aes

AESAESAAdvanced dvanced EEncryption ncryption SStandardtandard

CSC 7002 Computer SecurityCSC 7002 Computer Security

William Roche

http://ouray.cudenver.edu/~wrroche/Security/Presentation.html

Advanced Encryption StandardAdvanced Encryption Standard

• Adopted by National Institute of Standards and Technology (NIST) on May 26, 2002.

• AES is a simple design, a high speed algorithm, with low memory costs.

• AES is a symmetric block cipher. – The same key is used to encrypt and decrypt

the message.– The plain text and the cipher text are the

same size.

AES BlockAES Block

• AES has a fixed block size of 128 bits called a state

ABCDEFGHIJKLMNOP

A E I M 41 45 49 4D

B F J N 42 46 4A 4E

C G K O 43 47 4B 4F

D H L P 44 48 4C 50

(ASCII)

AES KeyAES Key

• AES key is either 128 bits, 192 bits or 256 bits

128 bits (4 words):11223344556677889900AABBCCDDEEFF

11 22 33 4455 66 77 8899 00 AA BBCC DD EE FF

AES KeyAES Keyor 192 bits (6 words)11223344556677889900AABBCCDDEEFF1122334455667788

11 22 33 4455 66 77 8899 00 AA BBCC DD EE FF11 22 33 4455 66 77 88

or 256 bits (8 words)11223344556677889900AABBCCDDEEFF11223344556677889900AABBCCDDEEFF

11 22 33 4455 66 77 8899 00 AA BBCC DD EE FF11 22 33 4455 66 77 8899 00 AA BBCC DD EE FF

ComparisonsComparisons

Key Length (Nk words)

Expanded Key Length

(words)Block Size (Nb words)

Number of Rounds Nr

AES-128 4 44 4 10AES-192 6 52 4 12AES-256 8 60 4 14

4 106 128 144 126 128 144 146 148 14

DES 2* 256 2 16* of 64 bits, only 56 are used

Rijndael - 128

Rijndael - 192

Rijndael - 256

4

6

8

44

52

60

SecuritySecurity• The key security feature is the size of the

key.

Assuming that one could build a machine that could recover a DES key in a second (i.e., try 255 keys per second), then it would take that machine approximately 149 thousand-billion (149 trillion) years to crack a 128-bit AES key. To put that into perspective, the universe is believed to be less than 20 billion years old.

• Accepting Moore's Law, doubling processor speed every 18 months, AES will be secure for another 109.5 years.

AES OperationsAES Operations

• AES Operates on the binary field GF(28).– This can be represented as a polynomial b(x)

with binary coefficients b {0,1}:

b7x7 + b6x6 + b5x5 + b4x4 + b3x3 + b2x2 + b1x + b0

• Multiplication in GF(28) consists of multiplying two polynomials modulo an irreducible polynomial of degree 8.– AES uses the following irreducible polynomial

m(x) = x8 + x4 + x3 + x + 1

AES AlgorithmAES Algorithm

KeyExpansion(byte key[4*Nk], word w[Nb*(Nr+1)],Nk) Cipher(byte in[4*Nb],byte out[4*Nb],word w[Nb*(Nr+1)]) begin

byte state[4,Nb] state = in AddRoundKey(state, w[0, Nb-1]) for round = 1 step 1 to Nr–1

SubBytes(state) ShiftRows(state) MixColumns(state) AddRoundKey(state,w[round*Nb,(round+1)*Nb-1])

end for SubBytes(state) ShiftRows(state) AddRoundKey(state, w[Nr*Nb, (Nr+1)*Nb-1]) out = state

end






end

AES Algorithm AES Algorithm Key Expansion Key Expansion

– Sample Key:

11223344556677889900AABBCCDDEEFF

• The first 4 (Nk) words are set equal to the key

w[0] 11 22 33 44 w[1] 55 66 77 88 w[2] 99 00 AA BB w[3] CC DD EE FF


For words 4 through 43

i = Nk // Nk = 4

while (i < Nb*(Nr+1)) { // Nb*(Nr+1)= 4*(10+1)= 44temp = w[ i – 1 ]

If ( i%Nk == 0 )

rotate word left 1 byte

process each byte through sbox

XOR with RCON[i/Nk-1] // just first byte of w[i]

w[ i ] = w[ i-4 ] XOR temp

i++}

i = Nk // Nk = 4

while (i < Nb*(Nr+1)) { // Nb*(Nr+1)= 4*(10+1)= 44

temp = w[ i - 1 ]



i = 4

temp = w[3] = CC DD EE FF


If ( i%Nk == 0 )



XOR with RCON[i/Nk-1]

temp = CC DD EE FFtemp = DD EE FF CC

temp = sbox[DD] sbox[EE] sbox[FF] sbox[CC] = C1 28 16 4B

RCON[0] = 01 temp = (C1 01) 28 16 4B

temp = C0 28 16 4B

rCon – rCon – rround ound ConConstantsstants

i 8 9 A B C D Exi

0x1b 0x36 0x6c 0xd8 0xab 0x4d 0x9a

Powers of x = 0x02

• rCon can be implemented with a look-up-table• 2i in GF(28)• Removes symmetry and linearity from key expansion.

i 0 1 2 3 4 5 6 7xi

0x01 0x02 0x04 0x08 0x10 0x20 0x40 0x80

Powers of x = 0x02



i = Nk // Nk = 4

while (i < Nb*(Nr+1)) {// Nb*(Nr+1)= 4*(10+1)= 44

temp = W[i-1]

If (i%Nk == 0)



XOR with RCON[i] // just first element of w

w[i] = w[i-4] XOR temp

i++}

i = 4

temp = C0 28 16 4Bw[i] = w[i-4] XOR temp

AES AlgorithmAES AlgorithmKey ExpansionKey Expansion

i = 4 temp = C0 28 16 4B w[i] = w[i-4] XOR temp


w[4] = (11 C0) (22 28) (33 16) (44 4B)

w[4] = D1 0A 25 0F

w[4] D1 0A 25 0F



i = Nk // Nk = 4 i = 5

while (i < Nb*(Nr+1)) { // Nb*(Nr+1)= 4*(10+1)= 44temp = w[i-1]

If (i%Nk == 0)



XOR with RCON[i/Nk-1] // just first element of W

w[i] = w[i-4] XOR temp

i++}

temp = w[4] = D1 0A 25 0F


i = 5 temp = D1 0A 25 0F w[i] = w[i-4] XOR temp


w[5] = (55 D1) (66 0A) (77 25) (88 0F)

w[4] D1 0A 25 0F

w[5] = 84 C6 52 87

Beginning Key Expansion for: 11 22 33 44 55 66 77 88 99 00 AA BB CC DD EE FF i subword rCon w[i-4]XORtemp w[i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






end






end

AES Algorithm AES Algorithm AddRoundKeyAddRoundKey

41 45 49 4D42 46 4A 4E43 47 4B 4F44 48 4C 50

11 22 33 4455 66 77 8899 00 AA BBCC DD EE FF

41 11 45 55 49 99 4D CC42 22 46 66 4A 00 4E DD 43 33 47 77 4B AA 4F EE44 44 48 88 4C BB 50 FF

50 10 D0 8160 20 4A 9370 30 E1 A100 C0 F7 AF

StateExpanded Key w[0] w[4]

After AddRoundKey






end

AES Algorithm AES Algorithm SubBytesSubBytes

• SubBytes is the SBOX for AES• This make AES a non-linear cryptographic system.• For every value of b there is a unique value for b’

– It is faster to use a substitution table (and easier).

x is the inverse value of the byte b

1 0 0 0 1 1 1 11 1 0 0 0 1 1 11 1 1 0 0 0 1 11 1 1 1 0 0 0 11 1 1 1 1 0 0 00 1 1 1 1 1 0 00 0 1 1 1 1 1 00 0 0 1 1 1 1 1

11000110

x0

x1

x2

x3

x4

x5

x6

x7

+=

b'0b'1b'2b'3b'4b'5b'6b'7

AES AlgorithmAES AlgorithmSubBytesSubBytes

0 1 2 3 4 5 6 7 8 9 a b c d e f0 63 7c 77 7b f2 6b 6f c5 30 1 67 2b fe d7 ab 761 ca 82 c9 7d fa 59 47 f0 ad d4 a2 af 9c a4 72 c02 b7 fd 93 26 36 3f f7 cc 34 a5 e5 f1 71 d8 31 153 4 c7 23 c3 18 96 5 9a 7 12 80 e2 eb 27 b2 754 9 83 2c 1a 1b 6e 5a a0 52 3b d6 b3 29 e3 2f 845 53 d1 0 ed 20 fc b1 5b 6a cb be 39 4a 4c 58 cf6 d0 ef aa fb 43 4d 33 85 45 f9 2 7f 50 3c 9f a87 51 a3 40 8f 92 9d 38 f5 bc b6 da 21 10 ff f3 d28 cd 0c 13 ec 5f 97 44 17 c4 a7 7e 3d 64 5d 19 739 60 81 4f dc 22 2a 90 88 46 ee b8 14 de 5e 0b dba e0 32 3a 0a 49 6 24 5c c2 d3 ac 62 91 95 e4 79b e7 c8 37 6d 8d d5 4e a9 6c 56 f4 ea 65 7a ae 8c ba 78 25 2e 1c a6 b4 c6 e8 dd 74 1f 4b bd 8b 8ad 70 3e b5 66 48 3 f6 0e 61 35 57 b9 86 c1 1d 9ee e1 f8 98 11 69 d9 8e 94 9b 1e 87 e9 ce 55 28 dff 8c a1 89 0d bf e6 42 68 41 99 2d 0f b0 54 bb 16

Y

X

AES Algorithm AES Algorithm SubBytesSubBytes

50 10 D0 8160 20 4A 9370 30 E1 A100 C0 F7 AF

Sbox( 50 ) Sbox( 10 ) Sbox( D0 ) Sbox( 81 )Sbox( 60 ) Sbox( 20 ) Sbox( 4A ) Sbox( 93 )Sbox( 70 ) Sbox( 30 ) Sbox( E1 ) Sbox( A1 )Sbox( 00 ) Sbox( C0 ) Sbox( F7 ) Sbox( AF )

53 CA 70 0CD0 B7 D6 DC51 04 F8 3263 BA 68 79

State


KeyExpansion(byte key[4*Nk],word w[Nb*(Nr+1)],Nk) Cipher(byte in[4*Nb],byte out[4*Nb],word w[Nb*(Nr+1)]) begin




end

AES AlgorithmAES AlgorithmShiftRowsShiftRows

53 CA 70 0CD0 B7 D6 DC51 04 F8 3263 BA 68 79

53 CA 70 0CB7 D6 DC D0F8 32 51 0479 63 BA 68

• Simple routine which performs a left shift rows 1, 2 and 3 by 1, 2 and 3 bytes respectively

Before Shift Rows After Shift Rows






end

AES Algorithm - MixColumnsAES Algorithm - MixColumns• This with shift rows provides diffusion • The columns are considered polynomials over

GF(28) and multiplied modulo x4+1 with a(x) where a(x) = {03}x3 + {01}x2 + {01}x + {02} NOTE: x4+1 is relatively prime to a(x)

a’j (aj*a(x))mod(x4+1)

• This can also be written as matrix multiplication.

02 03 01 01

01 02 03 01

01 01 02 03

03 01 01 02

a0

a1

a2

a3

a’0

a’1

a’2

a’3

=

AES Algorithm - MixColumnsAES Algorithm - MixColumns

02 03 01 01

01 02 03 01

01 01 02 03

03 01 01 02

a0

a1

a2

a3

a’0

a’1

a’2

a’3

=

a’0 = 2a0 + 3a1 + a2 + a3

a’1 = a0 + 2a1 + 3a2 + a3

a’2 = a0 + a1 + 2a2 + 3a3

a’3 = 3a0 + a1 + a2 + 2a3

Addition is easy in GF(28) : Addition is just the XOR operation

Multiplication by 1 is easy in GF(28) : Multiplication by one is the identity

Multiplication by 2 in GF(28) takes some work: . If multiplying by a value < 0x80 just shift all the bits left by 1 . If multiplying by a value ≥ 0x80 shift left by 1 and XOR with 0x1b . This prevents overflow and keeps the values within rangeTo Multiply by 3 in GF(28) : a * 0x03 = a * (0x02 + 0x01) = (a * 0x02) (a * 0x01)

a’0 = 2a0 3a1 a2 a3

a’1 = a0 2a1 3a2 a3

a’2 = a0 a1 2a2 3a3

a’3 = 3a0 a1 a2 2a3


KeyExpansion(byte key[4*Nk], word w[Nb* (Nr+1)],Nk) Cipher(byte in[4*Nb],byte out[4*Nb],word w[Nb*(Nr+1)]) begin




end

Sample ConversionsSample Conversions

Input String Key Output String (HEX)

ABCDEFGHIJKLMNOP 11223344556677889900AABBCCDDEEFF BC4784A37D6F46452656B993D53393F5

ABCDEFGHIJKLMNOP 01223344556677889900AABBCCDDEEFF 855866490543FDF6504FC84088FEDCA0

ABCDEFFHIJKLMNOP 11223344556677889900AABBCCDDEEFF 372CCA446C0D391C4381392344630EE1

Input String(HEX) Key Output String (HEX)

00000000000000000000000000000000 00000000000000000000000000000000 66E94BD4EF8A2C3B884CFA59CA342B2E

00000000000000000000000000000000 00000000000000000000000000000001 0545AAD56DA2A97C3663D1432A3D1C84

00000000000000000000000000000001 00000000000000000000000000000001 A17E9F69E4F25A8B8620B4AF78EEFD6F


AddRoundKey

SubBytes

ShiftRows

MixColumns

AddRoundKey

SubBytes

ShiftRows

AddRoundKey

1st Round

Repeat Nr -1 Round

Last Round

AddRoundKey

InvShiftRows

InvSubBytes

AddRoundKey

InvMixColumns

InvShiftRows

InvSubBytes

AddRoundKey

1st Round

Repeat Nr -1 Round

Last Round

PlainText Cipher Text

CipherText Plain Text

Encryption Decryption

RoundKey*RoundKey

RoundKey

RoundKey

RoundKey*

RoundKey*

* RoundKey Added in reverse order

Larger Plain TextsLarger Plain Texts

• How to avoid frequency analysis?

Cipher Block Chaining

PaddingPadding

• If plaintext messages are not divisible by 16 bytes. Padding may be a solution.

• An easy method is to add a single 1 bit at the end of the message followed by enough 0’s to fill the block.– If the block is filled, encode one more block

with a 1 followed by 0’s.

Attacks on AESAttacks on AES

• Differential Cryptanalysis – Study of how differences in input affect differences in output.– Greatly reduced due to high number of rounds.

• Linear Cryptanalysis – Study of correlations between input and output.– SBOX & Mix Columns are designed to frustrate

Linear Analysis

Attacks on AESAttacks on AES

• Side Channel Attacks – Attacks based on studying and measuring the actual implementation of the code.– For some implementations of AES the key has

been obtained in under 100 minutes.• Computer running AES was 850MHz, Pentium III

running FreeBSD 4.8

Types of Side Channel AttacksTypes of Side Channel Attacks

• Timing Attacks – Watches movement of data in and out of the CPU or memory.– It is difficult to retrieve an array element in a

time that is not dependent on the index value.

• Power Attacks – Watches power consumption by CPU or memory.– Changing one bit requires considerably less

power than changing all bits in a byte.

Attack PrecautionsAttack Precautions• Avoid use of arrays. Compute values in

SBOX and rCon. • Design algorithms and devices to work

with constant time intervals. (independent of key and plaintext.)– Hidden CPU timing data is a threat.

• Use same memory throughout, Cache is faster than DRAM

• Compute Key Expansion on the fly.• Utilize pipelining to stabilize CPU power

consumption.

Joan Daemen & Vincent Rijmen’s Joan Daemen & Vincent Rijmen’s AES Selling Points AES Selling Points

• Extremely fast compared to other block ciphers. (tradeoff between size and speed)

• The round transformation is parallel by design. Important in dedicated hardware.

• Amenable to pipelining

• The cipher does not use arithmetic operations so has no bias towards big or little endian architectures.

Joan Daemen & Vincent Rijmen’s Joan Daemen & Vincent Rijmen’s AES Selling Points AES Selling Points

• Fully Self-supporting. Does not use Sboxes of other ciphers, bits from Rand tables, digits of or any other such jokes.

• Is not based on obscure or not well understood processes

• The tight cipher design does not leave enough room to hide a trap door.

Joan Daemen & Vincent Rijmen’s Joan Daemen & Vincent Rijmen’s AES Limitations AES Limitations

• The inverse cipher is less suited to smart cards, as it takes more codes and cycles.

• The cipher and inverse cipher make use of different codes and/or tables.

• In hardware, The inverse cipher can only partially re-use circuitry which implements the cipher.

ReferencesReferencesAbout AES

AES Proposal : RijndaelJoan Daemen, Vincent Rijmen,2nd version of document to NIST

Polynomials in the Nations Service: Using Algebra to Design the Advanced Encryption Standard Susan LandauThe Mathmatical Association of America,Monthly 111 Feb 2004 Page(s):89-117

Selecting the Advanced Encryption Standard Burr, W.E.;Security & Privacy Magazine, IEEEVolume 1, Issue 2, Mar-Apr 2003 Page(s):43 - 52

Title: Introduction to CryptographyAuthor: Johannes A BuchmanPublisher: Springer; 2 edition (July 13, 2004)

ReferencesReferencesSecurity and Attacking AES

Power-analysis attack on an ASIC AES implementationOrs, S.B.; Gurkaynak, F.; Oswald, E.; Preneel, B.;Information Technology: Coding and Computing, 2004. Proceedings. ITCC 2004. International Conference onVolume 2, 2004 Page(s):546 - 552 Vol.2

Algebraic attacks on cipher systemsPenzhorn, W.T.;AFRICON, 2004. 7th AFRICON Conference in AfricaVolume 2, 2004 Page(s):969 - 974 Vol.2

Cache-Timing attacks on AES Daniel J Bernstein Preliminary version of report to National Science Foundation, grant CCR-9983950

ReferencesReferencesApplications / Implementations: AES

A high throughput low cost AES processorChih-Pin Su; Tsung-Fu Lin; Chih-Tsiun Huang; Cheng-Wen Wu;Communications Magazine, IEEEVolume 41, Issue 12, Dec. 2003 Page(s):86 - 91

An efficient FPGA implementation of advanced encryption standard algorithmShuenn-Shyang Wang; Wan-Sheng Ni;Circuits and Systems, 2004. ISCAS '04. Volume 2, 23-26 May 2004 Page(s):II - 597-600 Vol.2

High-speed VLSI architectures for the AES algorithmXinmiao Zhang; Parhi, K.K.;Very Large Scale Integration (VLSI) SystemsVolume 12, Issue 9, Sept. 2004 Page(s):957 – 967

Fast implementation of AES cryptographic algorithms in smart cardsChi-Feng Lu; Yan-Shun Kao; Hsia-Ling Chiang; Chung-Huang Yang;Security Technology, 2003. 14-16 Oct. 2003 Page(s):573 - 579

ReferencesReferencesApplications / Implementations : AES

A new VLSI implementation of the AES algorithmLiang Deng; Hongyi Chen;Communications, Circuits and Systems and West Sino Expositions, IEEE 2002 International Conference onVolume 2, 29 June-1 July 2002 Page(s):1500 - 1504 vol.2

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