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CRYPTOGRAPHY AND NETWORK SECURITY

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Simple DES – Differential cryptoanalysis – DES – Modes of operation – Triple DES – AES – RC4 – RSA – Attacks – Primality test – factoring.

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CRYPTOGRAPHY AND NETWORK SECURITY

1. 1. IT2352 CRYPTOGRAPHY AND NETWORK SECURITY UNIT – II Dr.A.Kathirvel, Professor and Head, Dept of IT Anand Institute of Higher Technology, Chennai
2. 2. UNIT - II Simple DES – Differential cryptanalysis – DES – Modes of operation – Triple DES – AES – RC4 – RSA – Attacks – Primality test – factoring.
3. 3. DATA ENCRYPTION • Encryption Definition: – The action of disguising information so that it can be recovered easily by the persons who have the key, but is highly resistant to recovery by persons who do not have the key.
4. 4. DATA ENCRYPTION – A message is cleartext (plaintext) is encrypted (disguised) through the use of an encryption key to create a Ciphertext.
5. 5. DATA ENCRYPTION – The encryption key may be changed from time to time to make an intruder’s task more difficult. – Restoration of a ciphertext to cleartext is achieved by the action of decryption using a decryption key. • In symmetric (Single key) - The encryption and decryption keys are the same. • In asymmetric (two keys) - The encryption and decryption keys are different.
6. 6. DATA ENCRYPTION • Encryption Methods - Encryption is accomplished by scrambling the bits, characters, words, or phrases in the original message. Scrambling involves two activities: • Transposition - In which the order of the bits patterns, characters, words or phrases is rearranged. • Substitution - In which new bit patterns, characters, words, or phrases are substituted for the originals without changing their order.
7. 7. DATA ENCRYPTION • Data Encryption Standard (DES): – Most widely used algorithm – Pioneered by IBM – It is symmetric cryptosystem – Developed to protect sensitive, unclassified, US government, Computer data. – Used to provide authentication of electronic funds transfer messages.
8. 8. DATA ENCRYPTION  DES Algorithm  The algorithm accepts plaintext, P, and performs an initial permutation, IP, on P producing P0, The block is then broken into left and right halves, the Left (L0) being the first 32 bits of P0 and the right (R0) being the last 32 bits of P0.  With L0 and R0, 16 rounds are performed until L16 and R16 are generated.  The inverse permutation, IP-1, is applied to L16R16 to produce ciphertext C.
9. 9. DATA ENCRYPTION • Public Key Cryptosystem – It is an asymmetric cryptosystem. – First announced in 1976. – Offer a radically different approach to encryption. – The idea depends on the use of a pair of keys that differ in a complementary way. – Several algorithms are proposed – RSA algorithm is considered to be highly secure. • Public key encryption can achieved – Privacy – Authentication
10. 10. DIFFERENTIAL CRYPTANALYSIS • one of the most significant recent (public) advances in cryptanalysis • known in 70's with DES design • Murphy, Biham & Shamir published 1990 • powerful method to analyse block ciphers • used to analyse most current block ciphers with varying degrees of success • DES reasonably resistant to it
11. 11. DIFFERENTIAL CRYPTANALYSIS • a statistical attack against Feistel ciphers • uses cipher structure not previously used • design of S-P networks has output of function f influenced by both input & key • hence cannot trace values back through cipher without knowing values of the key • Differential Cryptanalysis compares two related pairs of encryptions
12. 12. DIFFERENTIAL CRYPTANALYSIS COMPARES PAIRS OF ENCRYPTIONS • Differential cryptanalysis is complex • with a known difference in the input • searching for a known difference in output
13. 13. DIFFERENTIAL CRYPTANALYSIS • have some input difference giving some output difference with probability p • if find instances of some higher probability input / output difference pairs occurring • can infer subkey that was used in round • then must iterate process over many rounds
14. 14. DIFFERENTIAL CRYPTANALYSIS • perform attack by repeatedly encrypting plaintext pairs with known input XOR until obtain desired output XOR • when found – if intermediate rounds match required XOR have a right pair – if not then have a wrong pair • can then deduce keys values for the rounds – right pairs suggest same key bits – wrong pairs give random values • larger numbers of rounds makes it more difficult • Attack on full DES requires an effort on the order of 247, requiring 247 chosen plaintexts to be encrypted
15. 15. LINEAR CRYPTANALYSIS • another recent development • also a statistical method • based on finding linear approximations to model the transformation of DES • can attack DES with 247 known plaintexts, still in practise infeasible
16. 16. • have considered: – terminology – classical cipher techniques – substitution ciphers • cryptanalysis using letter frequencies – transposition ciphers DATA ENCRYPTION STANDARD
17. 17. MODERN BLOCK CIPHERS • will now look at modern block ciphers • one of the most widely used types of cryptography algorithms • provide strong secrecy and/or authentication services • in particular will introduce DES (Data Encryption Standard)
18. 18. BLOCK VS STREAM CIPHERS • block ciphers process messages 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 • hence are focus of course
19. 19. BLOCK CIPHER PRINCIPLES • block ciphers look like an extremely large substitution • would need table of 264 entries for a 64-bit block • arbitrary reversible substitution cipher for a large block size is not practical – 64-bit general substitution block cipher, key size 264! • most symmetric block ciphers are based on a Feistel Cipher Structure • needed since must be able to decrypt ciphertext to recover messages efficiently
20. 20. C. SHANNON AND SUBSTITUTION- PERMUTATION CIPHERS • in 1949 Shannon introduced idea of substitution- permutation (S-P) networks – modern substitution-transposition product cipher • these form the basis of modern block ciphers • S-P networks are based on the two primitive cryptographic operations we have seen before: – substitution (S-box) – permutation (P-box) (transposition) • provide confusion and diffusion of message
21. 21. DIFFUSION AND CONFUSION • Introduced by Claude Shannon to thwart cryptanalysis based on statistical analysis – Assume the attacker has some knowledge of the statistical characteristics of the plaintext • cipher needs to completely obscure statistical properties of original message • a one-time pad does this • more practically Shannon suggested combining elements to obtain: • diffusion – dissipates statistical structure of plaintext over bulk of ciphertext • confusion – makes relationship between ciphertext and key as complex as possible
22. 22. FEISTEL CIPHER STRUCTURE • Horst Feistel devised the feistel cipher – implements Shannon’s substitution-permutation network concept • partitions input block into two halves – process through multiple rounds which – perform a substitution on left data half – based on round function of right half & subkey – then have permutation swapping halves
23. 23. FEISTEL CIPHER • n sequential rounds • A substitution on the left half Li – 1. Apply a round function F to the right half Ri and – 2. Take XOR of the output of (1) and Li • The round function is parameterized by the subkey Ki – Ki are derived from the overall key K
24. 24. FEISTEL CIPHER DESIGN PRINCIPLES • block size - increasing size improves security, but slows cipher • key size - increasing size improves security, makes exhaustive key searching harder, but may slow cipher • number of rounds - increasing number improves security, but slows cipher • subkey generation - greater complexity can make analysis harder, but slows cipher • round function - greater complexity can make analysis harder, but slows cipher • fast software en/decryption & ease of analysis - are more recent concerns for practical use and testing
25. 25. FEISTEL CIPHER DECRYPTION
26. 26. DATA ENCRYPTION STANDARD (DES) • most widely used block cipher in world • adopted in 1977 by NBS (now NIST) - as FIPS PUB 46 • encrypts 64-bit data using 56-bit key • has widespread use • IBM developed Lucifer cipher - by team lead by Feistel – used 64-bit data blocks with 128-bit key • then redeveloped as a commercial cipher with input from NSA and others • in 1973 NBS issued request for proposals for a national cipher standard • IBM submitted their revised Lucifer which was eventually accepted as the DES
27. 27. DES DESIGN CONTROVERSY • although DES standard is public • was considerable controversy over design -in choice of 56-bit key (vs Lucifer 128-bit) •subsequent events and public analysis show in fact design was appropriate •DES has become widely used, especially in financial applications
28. 28. INITIAL PERMUTATION IP • first step of the data computation • IP reorders the input data bits • quite regular in structure – see text Table 3.2 • example: IP(675a6967 5e5a6b5a) = (ffb2194d 004df6fb)
29. 29. DES ROUND STRUCTURE • uses two 32-bit L & R halves • as for any Feistel cipher can describe as: Li = Ri–1 Ri = Li–1 xor F(Ri–1, Ki) • takes 32-bit R half and 48-bit subkey and: – expands R to 48-bits using Expansion Permutation E (Table 3.2 c.) – adds to subkey – passes through 8 S-boxes to get 32-bit result – finally permutes this using 32-bit Permutation Function P (Table 3.2 d)
30. 30. THE ROUND FUNCTION F(R,K)
31. 31. SUBSTITUTION BOXES S • 8 S-boxes (Table 3.3 ) • Each S-Box mapps 6 to 4 bits – outer bits 1 & 6 (row bits) select the row – inner bits 2-5 (col bits) select the column – For example, in S1, for input 011001, • the row is 01 (row 1) • the column is 1100 (column 12). • The value in row 1, column 12 is 9 • The output is 1001. • result is 8 X 4 bits, or 32 bits
32. 32. DES Key Schedule • forms subkeys used in each round • 1. initial permutation of the key PC1 (Table 3.4b) • 2. divide the 56-bits in two 28-bit halves • 3. at each round – 3.1. Left shift each half (28bits) separately either 1 or 2 places based on the left shift schedule (Table 3.4d) • Shifted values will be input for next round – 3.2. Combine two halfs to 56 bits, permuting them by PC2 (Table 3.4c) for use in function f • PC2 takes 56-bit input, outputs 48 bits
33. 33. DES DECRYPTION • decrypt must unwind steps of data computation • with Feistel design, do encryption steps again • using subkeys in reverse order (SK16 … SK1) • note that IP undoes final FP step of encryption • 1st round with SK16 undoes 16th encrypt round • …. • 16th round with SK1 undoes 1st encrypt round • then final FP undoes initial encryption IP • thus recovering original data value
34. 34. DES Decryption (Reverse encryption)
35. 35. AVALANCHE EFFECT • key desirable property of encryption alg • DES exhibits strong avalanche • where a change of one input or key bit results in changing approx half output bits
36. 36. STRENGTH OF DES – KEY SIZE • 56-bit keys have 256 = 7.2 x 1016 values • brute force search looks hard • recent advances have shown is possible – in 1997 on Internet in a few months – in 1998 on dedicated hardware (EFF) in a few days – in 1999 above combined in 22hrs! • still must be able to recognize plaintext • now considering alternatives to DES
37. 37. STRENGTH OF DES – TIMING ATTACKS • attacks actual implementation of cipher • use knowledge of consequences of implementation to derive knowledge of some/all subkey bits • specifically use fact that calculations can take varying times depending on the value of the inputs to it
38. 38. STRENGTH OF DES – ANALYTIC ATTACKS • now have several analytic attacks on DES • these utilise some deep structure of the cipher – by gathering information about encryptions – can eventually recover some/all of the sub-key bits – if necessary then exhaustively search for the rest • generally these are statistical attacks • include – differential cryptanalysis – linear cryptanalysis – related key attacks
39. 39. MODES OF OPERATION • block ciphers encrypt fixed size blocks • eg. DES encrypts 64-bit blocks, with 56-bit key • need way to use in practise, given usually have arbitrary amount of information to encrypt • four were defined for DES in ANSI standard ANSI X3.106-1983 Modes of Use • subsequently now have 5 for DES and AES • have block and stream modes
40. 40. ELECTRONIC CODEBOOK BOOK (ECB) • message is broken into independent blocks which are encrypted • each block is a value which is substituted, like a codebook, hence name • each block is encoded independently of the other blocks Ci = DESK1 (Pi) • uses: secure transmission of single values
41. 41. ELECTRONIC CODEBOOK BOOAK (ECB)
42. 42. ADVANTAGES AND LIMITATIONS OF ECB • repetitions in message may show in ciphertext – if aligned with message block – particularly with data such graphics – or with messages that change very little, which become a code-book analysis problem • weakness due to encrypted message blocks being independent • main use is sending a few blocks of data
43. 43. CIPHER BLOCK CHAINING (CBC) • message is broken into blocks • but these are linked together in the encryption operation • each previous cipher blocks is chained with current plaintext block, hence name • use Initial Vector (IV) to start process Ci = DESK1(Pi XOR Ci-1) C-1 = IV • uses: bulk data encryption, authentication
44. 44. CIPHER BLOCK CHAINING (CBC)
45. 45. ADVANTAGES AND LIMITATIONS OF CBC • each ciphertext block depends on all message blocks • thus a change in the message affects all ciphertext blocks after the change as well as the original block • need Initial Value (IV) known to sender & receiver – however if IV is sent in the clear, an attacker can change bits of the first block, and change IV to compensate – hence either IV must be a fixed value (as in EFTPOS) or it must be sent encrypted in ECB mode before rest of message • at end of message, handle possible last short block – by padding either with known non-data value (eg nulls) – or pad last block with count of pad size • eg. [ b1 b2 b3 0 0 0 0 5] <- 3 data bytes, then 5 bytes pad+count
46. 46. CIPHER FEEDBACK (CFB) • message is treated as a stream of bits • added to the output of the block cipher • result is feed back for next stage (hence name) • standard allows any number of bit (1,8 or 64 or whatever) to be feed back – denoted CFB-1, CFB-8, CFB-64 etc • is most efficient to use all 64 bits (CFB-64) Ci = Pi XOR DESK1(Ci-1) C-1 = IV • uses: stream data encryption, authentication
47. 47. CIPHER FEEDBACK (CFB)
48. 48. Advantages and Limitations of CFB • appropriate when data arrives in bits/bytes • most common stream mode • limitation is need to stall while do block encryption after every n-bits • note that the block cipher is used in encryption mode at both ends • errors propagate for several blocks after the error
49. 49. OUTPUT FEEDBACK (OFB) • message is treated as a stream of bits • output of cipher is added to message • output is then feed back (hence name) • feedback is independent of message • can be computed in advance Ci = Pi XOR Oi Oi = DESK1(Oi-1) O-1 = IV • uses: stream encryption over noisy channels • Note: the OFB mode description presented in Fig 3.14 on page 96 of Stallings’ text is incorrect. Refer to the NIST Spl Pubs 800-38A - Fig 4/page 14
50. 50. ADVANTAGES AND LIMITATIONS OF OFB • used when error feedback a problem or where need to encryptions before message is available • superficially similar to CFB • but feedback is from the output of cipher and is independent of message • a variation of a Vernam cipher – hence must never reuse the same sequence (key+IV) • sender and receiver must remain in sync, and some recovery method is needed to ensure this occurs • originally specified with m-bit feedback in the standards • subsequent research has shown that only OFB-64 should ever be used
51. 51. COUNTER (CTR) • a “new” mode, though proposed early on • similar to OFB but encrypts counter value rather than any feedback value • must have a different key & counter value for every plaintext block (never reused) Ci = Pi XOR Oi Oi = DESK1(i) • uses: high-speed network encryptions
52. 52. Counter (CTR)
53. 53. ADVANTAGES AND LIMITATIONS OF CTR • efficiency – can do parallel encryptions – in advance of need – good for bursty high speed links • random access to encrypted data blocks • provable security (good as other modes) • but must ensure never reuse key/counter values, otherwise could break (cf OFB)
54. 54. TRIPLE DES • clearly a replacement for DES was needed – theoretical attacks that can break it – demonstrated exhaustive key search attacks • AES is a new cipher alternative • prior to this alternative was to use multiple encryption with DES implementations • Triple-DES is the chosen form
55. 55. WHY TRIPLE-DES? • why not Double-DES? – NOT same as some other single-DES use, but have • meet-in-the-middle attack – works whenever use a cipher twice – since X = EK1[P] = DK2[C] – attack by encrypting P with all keys and store – then decrypt C with keys and match X value – can show takes O(256) steps
56. 56. TRIPLE-DES WITH TWO-KEYS • hence must use 3 encryptions – would seem to need 3 distinct keys • but can use 2 keys with E-D-E sequence – C = EK1[DK2[EK1[P]]] – nb encrypt & decrypt equivalent in security – if K1=K2 then can work with single DES • standardized in ANSI X9.17 & ISO8732 • no current known practical attacks
57. 57. TRIPLE-DES WITH THREE-KEYS • although are no practical attacks on two-key Triple- DES have some indications • can use Triple-DES with Three-Keys to avoid even these – C = EK3[DK2[EK1[P]]] • has been adopted by some Internet applications, eg PGP, S/MIME
58. 58. RC4 • a proprietary cipher owned by RSA DSI • another Ron Rivest design, simple but effective • variable key size, byte-oriented stream cipher • widely used (web SSL/TLS, wireless WEP) • key forms random permutation of all 8-bit values • uses that permutation to scramble input info processed a byte at a time
59. 59. RC4 SECURITY • Claimed secure against known attacks – Have some analyses, none practical • Result is very non-linear • Since RC4 is a stream cipher, must never reuse a key • Have a concern with WEP, but due to key handling rather than RC4 itself
60. 60. ADVANCED ENCRYPTION STANDARD • Replacement for DES was needed – Theoretical attacks that can break it – Demonstrated exhaustive key search attacks • Can use Triple DES – but slow, small block size • NIST issued a call for a new AES in 1997 • 15 candidates accepted in Jun 1998 • 5 candidates were short-listed in Aug 1999 • Rijndael was selected as the AES in Oct 2000 • Published as FIPS PUB 197 standard in Dec 2001
61. 61. AES REQUIREMENTS • Symmetric block cipher • 128-bit data, 128/192/256-bit keys • Stronger & faster than triple DES • Active life of 20-30 years (+ archival use) • Provide full specification & design details • Both C & Java implementations • NIST have released all submissions & unclassified analyses
62. 62. AES EVALUATION CRITERIA • Initial criteria: – Security – effort for practical cryptanalysis – Cost – in terms of computational efficiency (speed, memory) – Algorithm & implementation characteristics • flexibility, algorithm simplicity • Final criteria – General security – Ease of software & hardware implementation – Restricted-space environments – Attacks on implementations • timing attack, power analysis – Flexibility (in en/decrypt, keying, other factors)
63. 63. AES SHORT-LIST • After testing and evaluation, short-list in Aug 1999: – MARS (IBM) - complex, fast, high security margin – RC6 (USA) - very simple, very fast, low security margin – Rijndael (Belgium) - clean, fast, good security margin – Serpent (Euro) - clean, slow, very high security margin – Twofish (USA) - complex, very fast, high security margin • Then subject to further analysis & comment • Saw contrast between algorithms with – Few complex rounds vs. many simple rounds – Refined existing ciphers vs. new proposals
64. 64. THE AES CIPHER - RIJNDAEL • Designed by Rijmen-Daemen in Belgium • Block length: 128 bits • Key length: 128/192/256 bits • Number of Rounds: 10/12/14 rounds • An iterated cipher (rather than Feistel cipher) – Processes data as block of 4 columns of 4 bytes – Operates on entire data block in every round • Designed to be: – Resistance against all known attacks – Speed and code compactness on a wide range of platforms – Design simplicity
65. 65. OVERALL AES STRUCTURE • Data block of 4 columns of 4 bytes is “state” • Key is expanded to array of words • Has 9/11/13 rounds in which state undergoes: – Substitute bytes (1 S-box used on every byte) – Shift rows (permute bytes between columns) – Mix columns (substitute using matrix multiplication of columns) – Add round key (XOR state with key material) – View as alternating XOR key & scramble data bytes • Initial XOR key material & incomplete last round • With fast XOR & table lookup implementation
66. 66. AES Encryption & Decryption
67. 67. AES DATA STRUCTURES Plaint ext Ciphert ext
68. 68. AES ENCRYPTION ROUND
69. 69. SUBSTITUTE BYTES (SUBBYTES) • Simple substitution on each byte of state independently • Use an S-box of 16x16 bytes containing a permutation of all 256 8-bit values • Each byte of state is replaced by a new byte indexed by row (left 4-bits) & column (right 4-bits) – eg. byte {95} is replaced by {2A} in row 9 column 5 • S-box constructed using defined transformation of values in GF(28) • Designed to be resistant to all known attacks
70. 70. SUBSTITUTE BYTES
71. 71. SUBSTITUTE BYTES • GF(28) = 2[x] / (x8+x4+x3+x+1) SubBytes(a7a6a5a4a3a2a1a0) 1. z ← BinaryToField(a7a6a5a4a3a2a1a0) 2. if z ≠ 0 3. then z ← FieldInv(z) 4. (a7a6a5a4a3a2a1a0) ← FieldToBinary(z) 5. (c7c6c5c4c3c2c1c0) ← (01100011) 6. for i ← 0 to 7 7. do bi ← (ai+ai+4+ai+5+ai+6+ai+7+ci) mod 2 8. return (b7b6b5b4b3b2b1b0)
72. 72. SUBSTITUTE BYTES • Example – Input: a = 0x53 = 01010011 (x6+x4+x3+1) – Multiplicative inverse a-1 = x7+x6+x3+x (mod x8+x4+x3+x+1) – (a7a6a5a4a3a2a1a0) = (11001010) – (c7c6c5c4c3c2c1c0) = (01100011) – b0 = a0+a4+a5+a6+a7+c0 mod 2 = 0+0+0+1+1+1 mod 2 = 1 – b1 = a1+a5+a6+a7+a8+c1 mod 2 = 1+0+1+1+0+1 mod 2 = 0 – (b7b6b5b4b3b2b1b0) = (11101101) = 0xED
73. 73. INVERSE SUBBYTES InvSubBytes(b7b6b5b4b3b2b1b0) 1. (d7d6d5d4d3d2d1d0) ← (00000101) 2. for i ← 0 to 7 do 3. ai ← (bi+2+bi+5+bi+7+di) mod 2 4. z ← BinaryToField(a7a6a5a4a3a2a1a0) 5. if z ≠ 0 then 6. z ← FieldInv(z) 7. (a7a6a5a4a3a2a1a0) ← FieldToBinary(z) 8. return (a7a6a5a4a3a2a1a0)
74. 74. SHIFT ROWS • A circular byte shift in each – 1st row is unchanged – 2nd row does 1 byte circular shift to left – 3rd row does 2 byte circular shift to left – 4th row does 3 byte circular shift to left • Decrypt inverts using shifts to right • Since state is processed by columns, this step permutes bytes between the columns
75. 75. MIX COLUMNS • Each column is processed separately • Each byte is replaced by a value dependent on all 4 bytes in the column • Effectively a matrix multiplication in GF(28) using irreducible polynomial m(x) = x8 + x4 + x3 + x + 1
76. 76. GF(28)[x] / (x4+1) • GF(28) = 2[x] / (x8+x4+x3+x+1) • Let a(x) = a3x3+a2x2+a1x+a0, ai 2 GF(28) b(x) = b3x3+b2x2+b1x+b0, bi 2 GF(28) • Addition a(x) + b(x) = (a3+b3)x3+(a2+b2)x2+(a1+b1)x+(a0+b0) • Multiplication c(x) = a(x)b(x) = c6x6 + c5x5 + c4x4 + c3x3 + c2x2 + c1x1 + c0 c0 = a0b0 c4 = (a3b1 + a2b2 + a1b3) c1 = (a1b0 + a0b1) c5 = (a3b2 + a2b3) c2 = (a2b0 + a1b1 + a0b2) c6 = a3b3 c3 = (a3b0 + a2b1 + a1b2 + a0b3)
77. 77. GF(28)[x] / (x4+1) d(x) = c(x) mod (x4+1) = c6x6 + c5x5 + c4x4 + c3x3 + c2x2 + c1x1 + c0 = c3x3 + (c2+c6)x2 + (c1+c5)x + (c0+c4) = d3x3 + d2x2 + d1x + d0 d0 = a0b0 + a3b1 + a2b2 + a1b3 d1 = a1b0 + a0b1 + a3b2 + a2b3 d2 = a2b0 + a1b1 + a0b2 + a3b3 d3 = a3b0 + a2b1 + a1b2 + a0b3 d0 a0 a3 a2 a1 b0 d0 02 03 01 01 b0 d1 a1 a0 a3 a2 b1 d1 01 02 03 01 b1 d2 a2 a1 a0 a3 b2 d2 01 01 02 03 b2 d3 a3 a2 a1 a0 b3 d3 03 01 01 02 b3 = a(x) = {03}x3+{01}x2+{01}x+{02} =
78. 78. INVERSE MIXCOLUMNS b0 0E 0B 0D 09 d0 b1 09 0E 0B 0D d1 b2 0D 09 0E 0B d2 b3 0B 0D 09 0E d3 = a(x) = {03}x3+{01}x2+{01}x+{02} a(x)-1 mod (x4+1) = {0B}x3+{0D}x2+{09}x+{0E}
79. 79. ADD ROUND KEY • XOR state with 128-bits of the round key • Again processed by column (though effectively a series of byte operations) • Inverse for decryption identical – Since XOR own inverse, with reversed keys • Designed to be as simple as possible – A form of Vernam cipher on expanded key – Complexity of other stages ensures security
80. 80. AES Key Expansion • Takes 128-bit (16-byte; 4-word) key and expands into array of 44 32-bit words • Start by copying key into first 4 words • Then loop creating words that depend on values in previous & 4 places back – In 3 of 4 cases just XOR these together – 1st word in 4 has rotate + S-box + XOR round constant on previous, before XOR 4th back
81. 81. AES Key Expansion KeyExpasion(key) 1. RCon[1] ← 0x01000000 RCon[2] ← 0x02000000 RCon[3] ← 0x04000000 RCon[4] ← 0x08000000 RCon[5] ← 0x10000000 RCon[6] ← 0x20000000 RCon[7] ← 0x40000000 RCon[8] ← 0x80000000 RCon[9] ← 0x01B00000 RCon[10] ← 0x36000000 2. for i ← 0 to 3 do 3. w[i] ← (key[4i],key[4i+1],key[4i+2],key[4i+3]) 4. for i ← 4 to 43 do 5. temp ← w[i-1] 6. if i  0 mod 4 then 7. temp ← SubWord(RotWord(temp))© Rcon[i/4] 8. w[i] ← w[i-4]© temp
82. 82. AES KEY EXPANSION • RotWord(B0,B1,B2,B3) = (B1,B2,B3,B0) • SubWord(B0,B1,B2,B3) = (B0’,B1’,B2’,B3’), where Bi’ = SubBytes(Bi), i = 0,1,2,3
83. 83. KEY EXPANSION RATIONALE • Designed to resist known attacks • Design criteria included – Knowing part key insufficient to find many more – Invertible transformation – Fast on wide range of CPU’s – Use round constants to break symmetry – Diffuse key bits into round keys – Enough non-linearity to hinder analysis – Simplicity of description
84. 84. AES DECRYPTION • AES decryption is not identical to encryption since steps done in reverse • But can define an equivalent inverse cipher with steps as for encryption – But using inverses of each step – With a different key schedule • Works since result is unchanged when – Swap byte substitution & shift rows – Swap mix columns & add (tweaked) round key
85. 85. IMPLEMENTATION ASPECTS • Can efficiently implement on 8-bit CPU – byte substitution works on bytes using a table of 256 entries – shift rows is simple byte shift – add round key works on byte XOR’s – mix columns requires matrix multiply in GF(28) which works on byte values, can be simplified to use table lookups & byte XOR’s • Can efficiently implement on 32-bit CPU – redefine steps to use 32-bit words – can precompute 4 tables of 256-words – then each column in each round can be computed using 4 table lookups + 4 XORs – at a cost of 4Kb to store tables • Designers believe this very efficient implementation was a key factor in its selection as the AES cipher
86. 86. PRIVATE-KEY CRYPTOGRAPHY • traditional private/secret/single-key cryptography uses one key • shared by both sender and receiver • if this key is disclosed communications are compromised • also is symmetric, parties are equal • hence does not protect sender from receiver forging a message & claiming it’s sent by sender • probably most significant advance in the 3000 year history of cryptography • uses two keys – a public & a private key • asymmetric since parties are not equal
87. 87. PUBLIC-KEY CRYPTOGRAPHY • uses clever application of number theoretic concepts to function • complements rather than replaces private key crypto • public-key/two-key/asymmetric cryptography involves the use of two keys: – a public-key, which may be known by anybody, and can be used to encrypt messages, and verify signatures – a private-key, known only to the recipient, used to decrypt messages, and sign (create) signatures • is asymmetric because – those who encrypt messages or verify signatures cannot decrypt messages or create signatures
88. 88. PUBLIC-KEY CRYPTOGRAPHY
89. 89. WHY PUBLIC-KEY CRYPTOGRAPHY? • developed to address two key issues: – key distribution – how to have secure communications in general without having to trust a KDC with your key – digital signatures – how to verify a message comes intact from the claimed sender • public invention due to Whitfield Diffie & Martin Hellman at Stanford Uni in 1976
90. 90. PUBLIC-KEY CHARACTERISTICS • Public-Key algorithms rely on two keys with the characteristics that it is: – computationally infeasible to find decryption key knowing only algorithm & encryption key – computationally easy to en/decrypt messages when the relevant (en/decrypt) key is known – either of the two related keys can be used for encryption, with the other used for decryption (in some schemes)
91. 91. PUBLIC-KEY CRYPTOSYSTEMS
92. 92. PUBLIC-KEY APPLICATIONS • can classify uses into 3 categories: – encryption/decryption (provide secrecy) – digital signatures (provide authentication) – key exchange (of session keys) • some algorithms are suitable for all uses, others are specific to one
93. 93. SECURITY OF PUBLIC KEY SCHEMES • like private key schemes brute force exhaustive search attack is always theoretically possible • but keys used are too large (>512bits) • security relies on a large enough difference in difficulty between easy (en/decrypt) and hard (cryptanalyse) problems • more generally the hard problem is known, its just made too hard to do in practise • requires the use of very large numbers • hence is slow compared to private key schemes
94. 94. RSA • by Rivest, Shamir & Adleman of MIT in 1977 • best known & widely used public-key scheme • based on exponentiation in a finite (Galois) field over integers modulo a prime – nb. exponentiation takes O((log n)3) operations (easy) • uses large integers (eg. 1024 bits) • security due to cost of factoring large numbers – nb. factorization takes O(e log n log log n) operations (hard)
95. 95. RSA Key Setup • each user generates a public/private key pair by: • selecting two large primes at random - p, q • computing their system modulus N=p.q – note ø(N)=(p-1)(q-1) • selecting at random the encryption key e • where 1<e<ø(N), gcd(e,ø(N))=1 • solve following equation to find decryption key d – e.d=1 mod ø(N) and 0≤d≤N • publish their public encryption key: KU={e,N} • keep secret private decryption key: KR={d,p,q}
96. 96. RSA Use • to encrypt a message M the sender: – obtains public key of recipient KU={e,N} – computes: C=Me mod N, where 0≤M<N • to decrypt the ciphertext C the owner: – uses their private key KR={d,p,q} – computes: M=Cd mod N • note that the message M must be smaller than the modulus N (block if needed)
97. 97. RSA Example 1. Select primes: p=17 & q=11 2. Compute n = pq =17×11=187 3. Compute ø(n)=(p–1)(q-1)=16×10=160 4. Select e : gcd(e,160)=1; choose e=7 5. Determine d: de=1 mod 160 and d < 160 Value is d=23 since 23×7=161= 10×160+1 6. Publish public key KU={7,187} 7. Keep secret private key KR={23,17,11}
98. 98. RSA EXAMPLE CONT • sample RSA encryption/decryption is: • given message M = 88 (nb. 88<187) • encryption: C = 887 mod 187 = 11 • decryption: M = 1123 mod 187 = 88
99. 99. RSA KEY GENERATION • users of RSA must: – determine two primes at random - p, q – select either e or d and compute the other • primes p,q must not be easily derived from modulus N=p.q – means must be sufficiently large – typically guess and use probabilistic test • exponents e, d are inverses, so use Inverse algorithm to compute the other
100. 100. SECURITY OF RSA • How to attack RSA? – we have public key (n,e) – compute (n) and get d • Easier said than done!! – If we have n and (n) then we can factor n – If we have e and d then we can factor n Adi Shamir
101. 101. Attacks on RSA • Various attacks on RSA – known digits attack – low exponent attack – short plaintext attack – timing attack – factoring Len Adleman
102. 102. KNOWN DIGITS ATTACK • Theorem n = pq – has m digits If we know the first or the last m/4 digits of either p or q, then we can efficiently factor n • Theorem (n,e) – RSA public key n has m digits, and we know the last m/4 digits of d We can find d in time linear in e log e Conclusion: Need care about the choice of p and q!
103. 103. LOW EXPONENT ATTACK • Theorem – p,q – RSA primes, q < p < 2q – 1 ≤ d, e < (n) – de = 1 (mod (n)) – If d < ⅓n1/4 , then d can be calculated quickly • Consequences – cannot optimize decryption via small exponents – how to choose good d’s?
104. 104. SHORT PLAINTEXT ATTACK • Scenario – DES – symmetric cipher, used in the past – Two banks exchange DES keys over RSA – DES key m: 56 bits (m < 256 < 1017) – RSA encryption • c = me (mod n) • m small, but c will have many digits • Defense: – do not use short messages! – pad with random bits – Optimal Assymetric Encryption Padding
105. 105. SHORT PLAINTEXT ATTACK • C = ME (MOD N) M < 256 < 1017 • EVE PREPARES TWO LISTS: – CX-E (MOD N), 1 ≤ X ≤ 109 – YE (MOD N) 1 ≤ Y ≤ 109 • IF THERE IS A MATCH ON THE LISTS THEN – C = (XY)E (MOD N) – THUS: M = XY (MOD N) • IF M IS A MULTIPLE OF TWO NUMBERS <109 THEN THIS ATTACK WILL SUCCEED
106. 106. OPTIMAL ASSYMETRIC ENCRYPTION PADDING • n – k bits • k0, k1 – two numbers s.t. k0 + k1 < k • Message can have k - k0 - k1 bits • r – random string of k0 bits • G: k 0  k-k 0 • H: k-k 0  k 0 • The method x1 = m0k 1  G(r) x2 = r  H(x1) The message is x1x2 Bob decrypts and gets m0k 1 = x1  G(H(x1)  x2)
107. 107. TIMING ATTACK • Within RSA computation we do perform exponentiaiton – quick exponentiation procedure – multiplications occur for each bit of the exponent that is 1 – these multiplications take „random” amounts of time (variation) • Very hard in practice! – Initiated a big discussion
108. 108. EXPONENTIATION ALGORITHM • Goal: yd (mod n) – d = b1b2...bw (in binary left-to-right) • Algorithm 1. k = 1, s1 = 1 2. if bk = 1 then rk = sky (mod n) else rk = sk 3. sk+1 = rk 2 (mod n) 4. if k = w, stop else set k to k+1, goto 2 5. output rw
109. 109. FACTORING AND PRIMALITY TESTING • Factoring – Input: n  N – Output: nontrivial factor of n • Primality testing – Input n  N – Output: • the number is composite • the number is probably prime • Is there a difference? – Yes! – primality testing much easier! – You do not need to factor the number to see it is composite
110. 110. MILLER-RABIN TEST • Generalization of the Fermat’s test • Principle – if p is a prime then x2 = 1 (mod p) has only two solutions: x = 1 and x = -1 • Why does the principle hold? • Gist of the MR test – find a number b such that b2 = 1 (mod p) – If b  {-1,1} then composite
111. 111. MILLER-RABIN TEST MR( int n ): let n-1 = 2km a  random in {2, 3, ..., n-2 } b0 = am (mod n) if b0 = ±1 (mod n) then declare prime for j = 1 to k-1 do bj = bj-1 2 (mod n) if bj = 1 (mod n) then declare composite if bj = -1 (mod n) then declare prime declare composite • What are we doing? – b0 = am (mod n) – b1 = a2m (mod n) – b2 = a4m (mod n) – ... – bj = a2jm (mod n) – ... – bk-1 = a(n-1)/2 (mod n)
112. 112. MILLER-RABIN TEST: EXAMPLES • n = 5*7*11 = 385 n -1 = 384 = 27*3 k = 7, m = 3 a = 9 b0 = 93 = 344 (mod 385) b1 = 93*2 = 141 (mod 385) b2 = 93*22 = 246 (mod 385) b3 = 93*23 = 71 (mod 385) b4 = 93*24 = 36 (mod 385) b5 = 93*25 = 141 (mod 385) • n = 3*11*17 = 561 n -1 = 560 = 24*35 k = 4, m = 35 a = 2 b0 = 235 = 263 (mod 561) b1 = 235*2 = 166 (mod 561) b2 = 235*22 = 67 (mod 561) b3 = 235*23 = 1 (mod 561) Proof of compositeness!
113. 113. MILLER-RABIN TEST: EXAMPLES • n = 401 n -1 = 400 = 24*25 k = 4, m = 25 a = 3 b0 = 325 = 268 (mod 401) b1 = 325*2 = 45 (mod 401) b2 = 325*22 = 20 (mod 401) b3 = 325*23 = 400 (mod 401) = -1 (mod 401) • n = 401 n -1 = 400 = 24*25 k = 4, m = 25 a = 2 b0 = 225 = 356 (mod 401) b1 = 225*2 = 20 (mod 401) b2 = 225*22 = 400 (mod 401) Evidence of primality!
114. 114. MILLER-RABIN TEST • if b0 = ±1 (mod n) – all bi’s (i > 0) will be 1 – can’t find nontrivial roots of 1 • i  {1, ..., k-1} – if bi = 1 (mod n) then • bi-1 is neither 1 nor -1 • bi-1 2 = 1 (mod n) • we found a nontrivial root – if bi = -1 (mod n) then • bi+1 through bk are all 1 (mod n) • can’t find nontrivial roots of 1 • Why this works? – n-1 = 2km – b0 = am (mod n) – b1 = a2m (mod n) – b2 = a4m (mod n) – ... – bj = a2jm (mod n) – ... – bk-1 = a(n-1)/2 (mod n)
115. 115. MILLER-RABIN TEST: QUALITY • MR test is probabilistic • Answer – composite – the number is certainly composite – prime – the number is prime with high probability • Errors – MR(n) says prime but n is composite – Pr[error] ≤ ¼ – Repeat the test to downgrade the prob. of error
116. 116. OTHER PRIMALITY TESTS • Solovay-Strassen Test – similar in nature to MR – uses so called Jacobi symbol – fast in practice – probabilistic • Deterministic test – Agrawal, Kayal, and Saxena 2002 – extremely slow • Tests that prove primality – MR tests compositeness! – fairly slow – needed in very few cases
117. 117. FACTORING • Huge amount of work on factoring! – we look at some simple algorithms • Some best algorithms – quadratic sieve – elliptic curve – number field sieve • Assumption – Factor an odd integer – produce one factor – how to get all of them? O(e(1+o(1))sqrt(lnn lnln n)) O(e(1+o(1))sqrt(lnp lnln p)) O(e(1.92+o(1))(lnn)1/3(lnlnn)2/3 )
118. 118. FACTORING • Factoring – Input: n  N – Output: nontrivial factor of n • There are about (n) = n / ln n primes ≤ n • Trivial methods – divide by all numbers in {2, ... , n-1} – or by all primes p p ≤ sqrt(n) – These are exponential!
119. 119. FERMAT’S METHOD • the principle – express n as a difference of squares – n = x2 - y2 – n = (x-y)(x+y) • the algorithm – compute: n + i2 for i  {1,2, ... } – stop when n + i2 is a square (i.e., x2 = n+i2) – then we have n = x2 – i2 • Examples 15 = 42 – 12 = (4-1)(4+1) = 3*5 21 = 52 – 22 = 25 - 4 = (5-2)(5+2) = 3*7
120. 120. Fermat’s Method • The principle – express n as a difference of squares – n = x2 - y2 – n = (x-y)(x+y) • The algorithm – Compute: n + i2 for i 2 {1,2, ... } – Stop when n + i2 is a square (i.e., x2 = n+i2) – Then we have n = x2 – i2 • Performance – depends on distance between x and y – could be very slow! • Conclusion for RSA – p and q should differ by a large value
121. 121. POLLARD’S P-1 METHOD • The method input: n choose a > 1 (e.g., a = 2) choose B let b = aB! (mod n) d = gcd( b - 1, n ) d is a factor of n • Goal of the method – factor n = pq... – provided p-1 has only small prime factors • Example n = 7 * 11 = 77 a = 2 B = 4, B! = 2*3*4 = 24 b = 224 = 71 (mod 77) gcd(b-1, n) = gcd(70,77) = 7
122. 122. Pollard’s p-1 Method • The method input: n choose a > 1 (e.g., a = 2) choose B let b = aB! (mod n) d = gcd( b - 1, n ) d is a factor of n • Goal of the method – factor n = pq... – provided p-1 has only small prime factors • Example n = 7 * 11 = 77 a = 2 B = 2, B! = 2 b = 22 = 4 (mod 77) gcd(b-1, n) = gcd(3,77) = 1
123. 123. POLLARD’S P-1 METHOD • The method input: n choose a > 1 (e.g., a = 2) choose B let b = aB! (mod n) d = gcd( b - 1, n ) d is a factor of n • Goal of the method – factor n = pq... – provided p-1 has only small prime factors • Example n = 7 * 11 = 77 a = 2 B = 6, B! = 2*3*4*5*6 = 720 b = 2720 = 1 (mod 77) gcd(b-1, n) = gcd(0,77) = 77
124. 124. POLLARD’S P-1 METHOD • In symbols: – b1 = a (mod n) – b2 = b1 2 (mod n) – ... – bi = bi-1 i – ... • How to compute aB! – B! – can be very big – 5! = 120 – 6! = 720 – 10! = 3628800 – 20! = 2432902008176640000 – n! – about n log2 n bits
125. 125. POLLARD’S P-1 METHOD • In symbols: – b1 = a (mod n) – b2 = b1 2 (mod n) – ... – bi = bi-1 i – ... • How to compute aB! –a = 2, B = 4, n = 77 –b1 = 2 (mod 77) –b2 = 22 = 4 (mod 77) –b3 = 43 = 64 (mod 77) –b4 = 644 = 1677721 = 71 (mod 77)
126. 126. POLLARD’S P-1 METHOD • The method input: n choose a > 1 (e.g., a = 2) choose B let b = aB! (mod n) d = gcd( b - 1, n ) d is a factor of n • Why does it work? – p – prime factor of n – suppose: p-1 has only small prime factors – Then likely p-1 | B! – Then B! = k(n-1) b = (ap-1)k (mod p) b = 1 (mod p) p | b - 1
127. 127. POLLARD’S P-1 METHOD • Potential problems – n = pq – both p and q have small factors • b = 1 (mod p) • b = 1 (mod q) – Method fails • Choice of B – too small  method won’t work – too big  works slowly or fails • Example n = 7 * 11 = 77 a = 2 B = 2  to small B = 4  worked B = 6  to big 7 - 1 = 6 = 2*3 2! – does not contain 3 4! – contains 2 and 3 6! – contains 2,3 and 5  covers both factors!
128. 128. POLLARD’S P-1 METHOD • Conclusions for RSA – n = pq – p-1 or q-1 has small prime factors? • then RSA can be broken • How to defend? – p0  chose a large prime – e.g., p0 > 1040 – try numbers of the form: • kp0 + 1 • k – needs to be even! • k > 1060 • test kp0+1 for primality
129. 129. FACTORING • Relation to squares – n – an integer – x,y – to integers s.t. • x2 = y2 (mod n) • x  y (mod n) – if such x, y exist then n is composite • gcd( x-y, n ) is a nontrivial factor • Examples – 112 = 121 = 1 (mod 12) – 52 = 25 = 1 (mod 12) – 11  5 (mod 12) – gcd(11-5, 12 ) = 6
130. 130. FACTORING • Relation to squares – n – an integer – x,y – to integers s.t. • x2 ´ y2 (mod n) • x  y (mod n) – if such x, y exist then n is composite • gcd( x-y, n ) is a nontrivial factor • Examples – 52 = 25 = 7 (mod 9) – 142 = 196 = 7 (mod 9) – 14 = 9+5 = 5 (mod 9) – gcd(14-5, 9 ) = 9
131. 131. QUADRATIC SIEVIE • Idea – try to apply the principle from the previous slide – find x,y such that x2 = y2 (mod n) x  y (mod n) x  -y (mod n) – finding such x, y  not obvious – Take “random” squares – Reduce modulo n – Factor (hope for small factors!) – Try to build squares from what you get
132. 132. QUADRATIC SIEVE: EXAMPLE n = 3837523 93982 = 55  19 (mod n) 190952 = 22  5  11  13  19 (mod n) 19642 = 32  133 (mod n) 170782 = 26  32  11 (mod n) 80772 = 2  19 ( mod n) 33972 = 25  5  132 (mod n) 142622 = 52  72  13 (mod n) (9398  19095  1964  17078)2 = 28  32  56  112  134  192 = (24  3  53  11  132  19)2 (mod n) 22303872 = 25867052 (mod n) gcd( 2230387 – 2586705, 3837523) = 1093
133. 133. QUADRATIC SIEVE: EXAMPLE n = 3837523 93982 = 55  19 (mod n) 190952 = 22  5  11  13  19 (mod n) 19642 = 32  133 (mod n) 170782 = 26  32  11 (mod n) 80772 = 2  19 ( mod n) 33972 = 25  5  132 (mod n) 142622 = 52  72  13 (mod n) (9398  8077  3397)2 = 26  56  132  192 = (23  53  13  19)2 (mod n) 35905232 = 2470002 (mod n) BUT: n – 247000 = 3590523  3590523 = -247000 (mod n)
134. 134. HOW TO FIND THE SQUARES? • What squares to use? – we want small prime factors? – so x2 should be slightly above n • Idea: Try integers close to: – sqrt(i  n) + j – small j, various i – (sqrt(i  n) + j)2 ≈ in + 2j sqrt(in) +j2 – approx: 2j sqrt(in) + j2 (mod n)
135. 135. Questions ?