1. CSCE 715: Network Systems Security Chin-Tser Huang huangct@cse.sc.edu University of South Carolina

2. 9/9/20042 After DES… More symmetric encryption algorithms Triple-DES Advanced Encryption Standards

3. 9/9/20043 Triple DES Clearly a replacement for DES was needed theoretical attacks that can break it demonstrated exhaustive key search attacks Use multiple encryptions with DES implementations Triple-DES is the chosen form

4. 9/9/20044 Why Triple-DES? Double-DES may suffer from meet-in-the- middle attack works whenever use a cipher twice assume C = E K2 [E K1 [P]], so X = E K1 [P] = D K2 [C] attack by encrypting P with all keys and store then decrypt C with keys and match X value can show attack takes O(2 56 ) steps

5. 9/9/20045 Triple-DES with Two Keys Must use 3 encryptions would seem to need 3 distinct keys But can use 2 keys with E-D-E sequence encrypt & decrypt equivalent in security C = E K1 [D K2 [E K1 [P]]] if K1=K2 then can work with single DES Standardized in ANSI X9.17 & ISO8732 No current known practical attacks

6. 9/9/20046 Triple-DES with Three Keys Some proposed attacks on two-key Triple-DES, although none of them practical Can use Triple-DES with Three-Keys to avoid even these C = E K3 [D K2 [E K1 [P]]] Has been adopted by some Internet applications, e.g. PGP, S/MIME

7. 9/9/20047 Origins of Advanced Encryption Standard Triple-DES is slow with small blocks US NIST issued call for ciphers in 1997 15 candidates accepted in Jun 1998 5 were shortlisted in Aug 1999 Rijndael was selected as the AES in Oct 2000 Issued as FIPS PUB 197 standard in Nov 2001

8. 9/9/20048 AES Requirements Private key symmetric block cipher 128-bit data, 128/192/256-bit keys Stronger and faster than Triple-DES Active life of 20-30 years (+ archival use) Provide full specification and design details Both C and Java implementations NIST has released all submissions and unclassified analyses

9. 9/9/20049 AES Evaluation Criteria Initial criteria security – effort to practically cryptanalyze cost – computational algorithm & implementation characteristics Final criteria general security software & hardware implementation ease implementation attacks flexibility (in en/decrypt, keying, other factors)

10. 9/9/200410 AES Shortlist Shortlist in Aug 99 after testing and evaluation 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) - slow, clean, very high security margin Twofish (USA) - complex, very fast, high security margin Subject to further analysis and comment Contrast between algorithms with few complex rounds verses many simple rounds refined existing ciphers verses new proposals

11. 9/9/200411 The Winner - Rijndael Designed by Rijmen-Daemen in Belgium Has 128/192/256 bit keys, 128 bit data An iterative rather than feistel cipher treats data in 4 groups of 4 bytes operates an entire block in every round Designed to be resistant against known attacks speed and code compactness on many CPUs design simplicity Use finite field

12. 9/9/200412 Abstract Algebra Background Group Ring Field

13. 9/9/200413 Group A set of elements or “numbers” With a binary operation whose result is also in the set (closure) Obey following axioms associative law: (a.b).c = a.(b.c) has identity e : e.a = a.e = a has inverses a -1 : a.a -1 = e Abelian group if commutative a.b = b.a

14. 9/9/200414 Ring A set of elements with two operations (addition and multiplication) which are: an abelian group with addition operation multiplication has closure is associative distributive over addition: a(b+c) = ab + ac Commutative ring if multiplication operation is commutative Integral domain if multiplication operation has identity and no zero divisors

15. 9/9/200415 Field A set of numbers with two operations abelian group for addition abelian group for multiplication (ignoring 0) integral domain multiplicative inverse: aa -1 = a -1 a= 1 Infinite field if infinite number of elements Finite field if finite number of elements

16. 9/9/200416 Modular Arithmetic Define modulo operator a mod n to be remainder when a is divided by n Use the term congruence for: a ≡ b mod n when divided by n, a and b have same remainder e.g. 100  34 mod 11 b is called the residue of a mod n if 0  b  n-1 with integers can write a = qn + b

17. 9/9/200417 Divisor A non-zero number b is a divisor of a if for some m have a=mb ( a,b,m all integers) That is, b divides a with no remainder Denote as b|a E.g. all of 1,2,3,4,6,8,12,24 divide 24

18. 9/9/200418 Modular Arithmetic Can do modular arithmetic with any group of integers Z n = {0, 1, …, n-1} Form a commutative ring for addition With a multiplicative identity Some peculiarities if (a+b)≡(a+c) mod n then b≡c mod n but (ab)≡(ac) mod n then b≡c mod n only if a is relatively prime to n

19. 9/9/200419 Modulo 8 Example

20. 9/9/200420 Greatest Common Divisor (GCD) GCD (a,b) of a and b is the largest number that divides evenly into both a and b e.g. GCD(60,24) = 12 It is often desirable to find numbers that are relatively prime, namely they have no common factors (except 1) e.g. 8 and 15 relatively prime as GCD(8,15) = 1

21. 9/9/200421 Euclid's GCD Algorithm Use following theorem GCD(a,b) = GCD(b, a mod b) Euclid's Algorithm to compute GCD(a,b) A=a, B=b while B>0 R = A mod B A = B, B = R return A

22. 9/9/200422 Galois Fields Finite fields play a key role in cryptography Number of elements in a finite field must be a power of a prime p n Known as Galois fields Denoted GF(p n ) In particular often use the following forms GF(p) GF(2 n )

23. 9/9/200423 Galois Fields GF(p) GF(p) is set of integers {0,1, …, p-1} with arithmetic operations modulo prime p Form a finite field since have multiplicative inverses Hence arithmetic is “well-behaved” and can do addition, subtraction, multiplication, and division without leaving the field GF(p)

24. 9/9/200424 Arithmetic in GF(7)

25. 9/9/200425 Finding Multiplicative Inverses By extending Euclid’s algorithm 1.(A1, A2, A3)=(1, 0, m); (B1, B2, B3)=(0, 1, b) 2. if B3 = 0 return A3 = gcd(m, b); no inverse 3. if B3 = 1 return B3 = gcd(m, b); B2 = b –1 mod m 4. Q = A3 div B3 5. (T1, T2, T3)=(A1 – Q B1, A2 – Q B2, A3 – Q B3) 6. (A1, A2, A3)=(B1, B2, B3) 7. (B1, B2, B3)=(T1, T2, T3) 8. goto 2

26. 9/9/200426 Polynomial Arithmetic Can compute using polynomials Several alternatives available ordinary polynomial arithmetic poly arithmetic with coords mod p poly arithmetic with coords mod p and polynomials mod M(x)

27. 9/9/200427 Ordinary Polynomial Arithmetic Add or subtract corresponding coefficients Multiply all terms by each other E.g. let f(x) = x 3 + x 2 + 2 and g(x) = x 2 – x + 1 f(x) + g(x) = x 3 + 2x 2 – x + 3 f(x) – g(x) = x 3 + x + 1 f(x) x g(x) = x 5 + 3x 2 – 2x + 2

28. 9/9/200428 Polynomial Arithmetic with Modulo Coefficients Compute value of each coefficient as modulo some value Could be modulo any prime But we are most interested in mod 2 i.e. all coefficients are 0 or 1 e.g. let f(x) = x 3 + x 2, g(x) = x 2 + x + 1 f(x) + g(x) = x 3 + x + 1 f(x) x g(x) = x 5 + x 2

29. 9/9/200429 Modular Polynomial Arithmetic Can write any polynomial in the form f(x) = q(x) g(x) + r(x) can interpret r(x) as being a remainder r(x) = f(x) mod g(x) If no remainder say g(x) divides f(x) If g(x) has no divisors other than itself and 1 say it is irreducible (or prime) polynomial Arithmetic modulo an irreducible polynomial forms a field

30. 9/9/200430 Polynomial GCD Can find greatest common divisor for polys c(x) = GCD(a(x), b(x)) if c(x) is the poly of greatest degree which divides both a(x), b(x) can adapt Euclid’s Algorithm to find it: EUCLID[a(x), b(x)] 1. A(x) = a(x); B(x) = b(x) 2. if B(x) = 0 return A(x) = gcd[a(x), b(x)] 3. R(x) = A(x) mod B(x) 4. A(x)  B(x) 5. B(x)  R(x) 6. goto 2

31. 9/9/200431 Modular Polynomial Arithmetic Can compute in field GF(2 n ) polynomials with coefficients modulo 2 whose degree is less than n hence must reduce modulo an irreducible poly of degree n (for multiplication only) Form a finite field Can always find an inverse can extend Euclid’s Inverse algorithm to find

32. 9/9/200432 Arithmetic in GF(2 3 )

33. 9/9/200433 Rijndael Process data as 4 groups of 4 bytes (State) Has 9/11/13 rounds in which state undergoes: byte substitution (1 S-box used on every byte) shift rows (permute bytes between groups/columns) mix columns (subs using matrix multiply of groups) add round key (XOR state with key material) Initial XOR key material & incomplete last round All operations can be combined into XOR and table lookups, hence very fast and efficient

34. 9/9/200434 Rijndael

35. 9/9/200435 AES Round

36. 9/9/200436 Byte Substitution A simple substitution of each byte Uses one table of 16x16 bytes containing a permutation of all 256 8-bit values Each byte of state is replaced by byte in corresponding row (left 4 bits) and column (right 4 bits) eg. byte {95} is replaced by row 9 col 5 byte, which is {2A} S-box is constructed using a defined transformation of the values in GF(2 8 )

37. 9/9/200437 Shift Rows Circular byte shift in each row 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 Decryption does shifts to right Since state is processed by columns, this step permutes bytes between the columns

38. 9/9/200438 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(2 8 ) using prime poly m(x) =x 8 +x 4 +x 3 +x+1

39. 9/9/200439 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 is identical since XOR is own inverse, just with correct round key Designed to be as simple as possible

40. 9/9/200440 AES Key Expansion Take 128-bit (16-byte) key and expand into array of 44/52/60 32-bit words Start by copying key into first 4 words Then loop creating words that depend on values in previous and 4 places back in 3 of 4 cases just XOR these together every 4th has S-box + rotate + XOR constant of previous before XOR together Designed to resist known attacks

41. 9/9/200441 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 and add (tweaked) round key

42. 9/9/200442 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 shifting add round key works on byte XORs mix columns requires matrix multiply in GF(2 8 ) which works on byte values, can be simplified to use a table lookup

43. 9/9/200443 Implementation Aspects 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 16Kb to store tables Designers believe this very efficient implementation was a key factor in its selection as the AES cipher

44. 9/9/200444 Next Class Confidentiality of symmetric encryption Asymmetric encryption: RSA Read Chapters 7, 8, 9