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+
+9 Cryptanalysis
+---------------
+
+Cryptanalysis is aimed at breaking ciphers in the sense of
+reconstructing a key from given or chosen pairs of plain- and
+ciphertext. In the first chapter, we saw how various historic ciphers
+can be broken using regularities in the statistical distribution of
+ciphertexts or rather using the fact that known properties of the
+statistical distribution of plaintexts shine through in the
+statistical distribution of plaintexts. Modern cryptographic schemes
+avoid such attacks by making sure that the statistical distribution of
+ciphertext looks close to uniform no matter what the statistical
+distribution of the plaintexts is. The principles of confusion and
+diffusion were introduced to ensure this property at least
+approximately in practice. The notion of pseudo random generator
+furnishes an ideal yet useful theoretical model.
+
+Attacks against cryptographic schemes that properly implement
+confusion and diffusion are much more complicated and in many cases,
+in particular DES and AES not known to exist except in the sense that
+the computational time needed to carry them out is close to that of a
+brute-force attack (trying out all keys) which is always
+possible. Nevertheless, it is important to know and study principles
+of such attacks in order to avoid them when designing a new cipher,
+e.g. a smaller variant of existing ones to be run in
+resource-constrained environments e.g. "Internet of Things".
+
+We have already seen that a cipher that only uses linear or affine
+linear operations is vulnerable to an attack based on solving systems
+of linear equations. In this chapter we take a brief look at three
+more principles of cryptanalysis. First, we show why feedback shift
+registers which are a popular device for generating pseudorandom
+numbers are not secure.
+
+Linear Feedback Shift Registers (LFSR): Such a device is given by a
+sequence of shift registers with some outputs fed back via XOR to the
+input. For instance:
+
+       >[D]-o->[D]-o->[D]---o------------->
+      |  |             |    |
+       -(+)-----------(+)----
+
+When the shift registers (here noted [D]) are initialised with some
+bits then a stream of bits is generated at the output. E.g. starting from 100
+we obtain the following sequence of states:
+
+100,110,111,011,101,010,001,100,...
+and the following output sequence 00111010...
+
+Here, from xyz we get to uxy where u=x+z (mod 2).  Of course, the
+sequence repeats after at most 8 steps, but until then looks rather
+random. With a chain 16 shift registers one can achieve a period
+length of 65535. In order to encrypt a stream of bits ("stream
+cipher") one can initialise the shift registers with the secret key
+and then add (XOR) the generated stream to the data stream. Due to the
+relative simplicity of implementation this has been used for a long
+time in esp. military applications.
+
+However, LFSR are linear (the sequence generated by the sum of two
+keys equals the sum of the sequences generated by the keys
+individually) and this can be exploited in various ways. The most
+interesting way uses the Berlekamp Massey algorithm which from a given
+sequence of bits computes the shortest LFSR which generates it. If one
+knows the beginning m of a plaintext and the corresponding ciphertext
+c, e.g. because the plaintext starts with the date or some formal
+greeting one can then apply the Berlekamp-Massey algorithm to m+c and
+use the computed LFSR to decode the remaining ciphertext.
+
+In order to circumvent this one can use nonlinear functions in the
+feedback loop, e.g. logical or use nonlinear combinations of several
+LFSR. The resulting stream ciphers, e.g. A5/1 and A5/2, E0 are still
+in use in GSM and Bluetooth but are known to have vulnerabilities
+too. Indeed, the design of secure stream ciphers is very difficult and
+for high security applications one resorts to block ciphers run,
+e.g. in CTR mode in order to encrypt bitstreams.
+
+
+Linear cryptanalysis
+- - - - - - - - - - - 
+
+Linear cryptanalysis was developed in the 90s by Matsui and apparently
+was not known to the DES developers. Nevertheless, it can only produce
+a theoretical attack on the latter and on AES.
+
+We illustrate linear cryptanalysis with the toy cryptosystem from
+Michael Heys' tutorial
+https://www.engr.mun.ca/~howard/PAPERS/ldc_tutorial.pdf.
+
+It is an S/P network with a block length of 16 and four proper rounds,
+i.e. four rounds and an additional key mixing step. We regard each
+block as a 4x4 bit matrix. The substitution step applies the first
+S-Box of DES S_1, i.e.
+
+0 1 2 3 4 5 6 7 8 9 A B C D E F
+-------------------------------
+E 4 D 1 2 F B 8 3 A 6 C 5 9 0 7
+
+separately to each of the four rows (regarded as a hexadecimal number). So
+  
+(1 0 1 0)             (A)         (6)          (0 1 1 0)
+(0 1 1 0)   which is  (6) becomes (B) which is (1 0 1 1) 
+(1 1 0 1)             (D)         (9)          (1 0 0 1)
+(0 0 0 1)             (1)         (4)          (0 1 0 0)
+
+The permutation step is simply matrix transposition, thus our example
+becomes
+
+(0 1 1 0)           (6)
+(1 1 0 0)  which is (C)
+(1 0 0 1)           (9)
+(0 1 1 0)           (6)
+
+As usual, each round consists of addition mod2 of the round key,
+substitution (S), and permutation (P), i.e. transposition. After four
+rounds we add one more key.  The master key simply consists of the
+concatenation of the five round keys k = k1 k2 k3 k4 k5.
+
+We can now observe that the S-box S is not completely balanced in the
+following sense. Let WX1 be (1 0 1 1)^T and WY1 = (0 1 0 0)^T. The ^T
+suffix means transposition so W1, W2 are actually columns. We now have
+
+              Pr(WX1^T x + WY1^T S(x) = 0) = 3/4
+
+where the probability is taken over uniformly distributed
+x:{0,1}^4. In other words, for 12 out of the 16 possible inputs x=(x1
+x2 x3 x4)^T and results S(x)=(y1 y2 y3 y4)^T one has x1+x3+x4=y2
+whereas for the remaining four one has x1+x3+x4=/=y2,
+i.e. x1+x3+x4=y2+1.
+
+One can find such an imbalance by exhaustive search. Another such
+imbalance is given by WX2=(0 1 0 0)^T WY2 = (0 1 0 1)^T:
+
+              Pr(WX2^T x + WY2^T S(x) = 0) = 1/4
+
+Chaining these imbalances through the whole cipher (using the first
+one on round 1 and the second one on the other rounds) one obtains a
+similar linear relationship between the plaintext, the ciphertext just
+after round four, and the four round keys that have been used until
+then: Letting P be the plaintext (now written as dim16 vector) and U
+the ciphertext prior to applying the last S-boxes, permuting, and
+mixing with the fifth and last round key we find dim16 vectors A, B
+D1, D2, D3, D4 such that
+
+    Pr(A^T P + B^T U + D1^T K1 + D2^T K2 + D3^T K3 + D4^T K4 = 0) = 15/32
+
+with keys K1 ... K4 fixed and probability taken over the plaintexts P. 
+
+
+Since D1^T K1 + D2^T K2 + D3^T K3 + D4^T K4 is either 0 or 1 we have that
+
+     Prob(A^T P + B^T U = 0) = 15/32 or Prob(A^T P + B^T U = 0) = 17/32
+
+irrespective of K1,..,K4. This allows us to work out a likely value
+for K5 or rather some bits of it.
+
+Namely, given a list of plaintexts P_1..P_N and corresponding
+ciphertexts C_1..C_n we can try out all 2^16 possible choices for K_5,
+for each one compute the corresponding U_i := S^{-1}(P^{-1}(C_i+K_5)),
+and compute the ratio
+
+   p(K_5) = {i | A^T P_i + B^T U_i = 0} / N 
+
+We choose a value K_5 for which this ratio is close to 15/32 or 17/32
+and discard all those where it is close to 1/2. Now, p(K_5) does not
+actually depend on all bits of K_5 but only on those where the
+corresponding bit in B^T is set for the other ones only influence
+components of U_i which are not used in p(K_5). Nevertheless, this
+allows us to considerably narrow down the possible options for
+K_5. For each of those, we can then continue in the same fashion and
+thus peel off the fourth round key etc.
+
+
+Differential cryptanalysis
+- - - - - - - - - - - - - -
+
+Differential cryptanalysis was developed by Biham and Shamir as an
+attack against DES but apparently already known to the DES
+developers. It has been used to successfully break other ciphers. It
+uses the fact that the effect of adding a round key cancels out when
+we consider differences between two plaintexts. However, this
+requires the preparation of chosen plaintexts. In more detail, the
+*differential* between two plaintexts X1 and X2 is simply X1+X2,
+i.e. the bitvector which has a 1 where X1 and X2 differ and a 0 where
+they are equal.
+
+Coming back to our 4bit example S-box for any given differential there
+then are 2^4=16 different ways of realising it. E.g. the differential
+1010 arises for the plaintext pair 0011,1001 and also for
+1111,0101. Indeed, for any plaintext X1 there is a unique X2 such that
+X1, X2 have a given differential.
+
+Given two differentials X* and Y* we are interested in the following
+question. For how many of the (16) plaintext pairs X1,X2 with
+differential X* is it the case that the results of applying the S-box
+Y1=S(X1), Y2=S(X2) have differential Y*. Tabulating these numbers by
+exhaustive search yields the following table:
+
+           Output differential
+     0 1 2 3 4 5 6 7 8 9 A B C D E F 
+I 0 16 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 
+n 1  0 0 0 2 0 0 0 2 0 2 4 0 4 2 0 0 
+p 2  0 0 0 2 0 6 2 2 0 2 0 0 0 0 2 0
+u 3  0 0 2 0 2 0 0 0 0 4 2 0 2 0 0 4
+t 4  0 0 0 2 0 0 6 0 0 2 0 4 2 0 0 0 
+  5  0 4 0 0 0 2 2 0 0 0 4 0 2 0 0 2
+d 6  0 0 0 4 0 4 0 0 0 0 0 0 2 2 2 2 
+i 7  0 0 2 2 2 0 2 0 0 2 2 0 0 0 0 4
+f 8  0 0 0 0 0 0 2 2 0 0 0 4 0 4 2 2 
+f 9  0 2 0 0 2 0 0 4 2 0 2 2 2 0 0 0 
+' A  0 2 2 0 0 0 0 0 6 0 0 2 0 6 0 0 
+i B  0 0 8 0 0 2 0 2 0 0 0 0 0 2 0 2 
+a C  0 2 0 0 2 2 2 0 0 0 0 2 0 6 0 0 
+l D  0 4 0 0 0 0 0 4 2 0 2 0 2 0 2 0 
+  E  0 0 2 4 2 0 0 0 6 0 0 0 0 0 2 0   
+  F  0 2 0 0 6 0 0 0 0 4 0 2 0 0 2 0
+
+
+The high entries 8 and 6 are of particular interest because they
+constitute a deviation of an intuitive pseudorandom permutation.
+Let us now see how we can exploit these in order to attack our cipher.
+
+We feed in a pair of 16bit plaintexts X1 X2 which differ in the second
+block by B=1011 and are equal everywhere else.  With probability 8/16
+this will result in differential 2=0010 at the output of the second
+S-box. The output of the other S-boxes will be equal for X1 and X2.
+This differential 2=0010 will be propagated through the P network and
+results in a differential of 4=0100 at the entry of the third S-box of
+the second round:
+
+              Differential before P   Differential after P
+                    0 0 0 0                     0 0 0 0 
+                    0 0 0 0   ------P---->      0 0 1 0 
+                    0 1 0 0                     0 0 0 0 
+                    0 0 0 0                     0 0 0 0
+
+
+There, with probability 6/16 it will produce differential 6=0110 at
+the output. This results in differential 2=0010 at the input of the
+second and third S-boxes of the third round. With probability 36/256
+this will lead to differential 5 at both outputs which propagates to
+differential 6=0110 at the entries to the second and fourth S-boxes of
+the final round.
+
+                     0 0 0 0                    0 0 0 0 
+                     0 0 1 0  -------P----->    0 0 0 0 
+                     0 0 1 0                    0 1 1 0 
+                     0 0 0 0                    0 0 0 0
+
+Summing up, when we encipher plaintexts with differential 0B00 we will
+obtain with probability 1/2*3/8*9/64=27/1024 a differential 0606 at
+the entry to the last round. We now have to set the relevant
+components (second and fourth block) of the final key in such a way
+that this ratio (after inverting the last layer of S-boxes) is
+actually reproduced. This allows us (with high probability) to guess
+hone half of the bits of the last round key. Continuing in this way
+with other differentials combined with exhaustive search allows us to
+undo the entire last round and then to proceed from there.
+