Information Security and Cryptography 資訊安全與密碼學. Lecture 6 April 8, 2015 洪國寶

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1 Information Security and Cryptography 資訊安全與密碼學 Lecture 6 April 8, 2015 洪國寶 1

2 Outline Review Cryptology Introduction and terminologies Definition of cryptosystem and cryptanalysis Types of encryption Symmetric encryption -- Classical techniques Symmetric encryption -- Modern techniques secure encryption schemes modern symmetric block encryption DES AES 2

3 Review: Definition of cryptosystems A cryptosystem is a five-tuple (P,C,K,E,D), where the following conditions are satisfied: 1. P is a finite set of possible plaintexts 2. C is a finite set of possible ciphertexts 3. K, the key space, is a finite set of possible keys 4. For each k K, there is an encryption rule e K E and a corresponding decryption rule d K D. Each e K :P C and d K : C P are functions such that d K (e K (x)) = x for every plaintext x P. Example: Caesar cipher 3

4 Example: Caesar (shift) cipher P = {0(A), 1(B),, 25(Z)} C = {0(A), 1(B),, 25(Z)} K = { 0, 1, 2,, 25} e k (x) x + k mod 26 d k (y) y k mod 26 4

5 Review: Attacking a cryptosystem Cryptanalysis approach: this type of attack exploits the characteristics of the algorithm plus perhaps some knowledge of the general characteristics of the plaintext or even some sample plaintext-ciphertext pairs. Brute force approach: an attacker tries every possible key on a piece of ciphertext until intelligible translation into plaintext is obtained. 5

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7 Review: Cryptographic systems The type of operations used for transforming plaintext to ciphertext Substitution The number of keys used Symmetric, single-key, secret-key, conventional encryption The way in which the plaintext is processed Block cipher Transposition Asymmetric, two-key, or public-key encryption Stream cipher 7

8 Review: Classical Substitution Ciphers where letters of plaintext are replaced by other letters or by numbers or symbols monoalphabetic: Single letter: Caesar Multiple letter: Playfair, Hill polyalphabetic: Vigenere tableau 8

9 Review: Monoalphabetic Cipher shuffle the letters arbitrarily each plaintext letter maps to a different random ciphertext letter hence key is 26 letters long, K =26! Plain: abcdefghijklmnopqrstuvwxyz Cipher: DKVQFIBJWPESCXHTMYAUOLRGZN Plaintext: ifwewishtoreplaceletters Ciphertext: WIRFRWAJUHYFTSDVFSFUUFYA 9

10 Review: Monoalphabetic Cipher Language Redundancy and Cryptanalysis human languages are redundant eg "th lrd s m shphrd shll nt wnt" letters are not equally commonly used in English e is by far the most common letter then T,R,N,I,O,A,S other letters Z,J,K,Q,X are fairly rare have tables of single, double & triple letter frequencies Single letter monoalphabetic substitution ciphers are insecure. 10

11 Review: Playfair Cipher: Key Matrix a 5X5 matrix of letters based on a keyword plaintext encrypted two letters at a time: 1. if a pair is a repeated letter, insert a filler like 'X', 2. if both letters fall in the same row, replace each with letter to right (wrapping back to start from end), 3. if both letters fall in the same column, replace each with the letter below it (again wrapping to top from bottom), 4. otherwise each letter is replaced by the one in its row in the column of the other letter of the pair. 11

12 Review: Hill cipher Hill 1929 The encryption algorithm takes m successive plaintext letters and substitutes for them m ciphertext letters. K = {m m invertible matrices over Z26 } Hill cipher completely hides single letter frequencies (i.e. Hill cipher is strong against ciphertext only attack.) Hill cipher can be easily broken with a known plaintext attack (only need m plaintextciphertext pairs). 12

13 Review: comparison Single/multiple letter substitution Vulnerable to Caesar cipher Single Ciphertext-only attack Playfair cipher Multiple Ciphertext-only attack (need more ciphertexts) Hill cipher Multiple Knownplaintext attack 13

14 Review: Polyalphabetic Ciphers use multiple cipher alphabets makes cryptanalysis harder with more alphabets to guess and flatter frequency distribution use a key to select which alphabet is used for each letter of the message use each alphabet in turn repeat from start after end of key is reached simplest polyalphabetic substitution cipher is the Vigenère Cipher effectively multiple caesar ciphers 14

15 Review: One-Time Pad One-Time pad (OTP) P = C = K = (Z2) n, n 1 k = (k 1, k 2,, k n ) x = (x 1, x 2,, x n ) y = (y 1, y 2,, y n ) e k (x) = (x 1 k 1, x 2 k 2,, x n k n ) d k (y) = (y 1 k 1, y 2 k 2,, y n k n ) One-Time pad is unbreakable In practice, two fundamental difficulties Supplying truly random keys of large volumn is a significant task Key distribution and protection are problematic 15

16 Review:Transposition Ciphers these hide the message by rearranging the letter order without altering the actual letters used A simple transposition write message letters out diagonally over a number of rows then read off cipher row by row a more complex scheme write letters of message out in rows over a specified number of columns then reorder the columns according to some key before reading off the rows 16

17 Outline Review Cryptology Introduction and terminologies Definition of cryptosystem and cryptanalysis Types of encryption Symmetric encryption -- Classical techniques Symmetric encryption -- Modern techniques secure encryption schemes modern symmetric block encryption DES AES 17

18 Modern Block Ciphers: introduction Modern block ciphers P = C = {binary strings of fixed length} Can be regarded as substitution ciphers Classical substitution is vulnerable to statistical analysis (of the plaintext) and brute force attacks Reason: P and K are too small 18

19 Modern Block Ciphers: introduction To make statistical analysis (of the plaintext) and brute force attacks infeasible P and K must be large For n-bit block, we need to choose Large n and Arbitrary reversible substitution between P and C Reason: need a large amount of plaintexts and ciphertexts and space for statistical analysis 19

20 Modern Block Ciphers: introduction Problem: To determine the specific mapping from all possible mappings requires K = 2 n! Equivalently, the size of a key is n 2 n For n = 64, the size of a key is = bits 20

21 Modern Block Ciphers: introduction Solution: confine ourselves to a subset of the 2 n! possible mappings. For example, Hill cipher Utilize the concept of product cipher Shannon substitution-permutation (S-P) networks Feistel cipher structure 21

22 Shannon and Substitution- Permutation Ciphers in 1949 Claude Shannon introduced idea of substitution-permutation (S-P) networks these form the basis of modern block ciphers S-P networks are based on the two primitive cryptographic operations: substitution (S-box) permutation (P-box) provide confusion and diffusion of message 22

23 Diffusion and Confusion Terms introduced by Claude Shannon to capture the two basic building blocks for any cryptographic system Shannon s concern was to thwart cryptanalysis based on statistical analysis Diffusion The statistical structure of the plaintext is dissipated into long-range statistics of the ciphertext This is achieved by having each plaintext digit affect the value of many ciphertext digits Confusion Seeks to make the relationship between the statistics of the ciphertext and the value of the encryption key as complex as possible Even if the attacker can get some handle on the statistics of the ciphertext, the way in which the key was used to produce that ciphertext is so complex as to make it difficult to deduce the key 23

24 Feistel Cipher Structure Horst Feistel devised the feistel cipher based on concept of invertible product cipher 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 24

25 Feistel Cipher Structure 25

26 Feistel Cipher Design Features Block size Larger block sizes mean greater security but reduced encryption/decryption speed for a given algorithm Key size Larger key size means greater security but may decrease encryption/decryption speeds Number of rounds The essence of the Feistel cipher is that a single round offers inadequate security but that multiple rounds offer increasing security Subkey generation algorithm Greater complexity in this algorithm should lead to greater difficulty of cryptanalysis Round function F Greater complexity generally means greater resistance to cryptanalysis Fast software encryption/decryption In many cases, encrypting is embedded in applications or utility functions in such a way as to preclude a hardware implementation; accordingly, the speed of execution of the algorithm becomes a concern Ease of analysis If the algorithm can be concisely and clearly explained, it is easier to analyze that algorithm for cryptanalytic vulnerabilities and therefore develop a higher level of assurance as to its strength

27 Outline Review Cryptology Introduction and terminologies Definition of cryptosystem and cryptanalysis Types of encryption Symmetric encryption -- Classical techniques Symmetric encryption -- Modern techniques secure encryption schemes modern symmetric block encryption DES AES 27

28 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 has been considerable controversy over its security On May 19, 2005 FIPS 46-3 (DES) was withdrawn and is no longer approved for Federal use. 28

29 DES History IBM developed Lucifer cipher by team led 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 29

30 DES Design Controversy although DES standard is public was considerable controversy over design in choice of 56-bit key (vs Lucifer 128-bit) and because design criteria were classified subsequent events and public analysis show in fact design was appropriate DES has become widely used, esp in financial applications 30

31 DES Round Structure uses two 32-bit L & R halves as for any Feistel cipher can describe as: L i = R i 1 R i = L i 1 xor F(R i 1, K i ) takes 32-bit R half and 48-bit subkey and: expands R to 48-bits using perm E adds to subkey passes through 8 S-boxes to get 32-bit result finally permutes this using 32-bit perm P 31

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33 Strength of DES Key Size 56-bit keys have 2 56 = 7.2 x values brute force search looks hard recent advances have shown is possible in 1997 on Internet in a few months in 1998 on dedicated h/w (EFF) in a few days in 1999 above combined in 22hrs! still must be able to recognize plaintext now have alternatives to DES 33

34 DES Variants clear 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 34

35 Double DES 35

36 Triple DES 36

37 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 = 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 takes O(2 56 ) steps Use blackboard 37

38 Meet in the middle technique k1 = k2 = P Enc k1 (P) = Dec k2 (C) C k1 = 11.1 k2 = 11.1

39 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 = E K1 [D K2 [E K1 [P]]] encrypt & decrypt equivalent in security if K1=K2 then can work with single DES standardized in ANSI X9.17 & ISO Rev1.pdf no current known practical attacks 39

40 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 = E K3 [D K2 [E K1 [P]]] has been adopted by some Internet applications, eg PGP, S/MIME 40

41 Outline Review Cryptology Introduction and terminologies Definition of cryptosystem and cryptanalysis Types of encryption Symmetric encryption -- Classical techniques Symmetric encryption -- Modern techniques secure encryption schemes modern symmetric block encryption DES AES 41

42 AES why? DES had a problem brute force attacks had been successful New theoretical attacks that may be succesful 3DES works is slow particularly with small blocks is at its upper limit short lifetime 42

43 AES how? U.S. NIST announced a cipher competition in candidates were accepted in June were chosen for further analyses in August 1999 Rijndael from Belgium was selected as the cryptographac algorithm for AES in October 2000 AES was published as a standard in FIPS PUB 197 in November

44 AES requirements Symmetric block cipher with private key 128-bit data, 128, 192 and 256-bit key selectable Securer and faster than 3DES Active lifespan of 20 to 30 years Publicly published with complete specification and design Code in both C and Java 44

45 AES the last candidates MARS (IBM) complex, fast, high security margin RC6 (USA - Rivest) very simple, very fast, low security margin Rijndael (Belgien) pure, fast, good security margin Serpent (EU) slow, pure, very high security margin Twofish (USA - Schneier) complex, very fast, high security margin Clear differences Few complex rounds versus many simple rounds Improvements of existing ciphers vs. completely new ciphers 45

46 AES Evaluation Criteria initial criteria: security effort to practically cryptanalyse cost computational algorithm & implementation characteristics final criteria general security software & hardware implementation ease implementation attacks flexibility (in en/decrypt, keying, other factors) 46

47 The AES Cipher - 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 47

48 AES / Rijndael processes 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 multipy 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 & efficient 48

49 AES / Rijndael 49

50 AES/Rijndael Encryption Process

51 AES/Rijndael Parameters

52 AES / Rijndael All bytes in the AES algorithm are interpreted as finite field elements Mathematical preliminary Finite field 52

53 Introduction to finite fields will now introduce finite fields of increasing importance in cryptography AES, Elliptic Curve, Public Key concern operations on numbers where what constitutes a number and the type of operations varies considerably start with concepts of groups, rings, fields from abstract algebra 53

54 Groups A set of elements with a binary operation denoted by that associates to each ordered pair (a,b) of elements in G an element (a b ) in G, such that the following axioms are obeyed: (A1) Closure: If a and b belong to G, then a b is also in G (A2) Associative: a (b c) = (a b) c for all a, b, c in G (A3) Identity element: There is an element e in G such that a e = e a = a for all a in G (A4) Inverse element: For each a in G, there is an element a in G such that a a = a a = e (A5) Commutative: a b = b a for all a, b in G

55 Rings A ring R, sometimes denoted by {R, +, * }, is a set of elements with two binary operations, called addition and multiplication, such that for all a, b, c in R the following axioms are obeyed: (A1 A5) R is an abelian group with respect to addition; that is, R satisfies axioms A1 through A5. For the case of an additive group, we denote the identity element as 0 and the inverse of a as a (M1) Closure under multiplication: If a and b belong to R, then ab is also in R (M2) Associativity of multiplication: a (bc ) = (ab)c for all a, b, c in R (M3) Distributive laws: a (b + c ) = ab + ac for all a, b, c in R (a + b )c = ac + bc for all a, b, c in R In essence, a ring is a set in which we can do addition, subtraction [a - b = a + (-b )], and multiplication without leaving the set 55

56 Rings (cont.) A ring is said to be commutative if it satisfies the following additional condition: (M4) Commutativity of multiplication: ab = ba for all a, b in R An integral domain is a commutative ring that obeys the following axioms. (M5) Multiplicative identity: There is an element 1 in R such that a 1 = 1a = a for all a in R (M6) No zero divisors: If a, b in R and ab = 0, then either a = 0 or b = 0 56

57 Fields A field F, sometimes denoted by {F, +,* }, is a set of elements with two binary operations, called addition and multiplication, such that for all a, b, c in F the following axioms are obeyed: (A1 M6) F is an integral domain; that is, F satisfies axioms A1 through A5 and M1 through M6 (M7) Multiplicative inverse: For each a in F, except 0, there is an element a -1 in F such that aa -1 = (a -1 )a = 1 In essence, a field is a set in which we can do addition, subtraction, multiplication, and division without leaving the set. Division is defined with the following rule: a /b = a (b -1 ) 57

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59 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 & b have same remainder eg. 100 = 34 mod 11 b is called the residue of a mod n since with integers can always write: a = qn + b usually have 0 b n-1-12 mod 7-5 mod 7 2 mod 7 9 mod 7 59

60 Modular Arithmetic Modular arithmetic exhibits the following properties: 1. [(a mod n) + (b mod n)] mod n = (a + b) mod n 2. [(a mod n) - (b mod n)] mod n = (a - b) mod n 3. [(a mod n) * (b mod n)] mod n = (a * b) mod n We demonstrate the first property: Define (a mod n) = r a and (b mod n) = r b. Then we can write a = r a + jn for some integer j and b = r b + kn for some integer k Then: (a + b) mod n = (r a + jn + r b + kn) mod n = (r a + r b + (k + j)n) mod n = (r a + r b ) mod n = [(a mod n) + (b mod n)] mod n 60

61 Properties of Modular Arithmetic for Integers in Z n 61

62 Finite Fields finite fields play a key role in cryptography can show 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 fields: GF(p) GF(2 n ) 62

63 Finite Fields GF(p) GF(p) is the set of integers {0,1,, p-1} with arithmetic operations modulo prime p these 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) 63

64 Arithmetic in GF(7) (a) Addition modulo 7 (c) Additive and multiplicative inverses modulo 7 (b) Multiplication modulo 7 64

65 Finite Fields We have shown how to construct GF(p) We will now show how to construct GF(p n ) for n > 1 65

66 Polynomial Arithmetic with Modulo Coefficients when computing value of each coefficient do calculation modulo some value could be modulo any prime but we are most interested in mod 2 i.e. all coefficients are 0 or 1 (i.e. over Z 2 ) eg. let f(x) = x 3 + x 2 and g(x) = x 2 + x + 1 f(x) + g(x) = x 3 + x + 1 f(x) g(x) = x 5 + x 2 66

67 Example of Polynomial Arithmetic Over GF(2) 67

68 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 have no remainder say g(x) divides f(x) if g(x) has no divisors other than itself & 1 say it is irreducible (or prime) polynomial arithmetic modulo an irreducible polynomial forms a field 68

69 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 69

70 Finding inverses (polynomials) Use extended Euclidean algorithm L.C. Calvez, S. Azou and P. Vilbé, Variation on Euclid's algorithm for polynomials, Electron. Lett. 33 (11) (1997), pp A. Goupil and J. Palicot, Variation on variation on Euclid's algorithm, IEEE Trans. Signal Process. Lett. 11 (5) (2004), pp

71 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 algorithm to find 71

72 Polynomial Arithmetic Modulo (x 3 + x + 1) (a) Addition (b) Multiplication 72

73 Computational Example in GF(2 3 ) have (x 2 +1) is & (x 2 +x+1) is so addition is (x 2 +1) + (x 2 +x+1) = x 101 XOR 111 = and multiplication is (x+1).(x 2 +1) = x.(x 2 +1) + 1.(x 2 +1) = x 3 +x+x 2 +1 = x 3 +x 2 +x = (101)<<1 XOR (101)<<0 = 1010 XOR 101 = polynomial modulo reduction (get q(x) & r(x)) is (x 3 +x 2 +x+1 ) mod (x 3 +x+1) = 1.(x 3 +x+1) + (x 2 ) = x mod 1011 = 1111 XOR 1011 =

74 Computational Considerations since coefficients are 0 or 1, can represent any such polynomial as a bit string addition becomes XOR of these bit strings multiplication is shift & XOR cf long-hand multiplication modulo reduction done by repeatedly substituting highest power with remainder of irreducible poly (also shift & XOR) 74

75 AES / Rijndael Mathematical preliminary 75

76 AES / Rijndael Mathematical preliminary 4.1 Addition The addition of two elements in a finite field is achieved by adding the coefficients for the corresponding powers in the polynomials for the two elements. The addition is performed with the XOR operation (denoted by ) - i.e., modulo 2 - so that 1 1 = 0, 1 0 = 1, and 0 0 = 0. Consequently, subtraction of polynomials is identical to addition of polynomials. 76

77 AES / Rijndael Mathematical preliminary 4.1 Addition 77

78 AES / Rijndael Mathematical preliminary 78

79 AES / Rijndael Mathematical preliminary Use blackboard 79

80 AES / Rijndael 80

81 A complete AES Round 81

82 AES / Rijndael substitution A simple substitution of every byte Uses a 16*16 byte table with permutations of all bit values Every byte in state is replaced with byte from row=left 4 bits and column=right 4 bits e.g. byte {95} is replaced with row 9 column 5 => {2A} S-boxes are constructed with a defined transformation of values in the finite field GF(2 8 ) Designed to withstand all known attacks 82

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84 AES / Rijndael substitution 84

85 Construction of S-box The S-box of AES is constructed in the following fashion: 1. Initialize the S-box with the byte values in ascending sequence row by row. 2. Map each byte in the S-box to its multiplicative inverse in the field GF(2 8 ) 3. Perform affine transformation (next slide) 85

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87 Construction of S-box Examples (use blackboard) {83} (next slide) 87

88 Extended Euclid [(x 8 + x 4 + x 3 + x + 1), (x 7 + x + 1)]

89 Shift Rows a circular byte shift in each 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 does shifts to right since state is processed by columns, this step permutes bytes between the columns 89

90 Mix Columns Consider each column of state to be a 4-term polynomial with coefficient in GF(2 8 ) Each column is multplied modulo (x 4 + 1) by the fixed polynomial a(x) = {03}x 3 + {01}x 2 + {01}x + {02} 90

91 Mix Columns Consider each column of state to be a 4-term polynomial with coefficient in GF(2 8 ) Each column is multplied modulo (x 4 + 1) by the fixed polynomial a(x) = {03}x 3 + {01}x 2 + {01}x + {02} Example (use blackboard) 91

92 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 92

93 AES Key Expansion 93

94 AES Key Expansion takes 128/192/256-bit 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 1 st word in 4 has rotate + S-box + XOR round constant on previous, before XOR 4 th back 94

95 AES Key Expansion 95

96 AES Key Expansion Key expansion can be done just-in-time No extra storage required 96

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98 Key expansion example 98

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100 Example 100

101 101

102 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 102

103 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 103

104 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 104

105 The AES/Rijndael block cipher (Recap) 105

106 The AES/Rijndael block cipher (Recap) Continued in the next slide 106

107 The AES/Rijndael block cipher (Recap) 107

108 The AES/Rijndael block cipher (Recap) 108

109 For more information Currently, there exist three (3) approved encryption algorithms: AES, Triple DES, and Skipjack (for Escrowed Encryption Standard). AES pdf Block Cipher Mode 109

110 Questions? 110