large book shared by transmitter and receiver. Initial key followed by previous messages themselves!! Random number sequence based on common shared and secret seed. Such a cipher is difficult to break but not very practical. Module 1 - Private Key Crypto 13

14 Binary Vernam Unconditional Security. If plaintext is binary string and key is binary string of equal length then encryption can be done by a simple xor operation. Plaintext: Key: Ciphertext: If plaintexts uniformly distributed and keys are random then such a system offers unconditional security perfect secrecy! (Under the right mathematical formulation and assumptions). Intuitively perfect secrecy can be seen from the fact that given any plaintext and ciphertext, there is a key which maps the selected plaintext to the selected ciphertext. So given a ciphertext, we get no information whatsoever on what key or plaintext could have been used. How do we obtain random bit-strings for shared secret keys? Again system is not practical. Module 1 - Private Key Crypto 14

15 Perfect Secrecy M 1 M 2 M 3 C 1 C 2 C 3 Four possible Keys K 1,K 2,K 3, And K 4 M 4 C 4 M 5 C 5 Given a ciphertext, cryptanalyst cannot reduce uncertainty. Property can be formulated in more mathematically rigorous manner. Module 1 - Private Key Crypto 15

16 Imperfect Secrecy M 1 C 1 M 2 C 2 M 3 C 3 C 4 C 5 Given C 1 we know Message is M 3!! Given C 5, only one bit of uncertainty. Module 1 - Private Key Crypto 16

17 Encryption by Transposition P O K E M O N M A S T E R O E N T E M P K M O A S R Harder to break than substitution ciphers Preserve first order statistics One can arrange plaintext in table and sort rows and columns. Module 1 - Private Key Crypto 17

18 Product Ciphers To get improved security one can encrypt the ciphertext again. If one uses same algorithm super encryption. May or may not be useful. For example, super-encryption with Caesar cipher is as good as single encryption! If one uses different algorithms product cipher. Product ciphers based on sequence of substitutions and transpositions are very popular. You will see one later DES. Module 1 - Private Key Crypto 18

19 Shannon Characteristics of Good Ciphers The amount of secrecy needed should determine the amount of labor appropriate for encryption and decryption. The set of keys and enciphering algorithms should be free from complexity. The implementation of the process should be as simple as possible. Errors in ciphering should not propagate and cause corruption of future information in the message. The size of enciphered text should be no longer than the text of the original message. Module 1 - Private Key Crypto 19

20 Confusion and Diffusion Confusion: The cryptanalyst should not be able to predict what changing one character in the plaintext will do to the ciphertext. Diffusion: Changes in the key should affect many parts in the ciphertext. Module 1 - Private Key Crypto 20

21 DES Data Encryption Standard Private key. Encrypts by series of substitution and transpositions. Worldwide standard for more than 20 years. Has a history of controversy. Designed by IBM (Lucifer) with later help (interference?) from NSA. No longer considered secure for highly sensitive applications. Replacement standard (AES) recently completed. Module 1 - Private Key Crypto 21

22 DES - Overview Module 1 - Private Key Crypto 22

23 DES Each iteration. Module 1 - Private Key Crypto 23

24 DES Computation of F(R i-1,k i ) Module 1 - Private Key Crypto 24

25 Computation of F: Expansion function E: maps bit string of length 32 to bit string of length 48. Permutes bits in a fixed way and duplicates certain bits Key schedule: each round uses a 48 bit key obtained by performing permutations, shifts, and discarding bits from the original 56 bit key. Fixed algorithm for each round resulting 48 bit string broken into 8 6-bit strings Module 1 - Private Key Crypto 25

26 S-boxes: S Sj Is the table entry from S( b1b 2b3b4b5b6) row : b 1 b 2 column : b 3 b 4 b 5 b 6 S( ) = table[1,9] = 6d = 0110 Module 1 - Private Key Crypto 26

27 x L0 R0 R 0 L0 F( R0, K 1) Plain text Initial permutation (IP) Round-1 (key K 1 ) Rounds 2-15 L15 R15 L15 F( R15, K 16) L15 F( R15, K 16) R15 y R15 Round-16 (key K 16 ) swap IP inverse Cipher text

28 L15 F( R15, K 16) R15 y L15 F( R15, K 16) R15 R15 = L IP inverse Cipher text IP Round-1 (K 16 ) F( R15, K 16) F( R15, 16) 15 K encrypt decrypt R15 L15 Since b b = 0 b 0 = b

29 DES Security S-Box design not well understood (secret). Has survived some recent sophisticated attacks (differential cryptanalysis). Key is too short (thanks to NSA!). Hence is vulnerable to brute force attack distributed attack took 3 months. \$1,000,000 machine will crack DES in 35 minutes 1997 estimate. 10, days. In 1999 EFF achieved 245 billion keys per second rate to crack in 22 hours. Module 1 - Private Key Crypto 29

30 DES Cracking machine Module 1 - Private Key Crypto 30

31 Super-encryption. If key length is a concern, then instead of encrypting once, encrypt twice!! C = E K2 (E K1 (P)) P = D K2 (D K1 (C)) Does this result in a larger key space? That is a new mapping that could not have been obtained by a single key? With Caesar cipher NO! With DES yes! Encrypting with multiple keys is known as superencryption. May not always be a good idea. Module 1 - Private Key Crypto 31

32 Double DES K 1 K 2 P E X E C Encryption K 2 K 1 C D X D P Decryption Double DES is almost as easy to break as single DES (Needs more memory though)! Module 1 - Private Key Crypto 32

33 Double DES Meet-in-the-middle Attack. Based on the observation that, if Then C = E K2 (E K1 (P)) X = E K1 (P) = D K2 (C). Given a known (P, C) pair, encrypt P with all possible values of K and store result in table T. Next, decrypt C with all possible keys K and check result. If match occurs then check key pair with new known (P, C) pair. If match occurs, you have found the keys. Else continue as before. Process will terminate successfully. Module 1 - Private Key Crypto 33

34 Meet-in-the-middle Explanation. The first match does not say anything as we have 2 64 ciphertexts and keys. On the average / 2 64 = 2 48 keys will produce same ciphertext. So there could be 2 48 false alarms. However, with second known (P, C) pair, probability that E K1 (P) = D K2 (C) is So, probability that false alarm will survive two known (P, C) pairs is 2 48 / 2 64 = One can always check a third pair to further reduce the chance of a false alarm. Module 1 - Private Key Crypto 34

35 Triple DES K 1 K 2 K 1 P A B E D E Encryption C K 1 K 2 K 1 C B A D E D Decryption P Triple DES (2 keys) requires search. Is reasonably secure. 3 keys requires Module 1 - Private Key Crypto 35

36 Encryption Modes - ECB. Although DES encrypts 64 bits (a block) at a time, it can encrypt a long message (file) in Electronic Code Book (ECB) mode. Time = 1 Time = 2 Time = N P1 P2 PN K DES Encrypt K DES Encrypt K DES Encrypt C1 C2 CN C1 C2 CN K DES Decrypt K DES Decrypt K DES Decrypt P1 P2 PN If same key is used then identical plaintext blocks map to identical ciphertext. Module 1 - Private Key Crypto 36

37 Block Ciphers and Stream Ciphers Block ciphers partition plaintext into blocks and encrypt each block independently (with the same key) to produce ciphertext blocks. Caesar, Vigenere, DES etc. A stream cipher generates a keystream and encrypts by combining the keystream with the plaintext, usually with the bitwise XOR operation. Generation of the keystream can be independent of the plaintext and ciphertext (a synchronous stream cipher) or it can depend on the data and its encryption (self-synchronizing stream cipher ). Module 1 - Private Key Crypto 37

38 Stream Ciphers A Simple Form In their simplest form, a stream ciphers convert plaintext to ciphertext 1 bit at a time. A keystream generator outputs a stream of bits: k 1, k 2,, k n. Module 1 - Private Key Crypto 38

39 Stream Ciphers - Security Issues The security of this approach depends totally on the keystream generator. If it outputs an endless stream of zeros, the ciphertext will equal the plaintext and the operation will be meaningless. If it outputs an endless stream of random bits you have a one-time pad and perfect security. The reality lies between the simple XOR and the onetime pad. So how do we generate random bitstreams? One way is Linear Feedback Shift Register (LFSR) Another way is using a block cipher technique in a feedback mode. Module 1 - Private Key Crypto 39

40 Generating Random Key Streams Linear feedback shift registers (LFSR) b i are bits and f is a function depending of b 1, b 2,..., b n. The result f(b 1, b 2,..., b n ) enters the nth position of the register and all other bits b n,..., b r are shifted right one position, b 1 being the output bit. Module 1 - Private Key Crypto 40

41 Example LFSR Let n = 4 and f(b1, b2, b3, b4) = b1 b4. Let 0110 be the starting position (the seed). Then this LFSR will generate the sequence Stream ciphers based on LFSR s are well studied Module 1 - Private Key Crypto 41

42 Cipher Block Chain (CBC) Mode. IV Time = 1 P1 Time = 2 P2 Time = N PN + + CN-1 + K DES Encrypt K DES Encrypt K DES Encrypt C1 C2 CN C1 C2 CN K DES Decrypt K DES Decrypt K DES Decrypt IV + + CN-1 + P1 P2 PN Module 1 - Private Key Crypto 42

43 CBC Pros and Cons. If IV is different then different instances of same message (or block) will get encrypted differently. How does receiver know IV? Choose at random and send encrypted as first block. What happens if k th cipher block C K gets corrupted in transmission. With ECB Only decrypted P K is affected. With CBC? Only blocks P K and P K+1 are affected!! This can also allow some message tampering! What if one plaintext block P K is changed? With ECB only C K affected. With CBC all subsequent ciphertext blocks will be affected. This leads to an effective MAC based on DES CBC. Module 1 - Private Key Crypto 43

44 Cipher Feedback Mode (CFB). IV Shift register 64 - j bits j bits Shift register 64 - j bits j bits CM-1 Shift register 64 - j bits j bits 64 K DES Encrypt K DES Encrypt K DES Encrypt Select j bits j 64 Discard 64 - j bits Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits + j C 1 + C2 + CM j P 1 P 2 P M IV Shift register 64 - j bits j bits Shift register 64 - j bits j bits CM-1 Shift register 64 - j bits j bits K DES Encrypt K DES Encrypt K DES Encrypt Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits + C 1 + C 2 + C M P 1 P 2 P M Module 1 - Private Key Crypto 44

45 CFB Properties J is normally 8. Change in one plaintext bit is going to affect all subsequent ciphertext bits. So can be used for MAC. Change in ciphertext bit results in??? Module 1 - Private Key Crypto 45

46 Output Feedback Mode (OFB). IV Shift register 64 - j bits j bits Shift register 64 - j bits j bits OM-1 Shift register 64 - j bits j bits K DES Encrypt K DES Encrypt K DES Encrypt Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits + C1 + C2 + CM P 1 P 2 P M (a) Encryption IV Shift register 64 - j bits j bits Shift register 64 - j bits j bits OM-1 Shift register 64 - j bits j bits K DES Encrypt K DES Encrypt K DES Encrypt Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits Select j bits Discard 64 - j bits + + C1 C2 CM + P1 (b) Decryption P2 PM Figure 3.14 J-Bit Output Feedback (OFB) Mode Module 1 - Private Key Crypto 46

47 OFB Properties. Bit errors in transmission do not propagate. One can selectively flip ciphertext bits to flip corresponding decrypted plaintext bits. Bad!! Module 1 - Private Key Crypto 47

48 AES History National Institute of Science and Technology DES is an aging standard that no longer addresses today s needs for strong encryption Triple-DES: Endorsed by NIST as today s defacto standard AES: The Advanced Encryption Standard Finalized in 2001 Goal To define Federal Information Processing Standard (FIPS) by selecting a new powerful encryption algorithm suitable for encrypting government documents AES candidate algorithms were required to be: Symmetric-key, supporting 128, 192, and 256 bit keys Royalty-Free Unclassified (i.e. public domain) Available for worldwide export Module 1 - Private Key Crypto 48

49 History (cont.) AES Round-3 Finalist Algorithms: MARS Candidate offering from IBM RC6 Developed by Ron Rivest of RSA Labs, creator of the widely used RC4 algorithm Twofish From Counterpane Internet Security, Inc. Serpent Designed by Ross Anderson, Eli Biham and Lars Knudsen Rijndael Designed by Joan Daemen and Vincent Rijmen Module 1 - Private Key Crypto 49

50 Rijndael The Winner: Rijndael Joan Daemen (of Proton World International) and Vincent Rijmen (of Katholieke Universiteit Leuven). (pronounced Rhine-doll ) Allows only 128, 192, and 256-bit key sizes (unlike the other candidates) Variable block length of 128, 192, or 256 bits. All nine combinations of key/block length possible. A block is the smallest data size the algorithm will encrypt Vast speed improvement over DES in both hardware and software implementations 8416 bytes/sec on a 20MHz CPI) 8.8 Mbytes/sec on a 200MHz Pentium Pro Module 1 - Private Key Crypto 50

51 Rijndael Structure Rijndael consists of an initial Round Key addition; Nr-1 Rounds; a final round. In pseudo C code, this gives: Rijndael(State,CipherKey) { KeyExpansion(CipherKey,ExpandedKey) ; AddRoundKey(State,ExpandedKey); For( i=1 ; i<nr ; i++ ) Round(State,ExpandedKey + Nb*i) ; FinalRound(State,ExpandedKey + Nb*Nr); } Module 1 - Private Key Crypto 51

52 Rijndael Key W KE Key Expansion Round Keys K 1 K 2 K 3 K n-2 K n-1 K n X R 1 R 2 R 3 R n-2 R n-1 R n Y Encryption Rounds r 1 r n Key is expanded to a set of n round keys Input block X undergoes n rounds of operations (each operation is i based on value of the nth round key), until it reaches a final round. r Strength relies on the fact that it s s difficult to obtain the intermediate result (or state) ) of round n from round n+1 without the round key. Module 1 - Private Key Crypto 52

53 Number of Rounds Number of rounds (Nr) as a function of the block (Nb) and key length (Nk) in 32 bit words. Nr Nb = 4 Nb = 6 Nb = 8 Nk = Nk = Nk = Module 1 - Private Key Crypto 53

54 Rijndael K i Detailed view of round i Result from round i-1 ByteSub ShiftRow MixColumn AddRoundKey Pass to round i+1 Each round performs the following operations: Non-linear Layer: No linear relationship between the input and output of a round Linear Mixing Layer: Guarantees high diffusion over multiple rounds Very small correlation between bytes of the round input and the bytes of the output Key Addition Layer: Bytes of the input are simply XOR ed with the expanded round key Module 1 - Private Key Crypto 54

55 Rijndael Three layers provide strength against known types of cryptographic attacks: Rijndael provides full diffusion after only two rounds Linear and differential cryptanalysis Known-key and related-key attacks Square attack Interpolation attacks Weak-keys Rijndael has been shown to be K-secure: No key-recovery attacks faster than exhaustive search exist No known symmetry properties in the round mapping No weak keys No related-key attacks: No two keys have a high number of expanded round keys in common Module 1 - Private Key Crypto 55

56 Rijndael: ByteSub Each byte at the input of a round undergoes a nonlinear byte substitution according to the following: 1. First, taking the multiplicative inverse in GF(28). 00 is mapped onto itself. 2. Then, applying an affine (over GF(2)) transformation. Substitution ( S )-box Affine Transform Module 1 - Private Key Crypto 56

57 Rijndael: ShiftRow Depending on the block length, each row of the block is cyclically shifted according to the above table Module 1 - Private Key Crypto 57

58 Rijndael: MixColumn Each column is multiplied by a fixed polynomial C(x) = 03 *X *X *X + 02 This corresponds to matrix multiplication b(x) = c(x) a(x): Module 1 - Private Key Crypto 58

59 Rijndael: Key Expansion and Addition Each word is simply XOR ed with the expanded round key Key Expansion algorithm: KeyExpansion(int* Key[4*Nk], int* EKey[Nb*(Nr+1)]) { for(i = 0; i < Nk; i++) EKey[i] = (Key[4*i],Key[4*i+1],Key[4*i+2],Key[4*i+3]); for(i = Nk; i < Nb * (Nr + 1); i++) { temp = EKey[i - 1]; if (i % Nk == 0) temp = SubByte(RotByte(temp)) ^ Rcon[i / Nk]; EKey[i] = EKey[i - Nk] ^ temp; } } Module 1 - Private Key Crypto 59

60 Rijndael: Implementations Rijndael is well suited for software implementations on 8- bit processors (important for Smart Cards ) Operations focus on bytes and nibbles, not 32 or 64 bit integers Layers such as ByteSub can be efficiently implemented using small tables in ROM (e.g. < 256 bytes). No special instructions are required to speed up operation For 32-bit implementations: An entire round can be implemented via a fast table lookup routine on machines with 32-bit or higher word lengths Considerable parallelism exists in the algorithm Each layer operates in a parallel manner on bytes of the round state, all four component transforms act on individual parts of the block Although the Key expansion is complicated and cannot be parallelised, it only needs to be performed once until the two parties switch keys. Module 1 - Private Key Crypto 60

61 Rijndael: Implementations Hardware Implementations Performs very well in software, but in some cases more performance is required (e.g. server and VPN applications). Multiple S-Box engines, round-key EXORs, and byte shifts can all be implemented efficiently in hardware when absolute speed is required Small amount of hardware can vastly speed up 8-bit implementations Inverse Cipher Except for the non-linear ByteSub step, each part of Rijndael has a straightforward inverse and the operations simply need to be undone in the reverse order. Same code that encrypts a block can also decrypt the same block simply by changing certain tables and polynomials for each layer. The rest of the operation remains identical. Module 1 - Private Key Crypto 61

62 Rijndael Future Rijndael is an extremely fast, state-of-the-art, highly secure algorithm Has efficient implementations in both hardware and software; it requires no special instructions to obtain good performance on any computing platform Despite being the chosen by NIST as the AES candidate winner, Rijndael is not yet automatically the new encryption standard Triple-DES, still highly secure and supported by NIST, is expected to be common for the foreseeable future. Module 1 - Private Key Crypto 62

63 Other Private Key Cryptosystems IDEA Twofish Blowfish RC4, RC5, RC6 Serpent MARS Feal Module 1 - Private Key Crypto 63

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