A Tutorial on Linear and Differential Cryptanalysis by Howard M. Heys

A Tutorial on
Linear and Differential Cryptanalysis

by
Howard M. Heys
Electrical and Computer Engineering
Faculty of Engineering and Applied Science
Memorial University of Newfoundland
St. John’s, NF, Canada A1B 3X5
email: howard@engr.mun.ca

Abstract: In this paper, we present a detailed tutorial on linear cryptanalysis and
differential cryptanalysis, the two most significant attacks applicable to symmetric-key
block ciphers. The intent of the paper is to present a lucid explanation of the attacks,
detailing the practical application of the attacks to a cipher in a simple, conceptually
revealing manner for the novice cryptanalyst. The tutorial is based on the analysis of a
simple, yet realistically structured, basic Substitution-Permutation Network cipher.
Understanding the attacks as they apply to this structure is useful, as the Rijndael cipher,
recently selected for the Advanced Encryption Standard (AES), has been derived from
the basic SPN architecture. As well, experimental data from the attacks is presented as
confirmation of the applicability of the concepts as outlined.

1. Introduction
In this paper, we present a tutorial on two powerful cryptanalysis techniques applied to
symmetric-key block ciphers: linear cryptanalysis [1] and differential cryptanalysis [2].
Linear cryptanalysis was introduced by Matsui at EUROCRYPT ’93 as a theoretical
attack on the Data Encryption Standard (DES) [3] and later successfully used in the
practical cryptanalysis of DES [4]; differential cryptanalysis was first presented by
Biham and Shamir at CRYPTO ’90 to attack DES and eventually the details of the attack
were packaged as a book [5]. Although the early target of both attacks was DES, the wide
applicability of both attacks to numerous other block ciphers has solidified the preeminence of both cryptanalysis techniques in the consideration of the security of all block
ciphers. For example, many of the candidates submitted for the recent Advanced
Encryption Standard process undertaken by the National Institute of Standards and
Technology [6] were designed using techniques specifically targeted at thwarting linear
and differential cryptanalysis. This is evident, for example, in the Rijndael cipher [7], the
encryption algorithm selected to be the new standard. The concepts discussed in this
paper could be used to form an initial understanding required to comprehend the design
principles and security analysis of the Rijndael cipher, as well as many other ciphers
proposed in recent years.
The paper is structured as a tutorial and, as such, is intended to not be rigorously
mathematical. It introduces the basic concepts of linear and differential cryptanalysis but
is by no means a definitive source for understanding all the many refinements and
improvements of the attacks over the years. The basic purpose of the paper is to use a
simple (yet somewhat realistic) cipher structure to study the most basic concepts of the
two attacks. Other more formal discussions exist on the topic. For example, overviews of
the attacks as applied to Substitution-Permutation Networks (the cipher structured to be
considered in this paper) are presented in [8] and [9]. For a general introduction to block
ciphers and their analysis, see [10].
The need for a tutorial on the attacks arises from the very difficult nature of both attacks
and the lack of simplified, yet detailed, reference material describing the attacks.
Conventional cryptographic references and texts [11][12][13][14] generally present
material on block ciphers in a very descriptive manner, with little detail illustrating the
concepts of the attacks. Consequently, most published material detailing the attacks has a
research focus and gives little intuition and explanation for the non-expert. When the
basic concepts of the attack are described in the literature (as in Matsui’s and Biham and
Shamir’s original papers), they are typically presented in reference to DES which is, in
nature, somewhat convoluted in a manner which interferes with the understanding the
cryptanalytic concepts.

2

2. A Basic Substitution-Permutation Network Cipher
The cipher that we shall use to present the concepts is a basic Substitution-Permutation
Network (SPN). We will focus our discussion on a cipher, illustrated in Figure 1, that
takes a 16-bit input block and processes the block by repeating the basic operations of a
round four times. Each round consists of (1) substitution, (2) a transposition of the bits
(i.e., permutation of the bit positions), and (3) key mixing. This basic structure was
presented by Feistel back in 1973 [15] and these basic operations are similar to what is
found in DES and many other modern ciphers, including Rijndael. So although, we are
considering a somewhat simplified structure, an analysis of the attack of such a cipher
presents valuable insight into the security of larger, more practical constructions.
2.1 Substitution
In our cipher, we break the 16-bit data block into four 4-bit sub-blocks. Each sub-block
forms an input to a 4×4 S-box (a substitution with 4 input and 4 output bits), which can
be easily implemented with a table lookup of sixteen 4-bit values, indexed by the integer
represented by the 4 input bits. The most fundamental property of an S-box is that it is a
nonlinear mapping, i.e., the output bits cannot be represented as a linear operation on the
input bits.
For our cipher, we shall use the same nonlinear mapping for all S-boxes. (In DES all the
S-boxes in a round are different, while all rounds use the same set of S-boxes.) The
attacks of linear and differential cryptanalysis apply equally to whether there is one
mapping or all S-boxes are different mappings. The mapping chosen for our cipher, given
in Table 1, is chosen from the S-boxes of DES. (It is the first row of the first S-box.) In
the table, the most significant bit of the hexadecimal notation represents the leftmost bit
of the S-box in Figure 1.
input 0
output E

1
4

2
D

3
1

4
2

5
F

6
B

7
8

8
3

9
A

A
6

B
C

C
5

D
9

E
0

F
7

Table 1. S-box Representation (in hexadecimal)
2.2 Permutation
The permutation portion of a round is simply the tranposition of the bits or the
permutation of the bit positions. The permutation of Figure 1 is given in Table 2 (where
the numbers represent bit positions in the block, with 1 being the leftmost bit and 16
being the rightmost bit) and can be simply described as: the output i of S-box j is
connected to input j of S-box i. Note that there would be no purpose for a permutation in
the last round and, hence, our cipher does not have one.
input 1
output 1

2
5

3
9

4
13

5
2

6
6

7
10

8
14

Table 2. Permutation

3

9
3

10
7

11
11

12
15

13
4

14
8

15
12

16
16

P1

...

plaintext

...

P16

subkey K1 mixing

S11

S12

S13

S14
round 1

subkey K2 mixing

S21

S22

S23

S24
round 2

subkey K3 mixing

S31

S32

S33

S34
round 3

subkey K4 mixing

round 4
S41

S42

S43

S44

subkey K5 mixing

C1

...

ciphertext

...

C16

Figure 1. Basic Substitution-Permutation Network (SPN) Cipher

4

2.3 Key Mixing
To achieve the key mixing, we use a simple bit-wise exclusive-OR between the key bits
associated with a round (referred to as a subkey) and the data block input to a round. As
well, a subkey is applied following the last round, ensuring that the last layer of
substitution cannot be easily ignored by a cryptanalyst that simply works backward
through the last round’s substitution. Normally, in a cipher, the subkey for a round is
derived from the cipher’s master key through a process known as the key schedule. In our
cipher, we shall assume that all bits of the subkeys are independently generated and
unrelated.
2.4 Decryption
In order to decrypt, data is essentially passed backwards through the network. Hence,
decryption is also of the form of an SPN as illustrated in Figure 1. However, the
mappings used in the S-boxes of the decryption network are the inverse of the mappings
in the encryption network (i.e., input becomes output, output becomes input). This
implies that in order for an SPN to allow for decryption, all S-boxes must be bijective,
that is, a one-to-one mapping with the same number input and output bits. As well, in
order for the network to properly decrypt, the subkeys are applied in reverse order and the
bits of the subkeys must be moved around according to the permutation, if the SPN is to
look similar to Figure 1. Note also that the lack of the permutation after the last round
ensures that the decryption network can be the same structure as the encryption network.
(If there was a permutation after the last substitution layer in the encryption, the
decryption would require a permutation before the first layer of substitution.)

5

3. Linear Cryptanalysis
In this section, we outline the approach to attacking a cipher using linear cryptanalysis
based on the example cipher of our basic SPN.
3.1 Overview of Basic Attack
Linear cryptanalysis tries to take advantage of high probability occurrences of linear
expressions involving plaintext bits, "ciphertext" bits (actually we shall use bits from the
2nd last round output), and subkey bits. It is a known plaintext attack: that is, it is
premised on the attacker having information on a set of plaintexts and the corresponding
ciphertexts. However, the attacker has no way to select which plaintexts (and
corresponding ciphertexts) are available. In many applications and scenarios it is
reasonable to assume that the attacker has knowledge of a random set of plaintexts and
the corresponding ciphertexts.
The basic idea is to approximate the operation of a portion of the cipher with an
expression that is linear where the linearity refers to a mod-2 bit-wise operation (i.e.,
exclusive-OR denoted by "⊕"). Such an expression is of the form:
X i1 ⊕ X i2 ⊕ ... ⊕ X iu ⊕ Y j1 ⊕ Y j2 ⊕ ... ⊕ Y jv = 0

(1)

where Xi represents the i-th bit of the input X = [X1, X2, ...] and Yj represents the j-th bit of
the output Y = [Y1, Y2, ...]. This equation is representing the exclusive-OR "sum" of u
input bits and v output bits.
The approach in linear cryptanalysis is to determine expressions of the form above which
have a high or low probability of occurrence. (No obvious linearity such as above should
hold for all input and output values or the cipher would be trivially weak.) If a cipher
displays a tendency for equation (1) to hold with high probability or not hold with high
probability, this is evidence of the cipher’s poor randomization abilities. Consider that if
we randomly selected values for u + v bits and placed them into the equation above, the
probability that the expression would hold would be exactly 1/2. It is the deviation or bias
from the probability of 1/2 for an expression to hold that is exploited in linear
cryptanalysis: the further away that a linear expression is from holding with a probability
of 1/2, the better the cryptanalyst is able to apply linear cryptanalysis. In the remainder of
the paper, we refer to the amount by which the probability of a linear expression holding
deviates from 1/2 as the linear probability bias. Hence, if the expression above holds
with probability pL for randomly chosen plaintexts and the corresponding ciphertexts,
then the probability bias is pL – 1/2. The higher the magnitude of the probability bias, |pL
– 1/2|, the better the applicability of linear cryptanalysis with fewer known plaintexts
required in the attack.
There are several ways to mount the attack of linear cryptanalysis. In this paper, we shall
focus on what Matsui calls Algorithm 2 [1]. We investigate the construction of a linear
approximation involving plaintext bits as represented by X in (1) and the input to the last

6

round of the cipher (or equivalently the output of the 2nd last round of the cipher) as
represented by Y in (1). The plaintext bits are random and consequently so are the input
bits to the last round.
Equation (1) could be equivalently reformulated to have the right side being the sum of a
number of subkey bits. However, in (1) as written with the right side of "0", the equation
implicitly has subkey bits involved: these bits are fixed but unknown (as they are
determined by the key under attack) and implicity absorbed into the "0" on the right side
of equation (1) and the probability pL that the linear expression holds. If the sum of the
involved subkey bits is "0", the bias of (1) will have the same sign (+ or −) as the bias of
the expression involving the subkey sum and, if the sum of the involved subkey bits is
"1", the bias of (1) will have the opposite sign.
Note that pL = 1 implies that linear expression (1) is a perfect representation of the cipher
behaviour and the cipher has a catastrophic weakness. If pL = 0, then (1) represents an
affine relationship in the cipher, also an indication of a catastrophic weakness. For mod-2
addition systems, an affine function is simply the complement of a linear function. Both
linear and affine approximations, indicated by pL > 1/2 and pL < 1/2, respectively, are
equally susceptible to linear cryptanalysis and we shall generally use the term linear to
refer to both linear and affine relationships.
The natural question to ask is: How do we construct expressions which are highly linear
and, hence, can be exploited? This is done by considering the properties of the cipher’s
only nonlinear component: the S-box. When the nonlinearity properties of the S-box are
enumerated, it is possible to develop linear approximations between sets of input and
output bits in the S-box. Consequently, it is possible to concatenate linear approximations
of the S-boxes together so that intermediate bits (i.e., data bits from within the cipher) can
be cancelled out and we are left with a linear expression which has a large bias and
involves only plaintext and the last round input bits.
3.2 Piling-Up Principle
Before we consider constructing a linear expression for the example cipher of this paper,
we need some basic tools. Consider two random binary variables, X1 and X2. We begin by
noting the simple relationships: X1⊕X2 = 0 is a linear expression and is equivalent to X1 =
X2; X1⊕X2 = 1 is an affine expression and is equivalent to X 1 ≠ X 2 .
Now, assume that the probability distributions are given by

 p
Pr ( X 1 = i ) =  1
1 − p1

,i = 0
,i = 1

 p
Pr ( X 2 = i ) =  2
1 − p 2

,i = 0
, i = 1.

and

7

If the two random variables are independent, then
p1 p 2
, i = 0, j = 0

 p (1 − p )
, i = 0, j = 1

1
2
Pr ( X 1 = i, X 2 = j ) = 
, i = 1, j = 0
 (1 − p1 ) p 2
(1 − p1 )(1 − p 2 ) , i = 1, j = 1

and it can be shown that
Pr(X1 ⊕ X2 = 0)

= Pr(X1 = X2)
= Pr(X1 = 0, X2 = 0) + Pr(X1 = 1, X2 = 1)
= p1p2 + (1−p1)(1−p2).

Another perspective is to let p1 = 1/2+ε1 and p2 = 1/2+ε2, where ε1 and ε2 are the
probability biases and −1/2 ≤ ε1,ε2 ≤ +1/2. Hence, it follows that
Pr(X1 ⊕ X2 = 0) = 1/2 + 2ε1ε2
and the bias ε1,2 of X1 ⊕ X2 = 0 is
ε1,2 = 2ε1ε2.
This can be extended to more than two random binary variables, X1 to Xn, with
probabilities p1 = 1/2+ε1 to pn = 1/2+εn. The probability that X1 ⊕ ... ⊕ Xn = 0 holds can
be determined by the Piling-Up Lemma which assumes that all n random binary variables
are independent.
Piling-Up Lemma (Matsui [1])
For n independent, random binary variables, X1, X2, ...Xn,
n

Pr(X1 ⊕ ... ⊕ Xn = 0) = 1/2 + 2 n −1 ∏ ε i
i =1

or, equivalently,
n

ε1,2,..,n = 2 n −1 ∏ ε i
i =1

where ε1,2,..,n represents the bias of X1 ⊕ ... ⊕ Xn = 0.
Note that if pi = 0 or 1 for all i, then Pr(X1 ⊕ ... ⊕ Xn = 0) = 0 or 1. If only one pi = 1/2,
then Pr(X1 ⊕ ... ⊕ Xn = 0) = 1/2.
In developing the linear approximation of a cipher, the Xi values will actually represent
linear approximations of the S-boxes. For example, consider four independent random

8

binary variables, X1, X2, X3 and X4. Let Pr(X1 ⊕ X2 = 0) = 1/2 + ε1,2 and Pr(X2 ⊕ X3 = 0) =
1/2 + ε2,3 and consider the sum X1 ⊕ X3 to be derived by adding X1 ⊕ X2 and X2 ⊕ X3
together. Hence,
Pr(X1 ⊕ X3 = 0) = Pr([X1 ⊕ X2] ⊕ [X2 ⊕ X3] = 0).
So we are combining linear expressions to form a new linear expression. Since we may
consider random variables X1 ⊕ X2 and X2 ⊕ X3 to be independent, we can use the PilingUp Lemma, to determine
Pr(X1 ⊕ X3 = 0) = 1/2 + 2ε1,2ε2,3
and, consequently,
ε1,3 = 2ε1,2ε2,3.
As we shall see, the expressions X1 ⊕ X2 = 0 and X2 ⊕ X3 = 0 are analogous to linear
approximations of S-boxes and X1 ⊕ X3 = 0 is analogous to a cipher approximation where
the intermediate bit X2 is eliminated. Of course, the real analysis will be more complex
involving many S-box approximations.
3.3 Analyzing the Cipher Components
Before considering the attack in any more detail on the overall cipher, we first require
knowledge of the linear vulnerabilities of an S-box. Consider the S-box representation of
Figure 2 with input X = [X1 X2 X3 X4] and a corresponding output Y = [Y1 Y2 Y3 Y4]. All
linear approximations can be examined to determine their usefulness by computing the
probability bias for each. Hence, we are examining all expressions of the form of
equation (1) where X and Y are the S-box input and outputs, respectively.
X1

X2

X3

X4

4×4
S-box

Figure 2. S-box Mapping

Y1 Y2 Y3 Y4
For example, for the S-box used in our cipher, consider the linear expression
X 2 ⊕ X 3 ⊕ Y1 ⊕ Y3 ⊕ Y4 = 0 or equivalently
X 2 ⊕ X 3 = Y1 ⊕ Y3 ⊕ Y4 .

9

Applying all 16 possible input values for X and examining the corresponding output
values Y, it may be observed that for exactly 12 out the 16 cases, the expression above
holds true. Hence, the probability bias is 12/16−1/2 = 1/4. This is presented in Table 3.
Similarly, for equation

X 1 ⊕ X 4 = Y2
it may be seen that the probability bias is 0 and for equation
X 3 ⊕ X 4 = Y1 ⊕ Y4
the probability bias is 2/16−1/2 = −3/8. In the last case, the best approximation is an
affine approximation as indicated by the minus sign. However, the success of the attack is
based on magnitude of the bias and, as we shall see, affine approximations can be used
equivalently to linear approximations.

X1

X2

X3

X4

Y1

Y2

Y3

Y4

0
0
0
0
0
0
0
0
1
1
1
1
1
1
1
1

0
0
0
0
1
1
1
1
0
0
0
0
1
1
1
1

0
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1

0
1
0
1
0
1
0
1
0
1
0
1
0
1
0
1

1
0
1
0
0
1
1
1
0
1
0
1
0
1
0
0

1
1
1
0
0
1
0
0
0
0
1
1
1
0
0
1

1
0
0
0
1
1
1
0
1
1
1
0
0
0
0
1

0
0
1
1
0
1
1
0
1
0
0
0
1
1
0
1

Table 3. Sample Linear Approximations of S-box

10

X2
Y1
X1
⊕X3 ⊕Y3 ⊕X4
⊕Y4
0
0
0
0
0
1
1
0
0
1
1
1
1
1
0
1
1
1
0
1
0
0
1
1
0
0
1
0
0
0
1
1
1
1
1
0
1
1
1
1
0
0
0
0
1
0
0
0

Y2

1
1
1
0
0
1
0
0
0
0
1
1
1
0
0
1

X3
Y1
⊕X4 ⊕Y4
0
1
1
0
0
1
1
0
0
1
1
0
0
1
1
0

1
0
0
1
0
0
0
1
1
1
0
1
1
0
0
1

I
n
p
u
t
S
u
m

0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F

0
+8
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

1
0
0
0
0
+2
−2
+2
−2
0
0
+4
+4
−2
+2
+2
−2

2
3
4
5
0
0
0
0
0
0
−2 −2
0
0
−2 −2
0
0
0
0
0 −2 −2 −4
0 −2
0
−2
+4 +2
0
−2
0 +2 +2 −4
0
0
0
0
0
0
−2 −2
0
−2 +2 −4
0 −4 +4
0
+4 −2 −2
0
+2
0 −2 +4
+2
0 −2 −4
0
−4 −2 −2

Output Sum
6
7
8
9
0 0
0
0
−2 +6 +2 +2
0
0
−2 −2
0
0 +2 −6
0
0 −2
−2
+4 +2 −2
0
0 +2
0 −2
+2
0 −2
0
0
0 −2 +2
0
−2 −2 −4
+2 −2 +2 +2
+4
0
0
0
+2
0 +2
0
0 +2 −4 −2
0 +2 −2
0
+2
0
0 −2

A
0
0
+2
−2
0
−4
+2
+2
+2
−2
0
0
+2
+2
0
+4

B
0
0
+2
−2
+2
+2
+4
0
−2
+2
0
0
+4
0
−2
−2

C
0
+2
0
+2
+2
0
−2
+4
+2
0
+2
0
0
+2
−4
−2

D E F
0
0
0
+2
0
0
0 −6 +2
+2 −2 −2
0
−4 +2
0
−2 −2
0
0 −2
+2
0 +2
−2 −2 −6
+4 +2 −2
+2
0
0
0
0
0
+2
0 −2
0
0 +2
+2 −2
0
0 +2
0

Table 4. Linear Approximation Table
A complete enumeration of all linear approximations of the S-box in our cipher is given
in the linear approximation table of Table 4. Each element in the table represents the
number of matches between the linear equation represented in hexadecimal as "Input
Sum" and the sum of the output bits represented in hexadecimal as "Output Sum" minus
8. Hence, dividing an element value by 16 gives the probability bias for the particular
linear combination of input and output bits. The hexadecimal value representing a sum,
when viewed as a binary value indicates the variables involved in the sum. For a linear
combination of input variables represented as a1⋅X1⊕a2⋅X2⊕a3⋅X3⊕a4⋅X4 where ai ∈ {0,1}
and "⋅" represents binary AND, the hexadecimal value represents the binary value
a1a2a3a4, where a1 is the most significant bit. Similarly, for a linear combination of output
bits b1⋅Y1 ⊕ b2⋅Y2 ⊕ b3⋅Y3 ⊕ b4⋅Y4 where bi ∈ {0,1}, the hexadecimal value represents the
binary vector b1b2b3b4. Hence, the bias of linear equation X3 ⊕ X4 = Y1 ⊕ Y4 (hex input 3
and hex output 9) is −6/16 = −3/8 and the probability that the linear equation holds true is
given by 1/2 − 3/8 = 1/8.
Some basic properties of the linear approximation table can be noted. For example, the
probability that any sum of a non-empty subset of output bits is equal to the sum
involving no input bits is exactly 1/2 since any linear combination of output bits must
have an equal number of zeros and ones for a bijective S-box. Also, the linear
combination involving no output bits will always equal the linear combination of no input
bits resulting in a bias of +1/2 and a table value of +8 in the top left corner. Hence, the
top row of the table is all zeros, except for the leftmost value. Similarly, the first column

11

is all zeros except for the topmost value. It can also be noted the sum of any row or any
column must be either +8 or −8. We leave the proof of this as an exercise to the reader.
3.4 Constructing Linear Approximations for the Complete Cipher
Once the linear approximation information has been compiled for the S-boxes in an SPN,
we have the data to proceed with determining linear approximations of the overall cipher
of the form of equation (1). This can be achieved by concatenating appropriate linear
approximations of S-boxes. By constructing a linear approximation involving plaintext
bits and data bits from the output of the second last round of S-boxes, it is possible to
attack the cipher by recovering a subset of the subkey bits that follow the last round. We
illustrate with an example.
Consider an approximation involving S12, S22, S32, and S34 as illustrated in Figure 3. Note
that this actually develops an expression for the first 3 rounds of the cipher and not the
full 4 rounds. We shall see how this is useful in deriving the subkey bits after the last
round in the next section.
We use the following approximations of the S-box:
S12: X1⊕X3⊕X4 = Y2
S22: X2 = Y2⊕Y4
S32: X2 = Y2⊕Y4
S34: X2 = Y2⊕Y4

with probability 12/16 and bias +1/4
with probability 4/16 and bias −1/4

Letting Ui (Vi) represent the 16-bit block of bits at the input (output) of the round i Sboxes and Ui,j (Vi,j) represent the j-th bit of block Ui (Vi) (where bits are numbered from 1
to 16 from left to right in the figure of the cipher). Similarly, let Ki represent the subkey
block of bits exclusive-ORed at the input to round i, with the exception that K5 is the key
exclusive-ORed at the output of round 4.
Hence, U1 = P ⊕ K1 where P represents the block of 16 plaintext bits and "⊕" represents
the bit-wise exclusive-OR. Using the linear approximation of the 1st round, we then have
V1,6

= U1,5 ⊕ U1,7 ⊕ U1,8
= (P5 ⊕ K1,5) ⊕ (P7 ⊕ K1,7) ⊕ (P8 ⊕ K1,8)

with probability 3/4. For the approximation in the 2nd round, we have
V2,6 ⊕ V2,8 = U2,6
with probability 1/4. Since U2,6 = V1,6 ⊕ K2,6, we can get an approximation of the form
V2,6 ⊕ V2,8 = V1,6 ⊕ K2,6

12

(2)

P5
K1,5

P7 P8
K1,7

S11

K1,8

S12

S13

S14

S22

S23

S24

K2,6

S21

K3,6

S31

S32

K4,6

S33

K4,8

U4,6
S41

K3,14

K4,14

U4,8
S42

S43

K5,5 ... K5,8

S34

U4,14
S44

K4,16
U4,16

K5,13 ... K5,16

Figure 3. Sample Linear Approximation

13

with probability 1/4 and combining this with (2) which holds with probability of 3/4
gives
V2,6 ⊕ V2,8 ⊕ P5 ⊕ P7 ⊕ P8 ⊕ K1,5 ⊕ K1,7 ⊕ K1,8 ⊕ K2,6 = 0

(3)

which holds with probability of 1/2 + 2(3/4−1/2)(1/4−1/2) = 3/8 (that is, with a bias of
−1/8) by application of the Piling-Up Lemma. Note that we are using the assumption that
the approximations of S-boxes are independent which, although not strictly correct,
works well in practice for most ciphers.
For round 3, we note that
V3,6 ⊕ V3,8 = U3,6
with probability 1/4 and
V3,14 ⊕ V3,16 = U3,14
with probability 1/4. Hence, since U3,6 = V2,6 ⊕ K3,6 and U3,14 =V2,8 ⊕ K3,14,
V3,6 ⊕ V3,8 ⊕ V3,14 ⊕ V3,16 ⊕ V2,6 ⊕ K3,6 ⊕ V2,8 ⊕ K3,14 = 0

(4)

with probability of 1/2 + 2(1/4−1/2)2 = 5/8 (that is, with a bias of +1/8). Again, we have
applied the Piling-Up Lemma.
Now combining (3) and (4), to incorporate all four S-box approximations, we get
V3,6 ⊕ V3,8 ⊕ V3,14 ⊕ V3,16 ⊕ P5 ⊕ P7 ⊕ P8
⊕ K1,5 ⊕ K1,7 ⊕ K1,8 ⊕ K2,6 ⊕ K3,6 ⊕ K3,14 = 0.
Noting that U4,6 = V3,6 ⊕ K4,6, U4,8 = V3,14 ⊕ K4,8, U4,14 = V3,8 ⊕ K4,14, and U4,16 = V3,16 ⊕
K4,16, we can then write
U4,6 ⊕ U4,8 ⊕ U4,14 ⊕ U4,16 ⊕ P5 ⊕ P7 ⊕ P8 ⊕ ΣK = 0.
where
ΣK = K1,5 ⊕ K1,7 ⊕ K1,8 ⊕ K2,6 ⊕ K3,6 ⊕ K3,14 ⊕ K4,6 ⊕ K4,8 ⊕ K4,14 ⊕ K4,16
and ΣK is fixed at either 0 or 1 depending on the key of the cipher. By application of the
Piling-Up
Lemma,
the
above
expression
holds
with
probability
3
3
1/2+2 (3/4−1/2)(1/4−1/2) = 15/32 (that is, with a bias of −1/32).
Now since ΣK is fixed, we note that

14

U4,6 ⊕ U4,8 ⊕ U4,14 ⊕ U4,16 ⊕ P5 ⊕ P7 ⊕ P8 = 0

(5)

must hold with a probability of either 15/32 or (1−15/32) = 17/32, depending on whether
ΣK = 0 or 1, respectively. In other words, we now have a linear approximation of the first
three rounds of the cipher with a bias of magnitude 1/32. We must now discuss how such
a bias can be used to determine some of the key bits.
3.5 Extracting Key Bits
Once an R−1 round linear approximation is discovered for a cipher of R rounds with a
suitably large enough linear probability bias, it is conceivable to attack the cipher by
recovering bits of the last subkey. In the case of our example cipher, it is possible to
extract bits from subkey K5 given a 3 round linear approximation. We shall refer to the
bits to be recovered from the last subkey as the target partial subkey. Specifically, the
target partial subkey bits are the bits from the last subkey associated with the S-boxes in
the last round influenced by the data bits involved in the linear approximation.
The process followed involves partially decrypting the last round of the cipher.
Specifically, for all possible values of the target partial subkey, the corresponding
ciphertext bits are exclusive-ORed with the bits of the target partial subkey and the result
is run backwards through the corresponding S-boxes. This is done for all known
plaintext/ciphertext samples and a count is kept for each value of the target partial
subkey. The count for a particular target partial subkey value is incremented when the
linear expression holds true for the bits into the last round’s S-boxes (determined by the
partial decryption) and the known plaintext bits. The target partial subkey value which
has the count which differs the greatest from half the number of plaintext/ciphertext
samples is assumed to represent the correct values of the target partial subkey bits. This
works because it is assumed that the correct partial subkey value will result in the linear
approximation holding with a probability significantly different from 1/2. (Whether it is
above or below 1/2 depends on whether a linear or affine expression is the best
approximation and this depends on the unknown values of the subkey bits implicitly
involved in the linear expression.) An incorrect subkey is assumed to result in a relatively
random guess at the bits entering the S-boxes of the last round and as a result, the linear
expression will hold with a probability close to 1/2.
Let’s now put this into the context of our example. The linear expression of (5) affects the
inputs to S-boxes S42 and S44 in the last round. For each plaintext/ciphertext sample, we
would try all 256 values for the target partial subkey [K5,5...K5,8, K5,13...K5,16]. For each
partial subkey value, we would increment the count whenever equation (5) holds true,
where we determine the value of [U4,5...U4,8, U4,13...U4,16] by running the data backwards
through the target partial subkey and S-boxes S24 and S44. The count which deviates the
largest from half of the number of plaintext/ciphertext samples is assumed to the correct
value. Whether the deviation is positive or negative will depend on the values of the
subkey bits involved in ΣK. When ΣK = 0, the linear approximation of (5) will serve as the
estimate (with probability < 1/2) and when ΣK = 1, (5) will hold with a probability > 1/2.

15

We have simulated attacking our basic cipher by generating 10000 known
plaintext/ciphertext values and following the cryptanalytic process described for partial
subkey values [K5,5...K5,8] = [0010] (hex 2) and [K5,13...K5,16] = [0100] (hex 4). As
expected, the count which differed the most from 5000 corresponded to target partial
subkey value [2,4]hex, confirming that the attack has successfully derived the subkey bits.
Table 5 highlights a partial summary of the data derived from the subkey counts. (The
complete data involves 256 data entries, one for each target partial subkey value.) The
values in the table indicate the bias magnitude derived from
| bias | = | count − 5000 | / 10000
where the count is the count corresponding to the particular partial subkey value.
As can be seen from the partial results in the table, the largest bias occurs for partial
subkey value [K5,5...K5,8, K5,13...K5,16] = [2,4] and this observation was, in fact, found to
be true for the complete set of partial subkey values.
partial subkey
[K5,5...K5,8, K5,13...K5,16]
1C
1D
1E
1F
20
21
22
23
24
25
26
27
28
29

| bias |

partial subkey
[K5,5...K5,8, K5,13...K5,16]
2A
2B
2C
2D
2E
2F
30
31
32
33
34
35
36
37

0.0031
0.0078
0.0071
0.0170
0.0025
0.0220
0.0211
0.0064
0.0336
0.0106
0.0096
0.0074
0.0224
0.0054

| bias |
0.0044
0.0186
0.0094
0.0053
0.0062
0.0133
0.0027
0.0050
0.0075
0.0162
0.0218
0.0052
0.0056
0.0048

Table 5. Experimental Results for Linear Attack
The experimentally determined bias value of 0.0336 is very close to the expected value of
1/32 = 0.03125. Note that, although the correct target partial subkey has clearly the
highest bias, other large bias values occur indicating that the examination of incorrect
target partial subkeys is not precisely equivalent to comparing random data to a linear
expression (where the bias could be expected to be very close to zero). Inconsistencies in
the experimental biases can occur for several reasons including the S-box properties
influencing the partial decryption for different partial subkey values, the imprecision of
the independence assumption required for use in the Piling-Up Lemma, and the influence
of linear hulls (to be discussed in the next section).

16

3.6 Complexity of Attack
We refer to the S-boxes involved in the linear approximation as active S-boxes. In Figure
3, the four S-boxes in rounds 1 to 3 influenced by the highlighted lines are active. The
probability that a linear expression holds true is related to the linear probability bias in
the active S-boxes and the number of active S-boxes. In general, the larger the magnitude
of the bias in the S-boxes, the larger the magnitude of the bias of the overall expression.
Also, the fewer active S-boxes, the larger the magnitude of the overall linear expression
bias.
Let ε represent the bias from 1/2 of the probability that the linear expression for the
complete cipher holds. In his paper, Matsui shows that the number of known plaintexts
required in the attack is proportional to ε−2 and, letting NL represent the number of known
plaintexts required, it is reasonable to approximate NL by
NL ≈ 1/ε2.
In practice, it is generally reasonable to expect some small multiple of ε−2 known
plaintexts are required. Although strictly speaking, the complexity of the cryptanalysis
could be characterized in both time and space (or memory) domains, we refer to the data
required to mount the attack when considering the complexity of the cryptanalysis. That
is, we assume that if we are able to acquire NL plaintexts, we are able to process them.
Since the bias is derived using the Piling-Up Lemma where each term in the product
refers to an S-box approximation, it is easy to see that the bias is dependent on the biases
of the S-box linear approximations and the number of active S-boxes involved. General
approaches to providing security against linear cryptanalysis have focused on optimizing
the S-boxes (i.e., minimizing the largest bias) and finding structures to maximize the
number of active S-boxes. The design principles of Rijndael are an excellent example of
such an approach.
It must be cautioned, however, the concept of a "proof" of security to linear cryptanalysis
is usually premised on the nonexistence of highly likely linear approximations. However,
the computation of the probability of such linear approximations is based on the
assumption that each S-box approximation is independent (so that the Piling-Up Lemma
can be used) and on the assumption that one linear approximation scenario (i.e., a
particular set of active S-boxes) is sufficient to determine the best linear expression
between plaintext bits and data bits at the input to the last round. The reality is that the Sbox approximations are not independent and this can have significant impact on the
computation of the probability. Also, linear approximation scenarios involving the same
plaintext and last round input bits but different sets of active S-boxes can combine to give
a linear probability higher than that predicted by one set of active S-boxes. This concept
is referred to as a linear hull [16]. Most notably for example, a number of linear
approximation scenarios may have very small biases and on their own seem to imply that
a cipher might be immune to a linear attack. However, when these scenarios are
combined, the resulting linear expression of plaintext and last round input bits might have

17

a very high bias. Nevertheless, the approach outlined in this paper, tends to work well for
many ciphers because the independence assumption is a reasonable approximation and
when one linear approximation scenario of a particular set of active S-boxes has a high
bias, it tends to dominate the linear hull.

18

4. Differential Cryptanalysis
In this section, we now turn our focus to the application of differential cryptanalysis to
the basic SPN cipher.
4.1 Overview of Basic Attack
Differential cryptanalysis exploits the high probability of certain occurrences of plaintext
differences and differences into the last round of the cipher. For example, consider a
system with input X = [X1 X2 ... Xn] and output Y = [Y1 Y2 ... Yn]. Let two inputs to the
system be X′ and X″ with the corresponding outputs Y′ and Y″, respectively. The input
difference is given by ∆X = X′ ⊕ X″ where "⊕" represents a bit-wise exclusive-OR of the
n-bit vectors and, hence,
∆X = [∆X 1 ∆X 2 ... ∆X n ]
where ∆X i = X i′ ⊕ X i′′ with Xi′ and Xi″ representing the i-th bit of X′ and X″, respectively.
Similarly, ∆Y = Y′ ⊕ Y″ is the output difference and
∆Y = [∆Y1 ∆Y2 ... ∆Yn ]
where ∆Yi = Yi′ ⊕ Yi′′ .
In an ideally randomizing cipher, the probability that a particular output difference ∆Y
occurs given a particular input difference ∆X is 1/2n where n is the number of bits of X.
Differential cryptanalysis seeks to exploit a scenario where a particular ∆Y occurs given a
particular input difference ∆X with a very high probability pD (i.e., much greater than
1/2n). The pair (∆X, ∆Y) is referred to as a differential.
Differential cryptanalysis is a chosen plaintext attack, meaning that the attacker is able to
select inputs and examine outputs in an attempt to derive the key. For differential
cryptanalysis, the attacker will select pairs of inputs, X′ and X″, to satisfy a particular ∆X,
knowing that for that ∆X value, a particular ∆Y value occurs with high probability.
In this paper, we investigate the construction of a differential (∆X, ∆Y) involving
plaintext bits as represented by X and the input to the last round of the cipher as
represented by Y. We shall do this by examining high likely differential characteristics
where a differential characteristic is a sequence of input and output differences to the
rounds so that the output difference from one round corresponds to the input difference
for the next round. Using the highly likely differential characteristic gives us the
opportunity to exploit information coming into the last round of the cipher to derive bits
from the last layer of subkeys.
As with linear cryptanalysis, to construct highly likely differential characteristics, we
examine the properties of individual S-boxes and use these properties to determine the
19

complete differential characteristic. Specifically, we consider the input and output
differences of the S-boxes in order to determine a high probability difference pair.
Combining S-box difference pairs from round to round so that the nonzero output
difference bits from one round correspond to the non-zero input difference bits of the
next round, enables us to find a high probability differential consisting of the plaintext
difference and the difference of the input to the last round. The subkey bits of the cipher
end up disappearing from the difference expression because they are involved in both
data sets and, hence, considering their influence on the difference involves exclusiveORing subkey bits with themselves, the result of which is zero.
4.2 Analyzing the Cipher Components
We examine now the difference pairs of an S-box. Consider the 4×4 S-box representation
of Figure 2 with input X = [X1 X2 X3 X4] and output Y = [Y1 Y2 Y3 Y4]. All difference pairs
of an S-box, (∆X, ∆Y), can be examined and the probability of ∆Y given ∆X can be
derived by considering input pairs (X′, X″) such that X′ ⊕ X″ = ∆X. Since the ordering of
the pair is not relevant, for a 4×4 S-box we need only consider all 16 values for X′ and
then the value of ∆X constrains the value of X″ to be X″ = X′ ⊕ ∆X.
Considering the S-box of our cipher given in Section 2, we can derive the resulting values
of ∆Y for each input pair (X′, X″=X′ ⊕ ∆X). For example, the binary values of X, Y, and
the corresponding values for ∆Y for given input pairs (X, X ⊕ ∆X) are presented in Table
6 for ∆X values of 1011 (hex B), 1000 (hex 8), and 0100 (hex 4). The last three columns
of the table represent ∆Y values for the value of X (as given by the row) and the particular
∆X value for each column. From the table, we can see that the number of occurrences of
∆Y = 0010 for ∆X = 1011 is 8 out of 16 possible values (i.e., a probability of 8/16); the
number of occurrences of ∆Y = 1011 given ∆X = 1000 is 4 out of 16; the number of
occurrences of ∆Y = 1010 given ∆X = 0100 is 0 out of 16. If the S-box could be "ideal",
the number of occurrences of difference pair values would all be 1 to give a probability of
1/16 of the occurrence of a particular ∆Y value given ∆X. (It turns out that such an "ideal"
S-box is not mathematically possible.)
We can tabularize the complete data for an S-box in a difference distribution table in
which the rows represent ∆X values (in hexadecimal) and the columns represent ∆Y
values (in hexadecimal). The difference distribution table for the S-box of Table 1 is
given in Table 7. Each element of the table represents the number of occurrences of the
corresponding output difference ∆Y value given the input difference ∆X. Note that,
besides the special case of (∆X = 0, ∆Y = 0), the largest value in the table is 8,
corresponding to ∆X = B and ∆Y = 2. Hence, the probability that ∆Y = 2 given an
arbitrary pair of input values that satisfy ∆X = B is 8/16. The smallest value in the table is
0 and occurs for many difference pairs. In this case, the probability of the ∆Y value
occurring given the ∆X value is 0.

20

X

Y

0000
0001
0010
0011
0100
0101
0110
0111
1000
1001
1010
1011
1100
1101
1110
1111

1110
0100
1101
0001
0010
1111
1011
1000
0011
1010
0110
1100
0101
1001
0000
0111

∆X = 1011
0010
0010
0111
0010
0101
1111
0010
1101
0010
0111
0010
0010
1101
0010
1111
0101

∆Y
∆X = 1000
1101
1110
0101
1011
0111
0110
1011
1111
1101
1110
0101
1011
0111
0110
1011
1111

∆X = 0100
1100
1011
0110
1001
1100
1011
0110
1001
0110
0011
0110
1011
0110
0011
0110
1011

Table 6. Sample Difference Pairs of the S-box

I
n
p
u
t
D
i
f
f
e
r
e
n
c
e

0
1
2
3
4
5
6
7
8
9
A
B
C
D
E
F

0
16
0
0
0
0
0
0
0
0
0
0
0
0
0
0
0

1
0
0
0
0
0
4
0
0
0
2
2
0
2
4
0
2

2
0
0
0
2
0
0
0
2
0
0
2
8
0
0
2
0

3
0
2
2
0
2
0
4
2
0
0
0
0
0
0
4
0

4
0
0
0
2
0
0
0
2
0
2
0
0
2
0
2
6

5
0
0
6
0
0
2
4
0
0
0
0
2
2
0
0
0

Output Difference
6
7
8
9
0
0
0
0
0
2
0
2
2
2
0
2
0
0
0
4
6
0
0
2
2
0
0
0
0
0
0
0
2
0
0
2
2
2
0
0
0
4
2
0
0
0
6
0
0
2
0
0
2
0
0
0
0
4
2
0
0
0
6
0
0
0
0
4

Table 7. Difference Distribution Table

21

A
0
4
0
2
0
4
0
2
0
2
0
0
0
2
0
0

B
0
0
0
0
4
0
0
0
4
2
2
0
2
0
0
2

C
0
4
0
2
2
2
2
0
0
2
0
0
0
2
0
0

D
0
2
0
0
0
0
2
0
4
0
0
2
6
0
0
0

E
0
0
2
0
0
0
2
0
2
0
4
0
0
2
2
2

F
0
0
0
4
0
2
2
4
2
0
0
2
0
0
0
0

There are several general properties of the difference distribution table that should be
mentioned. First, it should be noted that the sum of all elements in a row is 2n =16;
similarly the sum of any column is 2n =16. Also, all element values are even: this results
because a pair of input (or output) values represented as (X′, X″) has the same ∆X value as
the pair (X″, X′) since ∆X = X′ ⊕ X″ = X″ ⊕ X′. As well, the input difference of ∆X = 0
must lead to an output difference of ∆Y = 0 for the one-to-one mapping of the S-box.
Hence, the top right corner of the table has a value of 2n = 16 and all other values in the
first row and first column are 0. Finally, if we could construct an ideal S-box, which
gives no differential information about the output given the input value, the S-box would
have all elements in the table equal to 1 and the probability of occurrence of a particular
value for ∆Y given a particular value of ∆X would be 1/2n = 1/16. However, as the
properties discussed above must hold, this is clearly not achievable.
Before we proceed to discuss the combining of S-box difference pairs to derive a
differential characteristic and an estimate of a good differential to use in the attack, we
must discuss the influence of the key on the S-box differential. Consider Figure 4. The
input to the "unkeyed" S-box is X and the output Y. However, in the cipher structure we
must consider the keys applied at the input of each S-box. In this case, if we let the input
to the "keyed" S-box be W = [W1 W2 W3 W4], we can consider the input difference to the
keyed S-box to be
∆W = [W1′ ⊕ W1′′ W2′ ⊕ W2′′ ... Wn′ ⊕ Wn′′]
where W ′ = [W1′ W2′ ... Wn′ ] and W ′′ = [W1′′ W2′′ ... Wn′′] represent the two input values.
Since the key bits remain the same for both W′ and W″,
∆Wi

= Wi′ ⊕ Wi″ = (Xi′ ⊕ Ki) ⊕ (Xi″ ⊕ Ki)
= Xi′ ⊕ Xi″ = ∆Xi

since Ki ⊕ Ki = 0. Hence, the key bits have no influence on the input difference value and
can be ignored. In other words, the keyed S-box has the same difference distribution table
as the unkeyed S-box.
W1

W2

W3

W4
K1
K2

X1

X2

X3

K3

X4
K4

4×4
S-box
Figure 4. Keyed S-box
Y1 Y2 Y3 Y4
22

4.3 Constructing Differential Characteristics
Once the differential information has been compiled for the S-boxes in an SPN, we have
the data to proceed with determining a useful differential characteristic of the overall
cipher. This can be done by concatenating appropriate difference pairs of S-boxes. By
constructing a differential characteristic of certain S-box difference pairs in each round,
such that a differential involves plaintexts bits and data bits to the input of the last round
of S-boxes, it is possible to attack the cipher by recovering a subset of the subkey bits
following the last round. We illustrate the construction of a differential characteristic with
an example.
Consider a differential characteristic involving S12, S23, S32, and S33. As was the case for
linear cryptanalysis, it is useful to visualize the differential characteristic in the form of a
diagram as shown in Figure 5. The diagram illustrates the influence of non-zero
differences in bits as they traverse the network, highlighting the S-boxes that may be
considered as active (i.e., for which there is a non-zero differential). Note that this
develops a differential characteristic for the first 3 rounds of the cipher and not the full 4
rounds. We shall see how this is useful in deriving bits from the last subkey in the next
section.
We use the following difference pairs of the S-box:
S12: ∆X = B → ∆Y = 2
S23: ∆X = 4 → ∆Y = 6
S32: ∆X = 2 → ∆Y = 5
S33: ∆X = 2 → ∆Y = 5

with probability 8/16

All other S-boxes will have zero input difference and consequently zero output
difference.
The input difference to the cipher is equivalent to the input difference to the first round
and is given by

∆P = ∆U 1 = [0000 1011 0000 0000]
where again, as with our presentation of linear cryptanalysis in Section 3, we are using Ui
to represent the input to the i-th round S-boxes and Vi to represent the output of the i-th
round S-boxes. Hence, ∆Ui and ∆Vi represent the corresponding differences. As a result,

∆V1 = [0000 0010 0000 0000]
considering the difference pair for S12 listed above and following the round 1 permutation

∆U 2 = [0000 0000 0100 0000]

23

∆P = [0000 1011 0000 0000]

S11

S12

S13

S14

S21

S22

S23

S24

S31

S32

S33

S34

S43

∆U4,13... ∆U4,16
S44

S41

∆U4,5... ∆U4,8
S42

K5,5 ... K5,8

K5,13 ... K5,16

Figure 5. Sample Differential Characteristic

24

with probability of 8/16 = 1/2 given the plaintext difference ∆P.
Now the second round differential using the difference pair for S23 results in

∆V2 = [0000 0000 0110 0000]
and the permutation of round 2 gives
∆U 3 = [0000 0010 0010 0000]
with probability 6/16 given ∆U2 and a probability of 8/16 × 6/16 = 3/16 given ∆P. In
determining the probability given plaintext difference ∆P, we have assumed that the
differential of the first round is independent of the differential of the 2nd round and,
hence, the probability of both occurring is determined by the product of the probabilities.
Subsequently, we can use the differences for the S-boxes of the third round, S32 and S33,
and the permutation of the third round to arrive at
∆V3 = [0000 0101 0101 0000]
and

∆U 4 = [0000 0110 0000 0110]

(6)

with a probability of (6/16)2 given ∆U3 and, hence, a probability of 8/16 × 6/16 × (6/16)2
= 27/1024 given plaintext difference ∆P, where again we have assumed independence
between the difference pairs of S-boxes in all rounds.
During the cryptanalysis process, many pairs of plaintexts for which ∆P = [0000 1011
0000 0000] will be encrypted. With high probability, 27/1024, the differential
characteristic illustrated will occur. We term such pairs for ∆P as right pairs. Plaintext
difference pairs for which the characteristic does not occur are referred to as wrong pairs.
4.4 Extracting Key Bits
Once an R−1 round differential characteristic is discovered for a cipher of R rounds with
a suitably large enough probability, it is conceivable to attack the cipher by recovering
bits from the last subkey. In the case of our example cipher, it is possible to extract bits
from subkey K5. The process followed involves partially decrypting the last round of the
cipher and examining the input to the last round to determine if a right pair has probably
occurred. We shall refer to the subkey bits following the last round at the output of Sboxes in the last round influenced by non-zero differences in the differential output as the
target partial subkey. A partial decryption of the last round would involve, for all Sboxes in the last round influenced by non-zero differences in the differential, the
exclusive-OR of the ciphertext with the target partial subkey bits and running the data

25

backwards through the S-boxes, where all possible values for the target subkey bits
would be tried.
A partial decryption is executed for each pair of ciphertexts corresponding to the pairs of
plaintexts used to generate the input difference ∆P for all possible target partial subkey
values. A count is kept for each value of the target partial subkey value. The count is
incremented when the difference for the input to the last round corresponds to the value
expected from the differential characteristic. The partial subkey value which has the
largest count is assumed to indicate the correct values of the subkey bits. This works
because it is assumed that the correct partial subkey value will result in the difference to
the last round being frequently as expected from the characterstic (i.e., the occurrence of
a right pair) since the characteristic has a high probability of occurring. (When a wrong
pair has occurred, even with the partial decryption with the correct subkey, the count for
the correct subkey will likely not be incremented.) An incorrect subkey is assumed to
result in a relatively random guess at the bits entering the S-boxes of the last round and as
a result, the difference will be as expected from the characteristic with a very low
probability.
Considering the attack on our example cipher, the differential characteristic affects the
inputs to S-boxes S42 and S44 in the last round. For each ciphertext pair, we would try all
256 values for [K5,5...K5,8, K5,13...K5,16]. For each partial subkey value, we would
increment the count whenever the input difference to the final round determined by the
partial decryption is the same as (6), where we determine the value of [∆U4,5... ∆U4,8,
∆U4,13... ∆U4,16] by running the data backwards through the partial subkey and S-boxes
S24 and S44. For each partial subkey value, the count represents the number of occurrences
of differences that are consistent with right pairs (assuming that the partial subkey is the
correct value). The count that is the largest is taken to be the correct value since we
assume that we are observing the high probability occurrence of the right pair.
Note that it is not necessary to execute the partial decryption for every ciphertext pair.
Since the input difference to the last round only influences 2 S-boxes, when the
characteristic has occurred (i.e., for right pairs), the ciphertext bit differences
corresponding to S-boxes S41 and S43 must be zero. Hence, we can filter out many wrong
pairs by discarding ciphertext pairs for which zeros do not appear in the appropriate subblocks of the ciphertext difference. In these cases, since the ciphertext pair cannot
correspond to a right pair, it is not necessary to examine [∆U4,5... ∆U4,8, ∆U4,13... ∆U4,16].
We have simulated attacking our basic cipher keyed using randomly generated subkeys
by generating 5000 chosen plaintext/ciphertext pairs (i.e., 10000 encryptions with
plaintext pairs satisfying ∆P = [0000 1011 0000 0000]) and following the process
described above. The correct target partial subkey value was [K5,5...K5,8, K5,13...K5,16] =
[0010,0100] = [2,4]hex. As expected, the largest count was observed for partial subkey
value [2,4]hex, confirming that the attack successfully derived the subkey bits. Table 8
highlights a partial summary of the data derived from the subkey counts. (The complete
data involves 256 data entries, one for each partial subkey value.) The values in the table

26

indicate the estimated probability of the occurrence of right pairs for the candidate partial
subkey derived from
prob = count / 5000.
where the count is the count corresponding to the particular partial subkey value.
As can be seen from the sample results of the table, the largest probability occurs for
partial subkey value [K5,5...K5,8, K5,13...K5,16] = [2,4]hex and this observation was, in fact,
found to be true for the complete set of partial subkey values.
In our example, we would expect the probability of the occurrence of the right pair to be
pD = 27/1024 = 0.0264 and we found experimentally the probability for the correct
subkey value [2,4] gave pD = 0.0244. Note that sometimes other large count values occur
for incorrect target partial subkeys. This indicates that the examination of incorrect target
partial subkeys is not precisely equivalent to comparing random differences to the
expected differential value. There are several factors which influence the counts to be
different then our theorized expectations including the S-box properties influencing the
partial decryption for different partial subkeys, the imprecision of the independence
assumption required for determination of the characteristic probability, and the concept
that differentials are composed of multiple differential characteristics (to be discussed in
the next section).
partial subkey
[K5,5...K5,8, K5,13...K5,16]
1C
1D
1E
1F
20
21
22
23
24
25
26
27
28
29

prob

partial subkey
[K5,5...K5,8, K5,13...K5,16]
2A
2B
2C
2D
2E
2F
30
31
32
33
34
35
36
37

0.0000
0.0000
0.0000
0.0000
0.0000
0.0136
0.0068
0.0068
0.0244
0.0000
0.0068
0.0068
0.0030
0.0024

Table 8. Experimental Results for Differential Attack

27

prob
0.0032
0.0022
0.0000
0.0000
0.0000
0.0000
0.0004
0.0000
0.0004
0.0004
0.0000
0.0004
0.0000
0.0008

4.5 Complexity of the Attack
For differential cryptanalysis, we refer to the S-boxes involved in a characteristic which
have a non-zero input difference (and hence a non-zero output difference) as active Sboxes. In general, the larger the differential probabilities of the active S-boxes, the larger
the characteristic probability for the complete cipher. Also, the fewer active S-boxes, the
larger the characteristic probability. As with linear cryptanalysis, we refer to the data
required to mount the attack when considering the complexity of the cryptanalysis. That
is, we assume that if we are able to acquire ND plaintexts, we are able to process them.
In general it is very complex to determine exactly the number of chosen plaintext pairs
required to mount the attack. However, it can be shown that a good rule-of-thumb for the
number of chosen plaintext pairs, ND, required to distinguish right pairs when trying
subkey candidates is

N D ≈ c / pD

(7)

where pD is the differential characteristic probability for the R−1 rounds of the R-round
cipher and c is a small constant. Assuming that the occurrences of difference pairs in each
active S-box are independent, the differential characteristic probability is given by
γ

pD = ∏ β i

(8)

i =1

where the number of active S-boxes is represented by γ and the occurrence of the
particular difference pair in the i-th active S-box of the characteristic has a probability
represented by βi.
It is not difficult to rationalize that (7) is true. It simply indicates that a few occurrences
of the right pair are enough to give a count to the correct target partial subkey value that
is significantly greater than the counts for the incorrect target partial subkey values. Since
a right pair is expected to occur for about every 1/pD pairs examined, in practice, it is
generally reasonable to use some small multiple of 1/pD chosen plaintext pairs to
successfully mount the attack.
Approaches to providing resistance to differential cryptanalysis have focused on the Sbox properties (i.e., minimizing the difference pair probability of an S-box) and finding
structures to maximize the number of active S-boxes. Rijndael is a good example of a
cipher designed to provide high resistance to differential cryptanalysis.
As with linear cryptanalysis, caution must be exercised in "proving" immunity to
differential cryptanalysis. The computation of the differential characteristic probability is
premised on the independence of the S-boxes involved in the approximation and in a real
cipher, there is a dependence between the data entering different S-boxes. Hence, the

28

probability pD is an estimate only. In practice, in many ciphers it has proven to be
reasonably accurate.
Most importantly, different differential characteristics with the same input difference and
output difference (i.e., the same differential) can combine to imply a probability for the
differential that is larger than is implied by the consideration of one differential
characteristic alone [17]. (This is analogous to the concept of linear hulls.) In order to
prove security to differential cryptanalysis, it is necessary to prove that the probability of
all differentials are below some acceptable threshold, not just that the probability of all
differential characteristics are below some acceptable threshold. Generally, though, it is a
reasonable assumption that, when a differential characteristic has a high probability, it
dominates the occurrence of a differential and the probability of the characteristic gives a
good approximation of the differential probability.

29

5. Advanced Concepts
Several extensions and modifications to the basic attacks of linear and differential
cryptanalysis have been proposed and analyzed since the original presentation of the
attacks. We do not present these advanced concepts and analyses in detail here, but
encourage the reader to pursue these concepts further. Note the papers cited in this
section represent a small sampling of work built on the two attacks.
We have mentioned, for example, the concepts of linear hulls [16] and differentials [17]:
both concepts are integral to understanding the nature of provable security to the two
cryptanalysis methods (an elusive goal!). There have been discussions of the similarities
of concepts between the two attacks [18][19] and the analysis of the combination of the
attacks into what is referred to as linear-differential cryptanalysis [20].
Several refinements to the cryptanalyses have attempted to improve the attacks for some
circumstances. Truncated differential cryptanalysis [21] proposes the exploitation of
differences at the cipher output where only some of the ciphertext bits have their
differences predicted. Higher order differential cryptanalysis [21] attempts to exploit
higher order differentials and is applicable to ciphers where the ciphertext bits are
represented as functions of low nonlinear order. Impossible differential cryptanalysis [22]
uses the non-existence of differences to derive cipher subkey bits, as opposed to the
existence of highly likely differences that are exploited in normal differential
cryptanalysis. Also, it should be noted that, in general, in differential cryptanalysis,
differences need not be based on bit-wise exclusive-OR but may be a difference of
another form, such as a difference as in the subtraction of one word of size n from
another modulo 2n [17]. Extensions to linear cryptanalysis have included the utilization of
multiple linear approximations [23] and the use of nonlinear approximations in
combination with linear cryptanalysis [24].
Many papers in recent years have discussed the application of linear and differential
cryptanalysis to ciphers proposed before the existence of the attacks was known. As well,
many techniques in cipher design have been proposed to make the application of the
attacks difficult, focusing on the constructions of cipher components such as S-boxes
[25][8] and the interconnection between layers of S-boxes [8][26][27]. As a result, the
attacks and their extensions are now very well understood and proposals such as Rijndael
[7] have been especially constructed with security against the attacks in mind.
Finally, we note that our presentation of the attacks does not discuss the method for
determining the best linear approximation and differential characteristic. However, this is
discussed, for example, in [28].

30

6. Conclusion
In this paper, we have strived to present the fundamental concepts of linear and
differential cryptanalysis as applied to a basic cipher. This cipher is a basic SubstitutionPermutation Network and is not of a realistic scale to be used as a practical cipher.
However, the structure is useful in examining the applicability of the attacks and this
example cipher has formed the cornerstone for the explanation of the two attacks.

31

7. References
[1] M. Matsui, "Linear Cryptanalysis Method for DES Cipher", Advances in Cryptology
- EUROCRYPT ’93 (Lecture Notes in Computer Science no. 765),
Springer-Verlag, pp. 386-397, 1994.
[2] E. Biham and A. Shamir, "Differential Cryptanalysis of DES-like Cryptosystems",
Journal of Cryptology, vol. 4, no. 1, pp. 3-72, 1991.
[3] National Bureau of Standards, "Data Encryption Standard", Federal Information
Processing Standard 46, 1977.
[4] M. Matsui, "The First Experimental Cryptanalysis of the Data Encryption Standard",
Advances in Cryptology - CRYPTO ’94 (Lecture Notes in Computer Science
no. 839), Springer-Verlag, pp. 1-11, 1994.
[5] E. Biham and A. Shamir, Differential Cryptanalysis of the Data Encryption
Standard, Springer-Verlag, 1993.
[6] National Institute of Standards, Advanced Encryption Standard (AES) web site:
www.nist.gov/aes.
[7] J. Daemen and V. Rjimen, "AES Proposal: Rijndael", First Advanced Encryption
Standard (AES) Conference, California, Aug. 1998. (See also [6].)
[8] H.M. Heys and S.E. Tavares, "Substitution-Permutation Networks Resistant to
Differential and Linear Cryptanalysis", Journal of Cryptology, vol. 9, no.1,
pp. 1-19, 1996.
[9] L. Keliher, "Linear and Differential Cryptanalysis of SPNs", unpublished.
[10] L. Knudsen, "Block Ciphers: A Survey", State of the Art in Applied Cryptography:
Course on Computer Security and Industrial Cryptography (Lecture Notes in
Computer Science no. 1528), Springer-Verlag, pp. 18-48, 1998.
[11] D.R. Stinson, Cryptography: Theory and Practice, CRC Press, 1995.
[12] B. Schneier, Applied Cryptography: Protocols, Algorithms, and Source Code in C,
2nd ed., John Wiley & Sons, 1995.
[13] A. J. Menezes, P.C. van Oorschot, and S.A. Vanstone, Handbook of Applied
Cryptography, CRC Press, 1997.
[14] W. Stallings, Cryptography and Network Security: Principles and Practices,
2nd ed., Prentice Hall, 1999.
[15] H. Feistel, "Cryptography and Computer Privacy", Scientific American, vol. 228,
no. 5, pp. 15-23, 1973.
[16] K. Nyberg, "Linear Approximations of Block Ciphers", Advances in Cryptology
- EUROCRYPT ’94 (Lecture Notes in Computer Science no. 950),
[17] X. Lai, J.L. Massey, and S. Murphy, "Markov Ciphers and Differential
Cryptanalysis", Advances in Cryptology - EUROCRYPT ’91 (Lecture Notes in
[18] E. Biham, "On Matsui’s Linear Cryptanalysis", Advances in Cryptology EUROCRYPT ’94 (Lecture Notes in Computer Science no. 950), Springer-Verlag,
pp. 341-355, 1995.
[19] F. Chabaud and S. Vaudenay, "Links Between Differential and Linear

32

[20] M. Hellman and S. Langford, "Differential-Linear Cryptanalysis", Advances in
Cryptology - CRYPTO ’94 (Lecture Notes in Computer Science no. 839),
[21] L.R. Knudsen, "Truncated and Higher Order Differentials", Fast Software
Encryption (Lecture Notes in Computer Science no. 1008), Springer-Verlag,
pp. 196-211, 1995.
[22] E. Biham, A. Biryukov, and A. Shamir, " Cryptanalysis of Skipjack Reduced to 31
Rounds Using Impossible Differentials", Advances in Cryptology EUROCRYPT ’99 (Lecture Notes in Computer Science no. 1592), SpringerVerlag, pp. 55-64, 1996.
[23] M.J.B. Robshaw and B.S. Kaliski, "Linear Cryptanalysis Using Multiple
Approximations", Advances in Cryptology - CRYPTO ’94 (Lecture Notes in
[24] L. Knudsen and M.J.B. Robshaw, "Nonlinear Approximations in Linear
[25] K. Nyberg, "Differentially Uniform Mappings for Cryptography, Advances in
Cryptology - EUROCRYPT ’93 (Lecture Notes in Computer Science no. 765),
[26] E. De Win, A. Bosselaers, B. Preneel, J. Daemen, and V. Rijmen, "The Cipher
SHARK", Fast Software Encryption (Lecture Notes in Computer Science
no. 1039), Springer-Verlag, pp. 99-112, 1996.
[27] A.M. Youssef, S. Mister, and S.E. Tavares, "On the Design of Linear
Transformations for Substitution Permutation Encryption Networks", Workshop
on Selected Areas of Cryptography (SAC ’96): Workshop Record, pp. 40-48,
1997.
[28] M. Matsui, "On Correlation Between the Order of S-boxes and the Strength of
DES", Advances in Cryptology - EUROCRYPT ’94 (Lecture Notes in Computer
Science no. 950), Springer-Verlag, pp. 366-375, 1995.

33

A Tutorial on Linear and Differential Cryptanalysis by Howard M. Heys

Recommended

More Related Content

What's hot (20)

Viewers also liked (20)

Similar to A Tutorial on Linear and Differential Cryptanalysis by Howard M. Heys (20)

More from Information Security Awareness Group (20)

Recently uploaded (20)

A Tutorial on Linear and Differential Cryptanalysis by Howard M. Heys