The Advanced Encryption Standard: Four Years On

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1 The Advanced Encryption Standard: Four Years On Matt Robshaw Reader in Information Security Information Security Group Royal Holloway University of London September 21, 2004 The State of the AES 1

2 The Advanced Encryption Standard In October 2000 Rijndael was chosen as the Advanced Encryption Standard (AES) Published as FIPS 197 Available via A block cipher is a versatile primitive to have Symmetric encryption algorithm Can be used to construct a stream cipher Can be used to construct a hash function Can be used to construct a MAC Replaces DES Provides vastly increased security But without the software costs of 3DES Likely to be used widely around the world However full deployment will be slow The view from NIST NIST expects to get the world to AES by 2020 AES and 3DES will co-exist as FIPS-approved algorithms to 2030 [SP ] September 21, 2004 The State of the AES 2

3 The AES Process The search for the AES began in 1997 Full archives at There were two rounds of assessment 15 ciphers in Round 1 5 ciphers in Round 2 MARS (IBM) RC6 (RSA Laboratories + Rivest) Rijndael (Daemen + Rijmen) Serpent (Anderson, Biham + Knudsen) Twofish (Counterpane) Very different design philosophies Different architectural features Different approaches to security Different performance profiles Rijndael was an excellent best-fit candidate Rijndael appears to be a consistently good performer in both hardware and software across a wide range of computing environments NIST Final Report September 21, 2004 The State of the AES 3

4 The AES The AES is a very elegant cipher Novel construction Good performance The AES is a carefully constructed cipher Good levels of security against known attacks Differential cryptanalysis Linear cryptanalysis Rijndael is more versatile than the AES Rijndael allowed for different block sizes This might have been helpful for hash function construction The structure of the AES has led to some novel analytical approaches Might a well-structured cipher offer new advantages to an attacker? What is the current state of AES cryptanalysis? September 21, 2004 The State of the AES 4

5 AES Design Basics Shannon introduced the ideas of confusion and diffusion These are not rigorous notions but guides to some form of ideal behaviour During the design of a block cipher we typically choose cipher components to deliver these properties Confusion The relationship between the plaintext, ciphertext, and key should be complex Typically provided by substitution operations Diffusion All of the ciphertext should depend on all of the plaintext and all of the key Typically provided by permutation operations September 21, 2004 The State of the AES 5

6 SP-Networks Single substitution and permutation operations on their own are unlikely to yield a strong cipher This leads us to SP-networks September 21, 2004 The State of the AES 6

7 AES Description The AES has one block and three key lengths For the AES b=128 and k=128, 192, and 256 Referred to as AES-128, AES-192, AES-256 Here we concentrate on b=k=128 Encryption can be described as a sequence of operations on an array of bytes Some operations are described over GF(2 8 ) The Rijndael polynomial is X 8 +X 4 +X 3 +X+1 Here we are less interested in the key schedule For k=128 The 128-bit user-supplied key is expanded into a sequence of 11 round keys each of 128 bits The key schedule (like the rest of the cipher) is very simple and lightweight September 21, 2004 The State of the AES 7

8 AES Encryption There are four components to an AES round SubBytes ShiftRows MixColumns AddRoundKey The AES is best described using an array of bytes Pack the input m 0 m 15 into a (4 4) square array September 21, 2004 The State of the AES 8

9 SubBytes There are 16 parallel S-box look-ups The same S-box is used in each case September 21, 2004 The State of the AES 9

10 ShiftRows Each row is rotated a different number of byte positions Row i (0 i 3) is moved by i byte positions to the left September 21, 2004 The State of the AES 10

11 MixColumns View each column as a GF(2 8 ) column vector Create a replacement column by computing M c September 21, 2004 The State of the AES 11

12 AES AddKey We add the round key for the given round September 21, 2004 The State of the AES 12

13 AES-128 (k=b=128) There are nine full rounds There is a key-addition prior to the first round There is a tenth round without MixColumns AES-192 and AES-256 have 12 and 14 rounds respectively AddRoundKey SubBytes ShiftRows MixColumns Repeat 9 times AddRoundKey SubBytes ShiftRows AddRoundKey September 21, 2004 The State of the AES 13

14 The AES S/P Network September 21, 2004 The State of the AES 14

15 Rijndael In Context While Rijndael may look quite different to other cipher designs it has eminent predecessors The success of Rijndael has also inspired other designers September 21, 2004 The State of the AES 15

16 AES Overview AES is a very simple S/P network Gives a good performance profile Some sample figures include Software; e.g. 2.1 GHz Pentium 4 [Wei Dai 04] AES-128: 62 Mbyte/sec AES-192: 56 Mbyte/sec AES-256: 49 Mbyte/sec Hardware Space/performance/technology/implementation trade-offs High-end performance» 1.3 Gbyte/sec (FPGA)» 3.1 Gbyte/sec (ASIC) Very careful construction September 21, 2004 The State of the AES 16

17 Some Details September 21, 2004 The State of the AES 17

18 The AES S-Box The S-box is crucial to security There are three components to its design 1. Invert the input x in GF(2 8 ) [ with 0 fi 0 ] 2. Multiply x (-1) by an (8 8) GF(2) matrix L 3. XOR the constant c = September 21, 2004 The State of the AES 18

19 S-Box Design Rationale The S-box has been carefully constructed 1. Invert the input in GF(2 8 ) [with 0 fi 0] This operation has been shown to be very good against differential and linear cryptanalysis Maximum difference propagation probability 2-6 and maximum linear correlation Multiply by an (8 8) GF(2) matrix L The operation x fi x (-1) is algebraically simple Multiplying by L should hinder attacks that exploit the GF(2 8 )-based algebraic structure 3. XOR the constant We remove the fixed point 0 fi 0 by adding a non-zero constant The mix of incompatible operations over GF(2 8 ) and GF(2) should help resist cryptanalysis September 21, 2004 The State of the AES 19

20 MixColumns The MixColumns operation provides mixing across bytes Introduce the concept of a branch number β for matrix M Denote the number of non-zero coefficients in column vector a by w b (a), then for a b β = min{ w b (a b) + w b (Ma Mb) } The MixColumns matrix M has β = 5 A non-zero difference in a single byte is spread to a nonzero difference in four bytes September 21, 2004 The State of the AES 20

21 Approaches to AES Analysis Statistical attacks Structural attacks Alternative representations Algebraic attacks September 21, 2004 The State of the AES 21

22 Statistical Attacks The AES is very resistant to statistical attacks The attacker attempts to construct statistical patterns via many cipher interactions Differential Cryptanalysis (DC) The statistical pattern depends on bitwise difference Linear Cryptanalysis (LC) The statistical pattern depends on the correlation between bits To illustrate, DC is thwarted by Careful S-box construction The probability p of a given bitwise non-zero difference propagation across an S-box is < 2-6 In an attack, an S-box supporting such a propagation is said to be an active S-box Carefully designed diffusion layer The number of active S-boxes n increases quickly The total differential probability behaves as p n Attack requirements are proportional to 1/p n September 21, 2004 The State of the AES 22

23 The AES S/P Network D September 21, 2004 The State of the AES 23

24 Statistical Attacks For differential and linear cryptanalysis Attacks over four rounds of the AES require at least 25 active S-boxes More careful analysis takes account of additional complicated phenomena Differentials, linear hulls, etc. Exploiting differential and linear techniques requires far more data than there is available A different cryptanalytic approach is required! September 21, 2004 The State of the AES 24

25 Approaches to AES Analysis Statistical attacks Structural attacks Alternative representations Algebraic attacks September 21, 2004 The State of the AES 25

26 Structural Attacks The AES is heavily optimised against statistical attacks Careful choice of S-box Carefully designed structure to quickly magnify the number of active S-boxes However this clean structure can be used to mount some innovative analysis Analysis is specific to AES-like ciphers Such attacks tend to have a similar form Identify a property over a few rounds that holds with a good probability Use special techniques to extend the attack a few rounds at the beginning and the end Best example is the so-called Square Attack But there are several others such as Impossible Differentials, Bottleneck Attacks, September 21, 2004 The State of the AES 26

27 Square Attack Suppose we have a set of 256 plaintexts The first byte in a text-set takes all possible values All other byte positions are fixed across the text-set Consider three rounds of encryption Round 1 Round 2 Round 3 September 21, 2004 The State of the AES 27

28 A Three Round Property September 21, 2004 The State of the AES 28

29 Structural Attacks Structural attacks are very effective over a moderate number of rounds However they do not extend well Since the number of rounds increases for different keysizes in some sense we re losing ground! Rounds AES % 70% 80% AES % 58% 75% AES % 50% 57% September 21, 2004 The State of the AES 29

30 Approaches to AES Analysis Statistical attacks Structural attacks Alternative representations Algebraic attacks September 21, 2004 The State of the AES 30

31 Alternative Representations The rich structure of the AES allows us to re-write and re-order components of the cipher There are a variety of reasons to consider alternative representations Different implementations Insights into algorithm design New approaches to cryptanalysis There have been a variety of proposals Continued fraction expansion Dual ciphers Algebraic structure September 21, 2004 The State of the AES 31

32 Algebraic Structure September 21, 2004 The State of the AES 32

33 One Round of the AES One round has the following form M September 21, 2004 The State of the AES 33

34 One Round of the AES We can move parts of the S-box into an augmented diffusion layer M* September 21, 2004 The State of the AES 34

35 Simplifying the AES The designers view of the AES: In one S-box mix operations in GF(2 8 ) and GF(2) Use a simple diffusion operation over GF(2) The unified view of the AES: Use an algebraically simple S-box in GF(2 8 ) Use a modified diffusion operation over GF(2) By grouping together similar operations The strategy of mixing operations in GF(2) and GF(2 8 ) within the S-box is unclear The issue of eliminating fixed points in the S-box is not relevant How complicated does this re-writing make the modified diffusion operation? September 21, 2004 The State of the AES 35

36 Simplifying the AES The modified diffusion layer can be represented as multiplication by a binary matrix M* Minimum polynomial for M* is (X + 1) 15 There are large fixed subspaces The modified GF(2) diffusion layer is very simple and preserves considerable structure However, have we really gained much? The S-boxes are defined over GF(2 8 ) and diffusion is defined over GF(2) This creates difficulties for the cryptanalyst September 21, 2004 The State of the AES 36

37 AES fi BES The unified AES consists of A layer of S-boxes over GF(2 8 ) A modified diffusion layer given by a GF(2) matrix M* Analysis techniques for the S-boxes don t work across diffusion and vice versa However it is possible to describe the actions of the the AES entirely with operations in GF(2 8 ) Embed the AES in a larger cipher, the BES AES: A A operates with a mix of GF(2) and GF(2 8 ) BES: B B operates exclusively in GF(2 8 ) B A = f(a) B September 21, 2004 The State of the AES 37

38 BES Each byte in the AES is represented by a set of conjugates in BES AES is a 16-byte block cipher BES is a 128-byte block cipher All AES operations can be replicated by simple operations on conjugates Even the GF(2) linear map L AES encryption can be described exclusively in terms of GF(2 8 ) operations The (slight) additional complexity allows us to avoid the tension between GF(2) and GF(2 8 ) September 21, 2004 The State of the AES 38

39 Comparing the AES and the BES S-box The AES S-box consists of 1 byte of input, inversion in GF(2 8 ), and mixing over GF(2) The BES S-box consists of 8 bytes of input, componentwise inversion in GF(2 8 ), and mixing over GF(2 8 ) September 21, 2004 The State of the AES 39

40 Approaches to AES Analysis Statistical attacks Structural attacks Alternative representations Algebraic attacks September 21, 2004 The State of the AES 40

41 Algebraic Attacks Algebraic analysis offers new approaches to symmetric cryptanalysis Algebraic techniques previously the preserve of public key cryptography Courtois and Pieprzyk proposed algebraic cryptanalysis against block ciphers Also valuable techniques against certain stream cipher designs September 21, 2004 The State of the AES 41

42 Algebraic Attacks Algebraic attacks require us to: Describe encryption as a system of equations Using key, plaintext, ciphertext, and internal variables Solve the system of equations (somehow!) Recover the key For most block ciphers The system of equations would be huge The system of equations would be complex For the AES this is not the case An algebraically simple S-box [x fi x -1 ] A simple and very structured diffusion layer September 21, 2004 The State of the AES 42

43 Algebraic Analysis for AES and BES There are two approaches to writing systems of equations across the S-box AES style: express inversion over GF(2) BES style: express the map L over GF(2 8 ) Courtois and Pieprzyk introduce a measure of S- box complexity, G s inputs, r equations, and t variables s r t G AES (i) AES (ii) BES September 21, 2004 The State of the AES 43

44 The BES System of Equations Consider the BES equivalent of AES-128 One single encryption provides 5,248 equations in 7,808 terms 3,840 sparse quadratic equations 1,408 linear equations 2,560 state variables 1,408 key variables The key schedule provides 2,560 equations in 3,308 terms 960 are sparse quadratic equations 1,600 linear equations 1,408 key variables and 640 auxiliary variables We can assume there is no zero-inversion (255/256) for encryption (255/256) for the key schedule September 21, 2004 The State of the AES 44

45 Solving Equations (I) Linearisation techniques Courtois and Pieprzyk proposed Extended Sparse Linearization (XSL) An extension of the XL algorithm XL is reasonably well understood Linearisation step Gaussian elimination step XSL adds an AES-specific enhancement to linearisation There are doubts over the full validity of XSL Experiments on some equation systems work But experiments on AES-like systems show that XSL might not be so successful All current claims for attacking the AES depend on the correctness of the XSL Beware of XL claims for XSL! September 21, 2004 The State of the AES 45

46 Solving Equations (II) Gröbner Basis algorithms Buchberger, F 4, (F 5 ) Small-scale experiments are successful The key can be recovered directly! However the algorithms quickly become inefficient The relation between these different techniques is becoming clearer F 4 should be better than XL Current implementations do not exploit the source of the equations Yet, the system of equations for the AES is very structured Ongoing research is considering how best to work with a very specific set of equations September 21, 2004 The State of the AES 46

47 Summary The AES is a very successful design Good performance Good security Traditional methods of attack are not successful However the AES is very structured AES-specific analysis techniques have been proposed These might provide new opportunities for the attacker in the future The current best approach (for the cryptanalyst) appears to be to use algebraic methods However they are exceptionally difficult to work with With what we know today there is no substantive reason to question the security of the AES September 21, 2004 The State of the AES 47

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