Cryptography and Network Security

Advanced Encryption Standard By William Stallings

Modified by M. Sakalli

Origins• clear a replacement for DES was needed

– have theoretical attacks that can break it– have demonstrated exhaustive key search attacks

• can use Triple-DES – but slow with small blocks• US NIST issued call for ciphers in 1997• 5 were short-listed in Aug-99

– MARS (IBM) - complex, fast, high security margin – RC6 (USA) - v. simple, v. fast, low security margin – Rijndael (Belgium) - clean, fast, good security margin – Serpent (Euro) - slow, clean, v. high security margin – Twofish (USA) - complex, v. fast, high security margin

• Rijndael was selected as the AES in Oct-2000• issued as FIPS PUB 197 standard in Nov-2001

AES Requirements• private key 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• Evaluation criteria of submitted ones

– General security – effort to practically cryptanalyse– algorithm & implementation characteristics– cost – computational, software & hardware implementation

ease, minimize implementation attacks– flexibility (in en/decrypt, keying, other factors)

Rijndael • processes data as 4 groups of 4 bytes (state)• has 9/11/13 rounds in which state undergoes:

1. byte substitution (1 S-box; byte to byte substitution) 2. shift rows (permutation of bytes) 3. mix columns (subs using gf28) 4. Add Round Key (XOR state with a portion of expended K)

• initial XOR key material & incomplete last round• all operations can be combined into XOR and table lookups - hence

very fast & efficient

The AES Cipher• 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

InvAddRoundKey• (A B) B = A • Key is used in reverse order

AddRoundKey• Each round uses four different words from the expanded key array.• Each column in the state matrix is XORed with a different word.• The heart of the encryption. All other functions’ properties are

permanent and known to all.

Substitution Byte (Subbyte)• It is a bytewise lookup process that returns a 4-

byte word in which each byte is the result of applying the Rijndael S-box. Designed to be resistant to all known attacks

• Simple substitution of each byte using one table of 16x16 bytes containing a permutation of all 256 8-bit values

• each byte of state is replaced by byte in row (left 4-bits) & column (right 4-bits)– eg. byte {95} is replaced by row 9 col 5 byte– which is the value {2A}

• S-box is constructed using a transformation of the values in GF(28)

Shift Rows

• a 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

• decrypt does shifts to right

• since state is processed by columns, this step permutes bytes between the columns

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 prime poly m(x)

=x8+x4+x3+x+1

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 simple

AES Round

Mathematical Review• Performing arithmetic operations on bytes requires to work

in a finite field and treat each byte as an element.• GF(28) - Finite field containing 256 elements. • Each element is a polynomial of degree 7 over Z2, hence

an element is defined by 8 binary values – a byte. • Addition – polynomial addition, over Z2, implemented using

XOR.• Multiplication – polynomial multiplication , over Z2, modulo

irreducible polynomial X8 + X4 + X3 + X + 1 Implemented using repetitive left shifts and XOR.

SubBytes - 16 X 16 table• Each byte is considered as an element in GF(28)• Called S-BoxA. 16 X 16 table contains all possible 256

elements.• Row Column Indices: Left and Right halves of the byte.• Each byte B in the state matrix is substituted with f(B).

SubBytes, S-Box computationComputing S-Box cells in three stages:-The cells are numbered in ascending order.-Each cell’s number is substituted with its multiplicative inverse over GF(28).- The cell’s bits go through the following transformation:

bi’ = bi b(i+4)mod(8) b(i+5)mod(8) b(i+6)mod(8) b(i+7)mod(8) ci

bi’ = new bit value, ci = the i’th bit of 63={11000110} irreducible polynomial

S-Box eg. byte {95} is replaced by row 9 col 5 byte which is the value {2A}

InvSubBytes• Same routine as SubBytes, but uses the inverse S-Box.

• Inverse S-box is computed by applying the inverse affine transformation and then substituting with the multiplicative inverse, of the cell’s value in the S-Box.

• The Inverse transformation:

bi’ = b(i+2)mod8 b(i+5)mod8 b(i+7)mod8 dibi’ = new bit value, di = the i’th bit of

05={00000101}.

SubBytes, crypto properties • S-Box design makes it resistant to cryptanalitic attacks. • Conditions:

– No fixed points S(a) a, no opposite fixed points IS(a) a complement.

– Invertible s box, IS[S(a)] = a;– but not self invertible, which means

S(a) IS(a), ie. S({95}) = {2A}, but IS({95}) = {AD} think S({2A})=?{95}

• To see that InvSubBytes is the inverse of SubBytes, – label the matrices in SubBytes and InvSubBytes as X and Y,

respectively, and the vector versions of constants c and d as C and D, respectively.

– For some 8-bit vector B B' = XB C. – To show that Y(XB C) D = B. – Must show YXB YC D = B.

• B’=XBC; (Y(XBC)D) = [YX][B][YC][D] = B

– Which means …

ShiftRows• Rows 2-4 in the state matrix are left shifted by different

offsets of 1-3 bytes respectively.• Strong diffusion effect. Separation of each four,

originally consecutive, bytes.

• A transformation which operates on individual columns – 32 bits/4 bytes.• Each column is treated as a 3 degree polynomial over GF(23)• Multiplied by the fixed polynomial:

a(x)=({03}X3 + {01}X2 + {01}X + {02})mod(x4+1)• a(x) was chosen so the multiplication/transformation is invertible. • Generally, multiplication in the above group mod(x4+1) doesn’t provide

inverse for each element. *coefficients multiplication is the GF(28) multiplication mentioned earlier.

MixColumn, props- The transformation is a linear code with a maximal distance between code words.- Combined with ShiftRows, after several rounds all output bits depend on all input bits.

• In GF(28), irreducible polynomial mod(x4+x3+x+1) ─ ({02} · {87}) ({03} · {6E}) {46} {A6} = {47}─ {87} ({02} · {6E}) ({03}· {46}) {A6} = {37}─ {87} {6E} ({02} · {46} ({03} · {A6}) = {94}─ ({03} · {87}) {6E} {46} ({02} · {A6} = {ED}• For the first equation,

– {02} · {87} = x*(x7 + x2+x+1) = (1 0000 1110) l because of the most left 1, (0000 1110) (0001 1011) = (0001 0101);

• and – {03} · {6E} = (x+1)*(x6 +x5 + x3+ x2+x)

= (x6 +x5 + x3+ x2+x) (x)*(x6 +x5 + x3+ x2+x), the same statement for the second side.

= {6E} ({02} · {6E}) = (0110 1110) (1101 1100) = (1011 0010).

• {02} · {87} = 0001 0101  • {03} · {6E} = 1011 0010  • {46} = 0100 0110  • {A6} = 1010 0110   • Total 0100 0111 = {47}

InvMixColumn

• Same routine as MixColumn, only instead of a(x) the inverse of a(x) is used:

a-1(x)={0B}x3{0D}x2{09}x{0E}

AES Key Expansion

• takes 128-bit (16-byte) key and expands 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 & 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

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

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(28) which

works on byte values, can be simplified to use a table lookup

• can efficiently implement on 32-bit CPU– redefine steps to use 32-bit words– can pre-compute 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

Summary

• have considered:– the AES selection process– the details of Rijndael – the AES cipher– looked at the steps in each round– the key expansion– implementation aspects