Improving Resistance to Differential Cryptanalysis and the Redesign of
LOKI
Lawrence BROWN Matthew KWAN Josef PIEPRZYK
Jennifer SEBERRY
Technical Report CS38/91
Department of Computer Science,
University College, UNSW, Australian Defence Force Academy,
Canberra ACT 2600. Australia.
Abstract
Differential Cryptanalysis is currently the most powerful
tool available for analysing block ciphers, and new block
ciphers need to be designed to resist it. It has been suggested
that the use of S-boxes based on bent functions, with a flat
XOR profile, would be immune. However our studies of
differential cryptanalysis, particularly applied to the LOKI
cipher, have shown that this is not the case. In fact, this
results in a relatively easily broken scheme. We show that an
XOR profile with carefully placed zeroes is required. We also
show that in order to avoid some variant forms of differential
cryptanalysis, permutation P needs to be chosen to prevent
easy propagation of a constant XOR value back into the same
S-box. We redesign the LOKI cipher to form LOKI91, to
illustrate these results, as well as to correct the key schedule
to remove the formation of equivalent keys. We conclude
with an overview of the security of the new cipher.
1 Introduction
Cryptographic research is currently a very active field, with the need for
new encryption algorithms having spurred the design of several new
block ciphers [1]. The most powerful tool for analysing such block
ciphers currently known is differential cryptanalysis. It has been used to
find design deficiencies in many of the new ciphers. Some new design
criteria have been proposed which are claimed to provide immunity to
1
Figure 1: An XOR Profile
differential cryptanalysis. These involve the use of S-boxes based on bent
functions, selected so the resultant box has a flat XOR profile.
In this paper, after presenting a brief introduction to the key concepts in
differential cryptanalysis, we will show that these new criteria do not
provide the immunity claimed, but rather can result in the design of a
scheme which may be relatively easily broken. What we believe is
required, is an S-box with carefully placed zeroes, which significantly
hinder the differential cryptanalysis process.
We continue by documenting our analysis of the LOKI cipher. We report
on a previously discovered differential cryptanalysis attack, faster than
exhaustive search up to 11 rounds, as well as on a new attack, using an
alternate form of analysis, which is faster than exhaustive search up to
14 rounds. We also briefly note some design deficiencies in the key
schedule which resulted in the generation of equivalent keys. This
process highlighted some necessary additional design criteria needed to
strengthen block ciphers against these attacks.
We conclude by describing the redesign of the LOKI cipher, using it to
illustrate the application of these additional design principles, and make
some comments on what we believe is the security of the new scheme.
2 Differential Cryptanalysis
2.1 Overview
Differential Cryptanalysis is a dynamic attack against a cipher, using a
very large number of chosen plaintext pairs, which through a statistical
analysis of the resulting ciphertext pairs, can be used to determine the
key in use. In general, differential cryptanalysis is much faster than
exhaustive search for a certain number of rounds in the cipher, however
there is a breakeven point where it becomes slower than exhaustive
search. The lower the number of rounds this is, the greater the security
2
Figure 2: A 2-round Iterative Characteristic
of the cipher. Differential Cryptanalysis was first described by Biham
and Shamir in [2], and in greater detail in [3]. These described the
general technique, and its application to the analysis of the DES and the
Generalised DES. Subsequent papers by them have detailed its
application to FEAL and N-Hash [4], and to Snefru, Khafre, Redoc-II,
LOKI, and Lucifer [5].
In Differential Cryptanalysis, a table showing the distribution of the
XOR of input pairs against the XOR of output pairs is used to determine
probabilities of a particular observed output pair being the result of
some input pair. The general form of such a table is shown in Fig 1.
To attack a multi-round block cipher, the XOR profile is used to build n
round characteristics, which have a given probability of occurring. These
characteristics specify a particular input XOR, a possible output XOR,
the necessary intermediate XOR's, and the probability of this occurring.
In their original paper [3], Biham and Shamir describe 1,2,3 and 5 round
characteristics which may be used to directly attack versions of DES up
to 7 rounds. Knowing a characteristic, it is possible to infer information
about the outputs for the next two rounds. To utilise this attack, a
number of pairs of inputs, having the nominated input XOR, are tried,
until an output XOR results which indicates that the pattern specified in
the characteristic has occurred. Since an n round characteristic has a
probability of occurrence, for most keys we can state on average, how
many pairs of inputs need to be trialed before the characteristic is
successfully matched. Once a suitable pair, known as a right pair, has
been found, information on possible keys which could have been used, is
deduced. Once this is done we have two plaintext-ciphertext pairs. We
3
know from the ciphertext, the input to the last round. Knowing the
input XOR and output XOR for this round, we can thus restrict the
possible key bits used in this round, by considering those outputs with
an XOR of zero, providing information on the outputs of some of the
S-boxes. By then locating additional right pairs we can eventually either
uniquely determine the key, or deduce sufficient bits of it that an
exhaustive search of the rest may be done.
N round characteristics can be concatenated to form longer
characteristics if the output of the first supplies the input to the second,
with probabilities multiplied together. A particularly useful
characteristic is one whose output is a swapped version of its input, and
which hence may be iterated with itself. This may be used to analyse an
arbitrary number of rounds of the cipher, with a steadily increasing work
factor. A particularly useful form is one where a non-zero input XOR to
the F function results in a zero output XOR. Such a characteristic is
illustrated in Fig 2, and may be denoted as:
A. (x, 0) -> (0, x) always (ie Pr=1)
B. (0, x) -> (x, 0) with some probability p
It may be iterated as A B A B A B A B to 8 rounds for example, with
characteristic probability p4. This form of characteristic is then used in
the analysis of arbitrary n round forms of a cipher, until the work factor
exceeds exhaustive search. These techniques are described in detail in [3].
2.2 Why Flat XOR profiles Don't Work
Given the success of differential cryptanalysis in the analysis of block
ciphers, it has become important to develop design criteria to improve
the resistance of block ciphers to it, especially with several candidates
having performed poorly. Dawson and Tavares [6] have proposed that the
selection of S-boxes with equal probabilities for each output XOR given
an input XOR (except input 0) would result in a cipher that was immune
to differential cryptanalysis (see Fig. 3). However a careful study of
Biham and Shamir's attack on the 8 round version of DES [3], confirmed
by our own analyses of LOKI, have shown that this is not the case.
Indeed the selection of such S-boxes results in a cipher which is
significantly easier to cryptanalyse than normal. This is done by
constructing a 2 round iterative characteristic of the form in Fig. 2, with
an input XOR that changes bits to one S-box only. We know we can do
this, since the flat XOR profile implies that an output XOR of 0 for a
specified input XOR will occur with P r(1=2n ). When iterated over 2k
rounds, this will have a probability of P r(1=2(k-1)n ), since you get the
last round for free. Consider a 16 round DES style cryptosystem, but
with S-boxes having a flat XOR profile of the form in Fig. 3 with m = 6,
4
Figure 3: Flat XOR Profile
n = 4, and k = 8. This may be attacked by a 15-round characteristic,
chosen to alter inputs to a single S-box only. This gives a probability to
break with a given test pair of P r(1=228), implying that about 228 pairs
need to be tried to break the cipher, far easier than by exhaustive search.
2.3 Significance of Permutation P
Although differential cryptanalysis may be done independent of
permutation P, knowledge of a particular P may be used to construct
some other useful n round characteristics for cryptanalysing a particular
cipher. The most useful of these take the form of a 2 or 3 round
characteristic which generate an output XOR identical to the input
XOR, either directly in 2 rounds, or by oscillating between two alternate
XOR values over 3 rounds. The 3 round characteristic is sensitive to the
form of permutation P . This form of characteristic has been found in the
original version of LOKI, as detailed below. It thus indicates that care is
needed in the design of not just the S-boxes, but of all elements in
function F , in order to reduce susceptibility to differential cryptanalysis.
3 Analysis of LOKI
3.1 Overview
LOKI is one of several recently proposed new block ciphers. It was
originally detailed by Brown, Pieprzyk and Seberry in [7]. Its overall
structure is shown in Fig. 4. We will refer to this version of the cipher as
LOKI89 in the remainder of this paper. It is a 16 round Feistel style
cipher, with a relatively straightforward key schedule, a non-linear
function F = P (S(E(R K))), and four identical 12-to-8 bit S-boxes.
Permutation P in LOKI has a regular structure which distributes the 8
output bits from each S-box to boxes [+3 + 2 + 1 0 + 3 + 2 + 1 0] in
the next round. Its S-box consists of 16 1-1 functions, based on
5
Figure 4: LOKI89 Overall Structure and S-box Detail
exponentiation in a Galois field GF (28) (see [8]), with the general
structure shown in Fig. 4.
3.2 Security of LOKI89
Initial testing of the statistical and complexity properties of LOKI89
indicated that its properties were very similar to those exhibited by
DES, FEAL8 and Lucifer [9], results that were very encouraging. Initial
examination of the XOR profile of the LOKI89 S-box also suggested that
it should be more resistant than DES to differential cryptanalysis.
LOKI89 was then analysed in detail using differential cryptanalysis.
Biham [5] describes an attack, using a 2 round iterative characteristic
with P r(118=220) P r(2-13:12 ), with non-zero inputs to 2 S-boxes
resulting in the same output. There are four related variants (by
rotation) of the form:
A. (00000510,00000000) -> (00000000,00000510) always
B. (00000000,00000510) -> (00000510,00000000) Pr(118/2^20)
6
This characteristic is iterated to 8 rounds with P r(2-52:48 ), allowing up
to 10 rounds to be broken faster than by exhaustive key space search.
This is a significantly better result than for the DES. The authors have
verified this attack.
The authors have subsequently found an alternate 3 round iterative
characteristic, attacking S-box 3, with an output XOR identical to the
input XOR. Since permutation P in LOKI89 permutes some output bits
back to the same S-box in the next round, this allows the characteristic
to be iterated. It has the form:
A. (00400000,00000000) -> (00000000,00400000) always
B. (00000000,00400000) -> (00400000,00400000) Pr(28/4096)
C. (00400000,00400000) -> (00400000,00000000) Pr(28/4096)
Since 28=4096 2-7:2 if we concatenate these characteristics, we get a
13 round characteristic in the order A B C A B C A B C A B C A with
a probability of P r(2-7:2*8 ) P r(2-57:6 ). This may be used to attack a
14 round version of LOKI89, and requires O(259) pairs to succeed. This
is of the same order as exhaustive search (which is of O(260), as detailed
below), and is thus a more successful attack than that reported
previously. It has been verified by Biham. This still leaves the full 16
round version of LOKI89 secure, but with a reduced margin against that
originally believed.
Independently, the authors [10], Biham [5], and the members of the
RIPE consortium have discovered a weakness in the LOKI89 key
schedule. It results in the generation of 15 equivalent keys for any given
key, effectively reducing the key-space to 260 keys. A complementation
property also exists which results in 256 (key, plain, cipher) triples being
formed, related by LOKI(P; K) pppppppppppppppp =
LOKI(P pppppppppppppppp; K mmmmmmmmnnnnnnnn), where
p = m n for arbitrary hex values m; n. This may be used to reduce the
complexity of a chosen plaintext attack by an additional factor of 16.
These results were found by analysing the key schedule by regarding
each S-box input as a linear function of the key and plaintext, and
solving to form (key, plaintext) pairs which result in identical S-box
input values. This lead to solving the following equations:
RD KRD n:ROT 12(KLD) = 0 (1)
LD KLD n:ROT 12(KRD) = 0 (2)
where LD = L0 L, RD = R0 R, KLD = KL0 KL, and
KRD = KR0 KR describe the difference between the related (key,
plaintext) pairs. This method is detailed by Kwan in [10].
In the light of these results, the authors have devised some additional
design guidelines to those originally used in the design of LOKI, and
have applied them in the development of a new version, LOKI91.
7
4 Redesign of LOKI
4.1 Some Additional Design Guidelines
To improve the resistance of a cipher to differential cryptanalysis, and to
remove problems with the key schedule, the following guidelines were
used:
o analyse the key schedule to minimize the generation of equivalent
key, or related (key, plaintext) pairs.
o minimise the probability that a non-zero input XOR results in a
zero output XOR, or in an identical output XOR, particularly for
inputs that differ in only 1 or 2 S-boxes.
o ensure the cipher has sufficient rounds so that exhaustive search is
the optimal attack (ie have insufficient pairs to do differential
cryptanalysis).
o ensure that there is no way to make all S-boxes give 0 outputs, to
increase the ciphers security when used in hashing modes.
These criteria were used when selecting the changes made to the LOKI
structure, detailed below.
4.2 Design of LOKI91
LOKI91 is the revised version, developed to address the results detailed
above. Changes have been made to two aspects of the LOKI structure.
Firstly the key schedule has been amended in several places to
significantly reduce the number of weak keys. Secondly the function used
in the S-boxes has been altered to improve its utility in hashing
applications, and to improve its immunity to differential cryptanalysis.
In more detail, the four changes made to the original design were:
1. change key schedule to swap halves after every second round
2. change key rotations to alternate between ROT13 and ROT12
3. remove initial and final XORs of key with plaintext and ciphertext
4. alter the S-box function used in the LOKI S-box (Fig. 4) to
Sf n(row; col) = (col + ((row * 17) f f16)&f f16)31mod grow (3)
where + and * refer to arithmetic addition and multiplication, is
addition modulo 2, and the exponentiation is performed in GF (28).
The generator polynomials used (grow ) are as for LOKI89 [7].
8
Figure 5: LOKI91 Overall Structure
9
___________________________________________________
|__Encrypt__Key________|__Decrypt__Key___________|_
| 0000000000000000 | 0000000000000000 * |
| 00000000aaaaaaaa | aaaaaaaa00000000 |
| 0000000055555555 | 5555555500000000 |
| 00000000ffffffff | ffffffff00000000 |
| aaaaaaaa00000000 | 00000000aaaaaaaa |
| aaaaaaaaaaaaaaaa | aaaaaaaaaaaaaaaa * |
| aaaaaaaa55555555 | 55555555aaaaaaaa |
| aaaaaaaaffffffff | ffffffffaaaaaaaa |
| 5555555500000000 | 0000000055555555 |
| 55555555aaaaaaaa | aaaaaaaa55555555 |
| 5555555555555555 | 5555555555555555 * |
| 55555555ffffffff | ffffffff55555555 |
| ffffffff00000000 | 00000000ffffffff |
| ffffffffaaaaaaaa | aaaaaaaaffffffff |
| ffffffff55555555 | 55555555ffffffff |
|__ffffffffffffffff__|____ffffffffffffffff__*__|___
Table 1: LOKI91 Weak and semi-weak key pairs
The overall structure of LOKI91 that results from these changes is
shown in Fig. 5.
The key schedule changes remove all but a single bit complementation
property, leaving an effective search space of 263 keys under a
chosen-plaintext attack. It reduces key equivalences to a single bit
complementation_property,_similar_to that of the DES where
LOKI(P ; K ) = LOKI(P; K) , by reducing the solutions to each of Eq. 1
and Eq. 2 to one, and removing the independence between them by
altering the swapping to every two rounds. It also greatly reduces the
number of weak and semi-weak keys to those shown in Table 1 (where an
* denotes weak keys). The DES also has 16 weak and semi-weak keys.
The removal of the initial and final XORs became necessary with the
change in the swap locations, since otherwise it would have resulted in
cancellation of the keys bits at the input to E in some rounds. This
change does affect the growth of ciphertext dependence on keys bits (see
[11]), increasing it from 3 to 5 rounds. This still compares favorably with
the DES which takes either 5 or 7 rounds, dependent on the type of
dependency analysed. This change also greatly assisted in the reduction
of the number of equivalent keys.
The new Sfn uses arithmetic addition and multiplication, as these are
non-linear when used in GF (28). The addition modulo two of the row
10
with f f16 ensures that an all zero input gives a non-zero output, thus
removing the major deficiency of LOKI89 when used as a hash function.
The result of the addition and multiplication is masked with f f16 to
restrict the value to lie with GF (28), prior to the exponentiation in that
field. The new function reduces the probabilities of occurance of n round
iterative characteristics, useful for differential cryptanalysis, to be as low
as possible. With the new function, LOKI91 is theoretically breakable
faster than an exhaustive key space search in:
o up to 10 rounds using a 2 round characteristic with the
f (x0)- > 00 mapping occuring with P r(122=1048576)
o up to 12 rounds using a 3 round characteristic with the
f (x0)- > x0 mapping occuring with P r(16=4096) (used twice)
At 16 rounds, cryptanalysis is generally impossible, as insufficient pairs
are available to complete the analysis. It would require:
o 280 pairs using the 3 round characterisitic, or
o 292 pairs using the 2 round characteristic
compared to a total of 263 possible plaintext pairs.
5 Conclusion
In this paper, we have shown that a flat XOR profile does not provide
immunity to differential cryptanalysis, but in fact leads to a very
insecure scheme. Instead a carefully chosen XOR profile, with suitably
placed 0 entries is required to satisfy the new design guidelines we have
identified. We also note an analysis of key schedules, which can be used
to determine the number of equivalent keys. We conclude with the
application of these results to the design of LOKI91.
6 Acknowledgements
Thank you to the members of the crypt group for their support and
suggestions. This work has been supported by ARC grant A48830241,
ATERB, and Telecom Australia research contract 7027.
Appendix A - Specification of LOKI91
Encryption
The overall structure of LOKI91 is shown in Fig. 5, and is specified as
follows. The 64-bit input block X is partitioned into two 32-bit blocks L
11
and R. Similarly, the 64-bit key is partitioned into two 32-bit blocks KL
and KR.
L0 = L KL0 = KL
R0 = R KR0 = KR (4)
The key-dependent computation consists (except for a final interchange
of blocks) of 16 rounds (iterations) of a set of operations. Each iteration
includes the calculation of the encryption function f . This is a
concatenation of a modulo 2 addition and three functions E, S, and P .
Function f takes as input the 32-bit right data half Ri-1 and the 32-bit
left key half KLi produced by the key schedule KS (denoted Ki below),
and which produces a 32-bit result which is added modulo 2 to the left
data half Li-1 . The two data halves are then interchanged (except after
the last round). Each round may thus be characterised as:
Li = Ri-1
Ri = Li-1 f (Ri-1 ; KLi) (5)
f (Ri-1; Ki) = P (S(E(Ri-1 Ki)))
The component functions E, S, and P are described later.
The key schedule KS is responsible for deriving the sub-keys Ki, and is
defined as follows: the 64-bit key K is partitioned into two 32-bit halves
KL and KR. In each round i, the sub-key Ki is the current left half of
the key KLi-1 . On odd numbered rounds (1, 3, 5, etc), this half is then
rotated 12 bits to the left. On even numbered rounds (2, 4, 6, etc), this
half is then rotated 13 bits to the left, and the key halves are
interchanged.
This may be defined for odd numbered rounds as:
Ki = KLi-1
KLi = ROL(KLi-1; 13) (6)
KRi = KRi-1
This may be defined for even numbered rounds as:
Ki = KLi-1
KLi = KRi-1 (7)
KRi = ROL(KLi-1; 12)
Finally after the 16 rounds, the two output block halves L16 and R16 are
then concatenated together to form the output block Y . This is defined
as (note the swap of data halves to undo the final interchange in Eq.5):
Y = R16 | L16 (8)
12
___________________________________________________________________
| 3 2 1 0 31 30 29 28 27 26 25 24 |
| 27 26 25 24 23 22 21 20 19 18 17 16 |
| 19 18 17 16 15 14 13 12 11 10 9 8 |
|__11__10____9_____8____7_____6____5_____4____3_____2____1_____0__|_
Table 2: LOKI Expansion Function E
Decryption
The decryption computation is identical to that used for encryption, save
that the partial keys used as input to the function f in each round are
calculated in reverse order. The rotations are to the right, and an initial
pre-rotation of 8 places is needed to form the key pattern.
Function f
The encryption function f is a concatenation of a modulo 2 addition and
three functions E, S, and P , which takes as input the 32-bit right data
half Ri-1 and the 32-bit left key half KLi, and produces a 32-bit result
which is added modulo 2 to the left data half Li-1 .
f (Ri-1; Ki) = P (S(E(Ri-1 Ki))) (9)
The modulo 2 addition of the key and data halves ensures that the
output of f will be a complex function of both of these values.
The expansion function E takes a 32-bit input and produces a 48-bit
output block, composed of four 12-bit blocks which form the inputs to
four S-boxes in function f . Function E selects consecutive blocks of
twelve bits as inputs to S-boxes S(4), S(3), S(2), and S(1) respectively, as
follows:
[b3 b2 ::: b0 b31 b30 ::: b24]
[b27 b26 ::: b16]
[b19 b18 ::: b8]
[b11 b10 ::: b0]
This is shown in full in Table 2 which specifies the source bit for outputs
bits 47 to 0 respectively:
The substitution function S provides the confusion component in the
LOKI cipher. It takes a 48-bit input and produces a 32-bit output. It is
composed of four S-boxes, each of which takes a 12-bit input and
produces an 8-bit output, which are concatenated together to form the
32-bit output of S. The 8-bit output from S(4) becomes the most
significant byte (bits [31...24]), then the outputs from S(3) (bits[23...16]),
S(2) (bits[15...8]), and S(1) (bits [7...0]). In LOKI91 the four S-boxes are
13
___________________________
|__Row__|_|genrow__|_erow__|
| 0 | | 375 | 31 |
| 1 | | 379 | 31 |
| 2 | | 391 | 31 |
| 3 | | 395 | 31 |
| 4 | | 397 | 31 |
| 5 | | 415 | 31 |
| 6 | | 419 | 31 |
| 7 | | 425 | 31 |
| 8 | | 433 | 31 |
| 9 | | 445 | 31 |
| 10 | |451 | 31 |
| 11 | |463 | 31 |
| 12 | |471 | 31 |
| 13 | |477 | 31 |
| 14 | |487 | 31 |
|___15___|_|499____|__31___|
Table 3: LOKI S-box Irreducible Polynomials and Exponents
identical. The form of each S-box is shown in Fig 4. The 12-bit input is
partitioned into two segments: a 4-bit row value row formed from bits
[b11 b10 b1 b0], and an 8-bit column value col formed from bits
[b9 b8 ::: b3 b2]. The row value row is used to select one of 16 S-functions
Sf nrow (col), which then take as input the column value col and produce
an 8-bit output value. This is defined as:
Sf nrow (col) = (col + ((row * 17) f f16)r&f f16)erow mod grow (10)
where genrow is an irreducible polynomial in GF (28), and erow is the
(constant 31) exponent used in forming Sf nrow (col). The generators and
exponents to be used in the 16 S-functions in LOKI91 are specified in
Table 3. For ease of implementation in hardware, this function can also
be written as:
Sf nrow (col) = (col + ((_____row)|(_____row<< 4))&f f16)31mod grow (11)
The permutation function P provides diffusion of the outputs from the
four S-boxes across the inputs of all S-boxes in the next round. It takes
the 32-bit concatenated outputs from the S-boxes, and distributes them
over all the inputs for the next round via a regular wire crossing which
takes bits from the outputs of each S-box in turn, as defined in Table 4
which specifies the source bit for output bits 31 to 0 respectively.
14
__________________________________________
| 31 23 15 7 30 22 14 6 |
| 29 21 13 5 28 20 12 4 |
| 27 19 11 3 26 18 10 2 |
|__25__17____9____1____24___16____8____0__|
Table 4: LOKI Permutation P
Test Data
A single test triplet for the LOKI91 primitive is listed below.
# Single LOKI91 Certification triplet
# data is saved as (key, plaintext, ciphertext) hex triplets
#
3849674c2602319e 126898d55e911500 c86caec1e3b7b17e
15
References
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[3] E. Biham and A. Shamir, ``Differential Cryptanalysis of
DES-like Cryptosystems," Weizmann Institute of Science,
Rehovot, Israel, Technical Report, 19 July 1990.
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[9] L. Brown, J. Pieprzyk, R. Safavi-Naini and J. Seberry, ``A
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[10] M. Kwan and J. Pieprzyk, ``A General Purpose Technique for
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16
[11] L. Brown and J. Seberry, ``Key Scheduling in DES Type
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Springer Verlag, pp. 221--228, 1990.
17
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