CSCE 715: Network Systems Security Chin-Tser Huang University of South Carolina

09/05/20132 After DES… More symmetric encryption algorithms Triple-DES Advanced Encryption Standards

09/05/20133 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

09/05/20134 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] given a known pair (P, C), attack by encrypting P with all keys and store then decrypt C with keys and match X value can be shown that this attack takes O(2 56 ) steps

09/05/20135 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 is compatible with single DES Standardized in ANSI X9.17 & ISO8732 No current known practical attacks

09/05/20136 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

09/05/20137 Origins of Advanced Encryption Standard Triple-DES is slow with small blocks US NIST issued call for ciphers in candidates accepted in Jun were shortlisted in Aug 1999 Rijndael was selected as the AES in Oct 2000 Issued as FIPS PUB 197 standard in Nov 2001

09/05/20138 AES Requirements Private key symmetric block cipher 128-bit data, 128/192/256-bit keys Stronger and faster than Triple-DES Active life of years (+ archival use) Provide full specification and design details Both C and Java implementations NIST has released all submissions and unclassified analyses

09/05/20139 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)

09/05/ 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

09/05/ 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 on 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

09/05/ Abstract Algebra Background Group Ring Field

09/05/ Group A set of elements or “numbers” With a binary operation whose result is also in the set (closure) Obey the 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

09/05/ 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

09/05/ Field A set of numbers with two operations integral domain multiplicative inverse: aa -1 = a -1 a= 1 Infinite field if infinite number of elements Finite field if finite number of elements

09/05/ 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

09/05/ 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

09/05/ 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

09/05/ Modulo 8 Example

09/05/ 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 Two numbers are called relatively prime if they have no common factors (except 1) e.g. 8 and 15 relatively prime as GCD(8,15) = 1

09/05/ 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

09/05/ 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 )

09/05/ Galois Fields GF(p) GF(p) is set of integers {0,1, …, p-1} with arithmetic operations modulo prime p Form a finite field have multiplicative inverses Hence arithmetic is “well-behaved” and can do addition, subtraction, multiplication, and division without leaving the field GF(p)

09/05/ Arithmetic in GF(7)

09/05/ Finding Multiplicative Inverses By extending Euclid’s algorithm gcd(a, b) = d = ax + by a = q 1 b + r 1 r 1 = ax 1 + by 1 b = q 2 r 1 + r 2 r 2 = ax 2 + by 2 r 1 = q 3 r 2 + r 3 r 3 = ax 3 + by 3 …… r n-2 = q n r n-1 + r n r n = ax n + by n Can derive r i = r i-2 – r i-1 q i And x i = x i-2 – q i x i-1 y i = y i-2 – q i y i-1

09/05/ 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)

09/05/ Ordinary Polynomial Arithmetic Add or subtract corresponding coefficients Multiply all terms by each other E.g. let f(x) = x 3 + x 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

09/05/ 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

09/05/ 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 Polynomial arithmetic modulo an irreducible polynomial forms a field

09/05/ Polynomial GCD Can find greatest common divisor for polynomials 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

09/05/ 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

09/05/ Arithmetic in GF(2 3 )

09/05/ AES Cipher – 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

09/05/ AES Encryption and Decryption

09/05/ AES Data Structure

09/05/ AES Encryption Round

09/05/ Byte Substitution A simple substitution of each byte Uses one table of 16x16 bytes containing a permutation of all 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 )

Byte Substitution Rationale S-box is designed to be resistant to known cryptanalytic attacks The Rijndael developers sought a design that has a low correlation between input bits and output bits and the property that the output is not a linear mathematical function of the input Nonlinearity is due to the use of the multiplicative inverse 09/05/201338

09/05/ 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

Shift Row Transformation 09/05/201340

Shift Row Rationale  More substantial than it may first appear  The State, as well as the cipher input and output, is treated as an array of four 4-byte columns  On encryption, the first 4 bytes of the plaintext are copied to the first column of State, and so on  The round key is applied to State column by column  Thus, a row shift moves an individual byte from one column to another, which is a linear distance of a multiple of 4 bytes  Shift row ensures that the 4 bytes of one column are spread out to four different columns 09/05/201341

09/05/ 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

MixColumn Transformation 09/05/201343

Mix Columns Rationale Coefficients of a matrix based on a linear code with maximal distance between code words ensures a good mixing among the bytes of each column The mix column transformation combined with the shift row transformation ensures that after a few rounds all output bits depend on all input bits 09/05/201344

09/05/ 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 but ensured to affect every bit of State

09/05/ AES Key Expansion Take 128/192/256-bit 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

09/05/ AES Key Expansion

09/05/ AES Decryption AES decryption is not identical to encryption because steps are done in reverse But can define an equivalent inverse cipher with steps as for encryption use inverses of each step But with a different key schedule Works since result is unchanged when swap byte substitution & shift rows swap mix columns and add (tweaked) round key

AES Example 09/05/201349

Avalanche Effect of AES 09/05/201350

09/05/ 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

09/05/ 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

09/05/ Next Class Confidentiality of symmetric encryption Read Chapter 14