Stream Ciphers
RSA Laboratories Technical Report TR 701
Version 2.0 July 25 1995
M.J.B. Robshaw
matt rsa. com
RSA Laboratories
100 Marine Parkway
Redwood City CA 94065 1031
c
Copyright 1995 RSA Laboratories a division of RSA Data Security Inc.
All rights reserved.
003 903040 200 000 000
i
Contents
1 Introduction 1
2 General background 2
3 Classi cation 3
4 Analysis 5
4.1 Appearance 5
4.1.1 Period 5
4.1.2 Statistical measures 7
4.2 Measures of complexity 8
4.2.1 Linear complexity 8
4.2.2 Other measures of complexity 10
4.3 Some theoretical results 12
5 Congruential generators 13
6 Shift register based schemes 14
6.1 Linear feedback shift registers 14
6.2 Combination and lter generators 16
6.2.1 Correlation attacks 16
6.2.2 Two weak generators 17
6.2.3 Boolean functions 18
6.2.4 Three more attacks 19
6.3 Multiplexers 19
6.4 Clock control 20
6.4.1 Stop and go with variants 21
6.4.2 Cascades 22
6.5 Shrinking and self shrinking generator 23
6.6 Summation generator 24
7 Alternative designs 25
7.1 RC4 25
7.2 SEAL 25
7.3 Number theoretic techniques 26
7.4 Other schemes 27
7.4.1 1 p generator 27
7.4.2 Knapsack generator 27
7.4.3 PKZIP 28
ii Stream Ciphers
7.5 Final examples 28
7.5.1 Randomized ciphers 28
7.5.2 Cellular automata 29
8 Conclusions 29
1. Introduction 1
1 Introduction
Cryptosystems are divided between those that are secret key or symmetric
and those that are public key or asymmetric . With the latter the sender
uses publicly known information to send a message to the receiver. The
receiver then uses secret information to recover the message. In secret
key cryptography the sender and receiver have previously agreed on some
private information that they use for both encryption and decryption. This
information must be kept secret from potential eavesdroppers.
There is a further division of symmetric cryptosystems into block ciphers
and stream ciphers. The distinction between block and stream ciphers is
perhaps best summarized by the following quotation due to Rueppel 126
Block ciphers operate with a xed transformation on large blocks
of plaintext data stream ciphers operate with a time varying
transformation on individual plaintext digits.
In this technical report we provide a review of current stream cipher
techniques. Anyone looking through the cryptographic literature will be
struck by a great di erence in the treatment of block ciphers and stream
ciphers. Practically all work in the cryptanalysis of block ciphers is focused
on DES 94 and nearly all the proposed block ciphers are based in some way
on the perceived design goals of DES. There is no algorithm occupying an
equivalent position in the eld of stream ciphers. There are a huge variety
of alternative stream cipher designs and cryptanalysis tends to be couched
in very general terms.
While much of this distinction might be attributed to the publication of
DES as a Federal Standard and the subsequent high pro le of this algorithm
within both the business and cryptographic communities there may well be
an additional and more subtle factor to consider.
We will see in this report that many stream ciphers have been proposed
which use very basic building blocks. The mathematical analysis of these
components has been very advanced for some considerable time see Section
6.1 and intensive design and cryptanalysis over the years has resulted in the
formulation of a set of ground rules for the design of stream ciphers. It is
well known that highly developed analytic techniques facilitate both design
and cryptanalysis.
No such well developed list could be given until very recently for block
ciphers. The design criteria for DES were not published and cryptanaly
sis was for a long time frustratingly unsuccessful. There seemed to be
2 Stream Ciphers
little general theory available. With the advent of di erential cryptanal
ysis 7 and more recently linear cryptanalysis 80 both designers and
cryptanalysts had new and clear cut issues to consider and there has been
considerable recent activity in both the design and analysis of block ciphers.
Curiously research into stream ciphers seems to be a predominantly Eu
ropean a air. By comparing the proceedings of the two major cryptography
conferences we often see that Eurocrypt meetings dedicate several sessions
to stream cipher issues whereas a single session is more often the norm at the
US Crypto meetings. This imbalance in interest may well be a by product
of the pre occupation with DES in the US however an increased pro le for
stream ciphers can be expected as more developers look to stream ciphers
to provide the encryption speeds they need.
This report aims to provide a snapshot of the di erent techniques avail
able today and to report on the direction and status of both prior and
contemporary research. We will avoid considerable depth on the di erent
topics since we aim to cover the ground and prefer to point the reader
to sources of further details. For a more detailed source of information on
stream ciphers we strongly recommend the excellent article by Rueppel in
Contemporary Cryptology 126 though there are also other survey articles
in the literature 51 138 .
2 General background
Much of the popularity of stream ciphers can undoubtedly be attributed to
1
the work of Shannon in the analysis of the one time pad originally known
as the Vernam cipher 130 .
The one time pad uses a long string of keystream which consists of bits
that are chosen completely at random. This keystream is combined with
the plaintext on a bit by bit basis. The keystream is the same length as
the message and can be used only once as the name one time pad implies
clearly a vast amount of keystream might be required. We write the plain
text message m as a sequence of bits m m0 m1 mn 1 and the binary
keystream k which is the same length as the message as k k0k1 kn 1.
The ciphertext c c0c1 cn 1 is de ned by ci mi ki for 0 i n 1
where denotes bitwise exclusive or.
In his seminal paper 125 Shannon proved what many had previously
believed namely that the one time pad is unbreakable . In fact Shannon
1
This name was adopted following its use during the Second World War with the help
of a paper pad.
3. Classi cation 3
described the cryptosystem as being perfect even an adversary with an
in nite amount of computing power is unable to do better than guess the
value of a message bit since the ciphertext is statistically independent of the
plaintext.
Because of the practical problems involved with a system requiring such
a vast amount of key information the Moscow Washington hotline used
to be cited as perhaps the only place where the requirements for secrecy
outweighed the problems of key management. Somewhat disappointingly
Massey reports 126 that this is no longer the case and a conventional
secret key cipher requiring much less key is used instead.
A stream cipher attempts to capture the spirit of the one time pad by
using a short key to generate the keystream which appears to be random.
Sucha keystream sequence is often described as pseudo random and deciding
what constitutes a pseudo random sequence forms much of the work in the
eld of stream ciphers. We will say that the keystream is generated by the
keystream generator other terms in the literature include pseudo random
sequence generator and running key generator.
Stream ciphers can be very fast to operate they are generally much faster
than block ciphers. Since the keystream can often be generated indepen
dently of the plaintext or ciphertext such generators often have the advan
tage that the keystream can be generated prior to encryption or decryption
with only an easy combining step left when the message or ciphertext is to
be processed.
3 Classi cation
We will very informally describe the state of a cryptosystem as the values
of a set of variables that together provide a unique description of the status
of the device.
When we design a stream cipher there are essentially two concerns. The
rst is how to describe the next state of the cryptosystem in terms of the
current state and the second is how to express the ciphertext in terms of
the plaintext and the state.
The second issue is perhaps the easiest to solve since almost invariably
the ciphertext is expressed as the bit wise exclusive or of the plaintext and
a function of the state of the cryptosystem. Combining functions other
than exclusive or might also be considered. The sequence generated by the
function of the state of the cryptosystem is conventionally known as the
keystream.
4 Stream Ciphers
The choice of the expression of the next state of the cryptosystem which
constituted our rst design decision provides us with a classi cation of
stream ciphers into two types.
If the next state of the cryptosystem is de ned independently of both
plaintext and ciphertext then the stream cipher is termed synchronous.
In such a scheme each plaintext bit is encrypted independently of the
others and the corruption of a bit of the ciphertext during transmission will
not a ect the decryption of other ciphertext bits. The cipher is described
as having no error propagation and though this appears to be a desirable
property it has several implications. First it limits the opportunity to
detect an error when decryption is performed but more importantly an
attacker is able to make controlled changes to parts of the ciphertext knowing
full well what changes are being induced on the corresponding plaintext.
Of more practical signi cance both the encrypting and decrypting units
must remain in step since decryption cannot proceed successfully unless the
keystreams used to encrypt and decrypt are synchronized. Synchronization
is usually achieved by including marker positions in the transmission the
net e ect being that a bit of ciphertext missed during transmission results
in incorrect decryption until one of the marker positions is received.
In contrast self synchronizing or asynchronous stream ciphers have the
facility to resume correct decryption if the keystream generated by the de
crypting unit falls out of synchronization with the encrypting keystream.
For these stream ciphers the function that de nes the next state of the
cryptosystem takes as input some of the previously generated ciphertext.
The most common example of this is provided by some block cipher in what
is termed cipher feedback CFB mode 95 .
Suppose the encryption of a bit depends on c previous ciphertext bits.
The system demonstrates limited error propagation if one bit is received
incorrectly then decryption of the following c bits may be incorrect. Ad
ditionally however the system is able to resynchronize itself and produce
a correct decryption after c bits have been received correctly. This makes
such ciphers suitable for applications where synchronization is di cult to
maintain.
Self synchronizing stream ciphers have some limited error propagation
which may or may not be viewed as an advantage. Certainly any changes
made by an attacker to the ciphertext will have additional consequences on
other parts of the decrypted plaintext. However Rueppel suggests 114 that
there are two drawbacks to self synchronizing stream ciphers.
First an opponent knows some of the variables being used as input to
the generator since this input is taken from the ciphertext. Second these
4. Analysis 5
generators have a limited analyzability because the keystream depends on
the message. Nevertheless the design of self synchronizing stream ciphers
has been addressed to a limited extent in the literature 101 26 and Maurer
81 has provided some framework for a general assessment of the security
o ered by these stream ciphers. The rest of this report is concerned with
synchronous stream ciphers or keystream generators.
4 Analysis
There are many di erent considerations we must keep in mind when we
consider the suitability of a keystream generated by some stream cipher.
The criteria we list here provide only some of the necessary conditions for the
security of the keystream a keystream might well satisfy all these conditions
and yet still be vulnerable to some attack.
Over the years a vast number of di erent considerations have been high
lighted and they seem to fall into one of two camps. The rst group are
used to assess the appearance of the keystream is there some imbalance to
the way the sequence is generated that allows a cryptanalyst to guess the
next bit with some probability better than that of random guessing
The second group of criteria address the ability of a cryptanalyst to use
the bits of the keystream he might already have to construct a sequence
that replicates the keystream. In some way we are considering the inherent
complexity of the sequence and attempting to answer the question is it
hard to reproduce the sequence
Finally Section 4.3 describes attempts to provide a rm theoretical basis
for the security o ered by stream ciphers.
4.1 Appearance
4.1.1 Period
The rst attribute of a sequence and one of the most important to consider
is the length of the period. The usual model for a keystream generator is
provided by a nite state machine .
In this model the machine is regularly clocked and at each clocking in
stant the internal state of the machine is updated in a way that is determined
by its current state. At the same time some of the keystream is output. Since
there are a nite number of states available it is clear that eventually some
internal state will occur twice. Since the successor state and the output are
determined by the current state then from the point of repetition on the
6 Stream Ciphers
keystream will be identical to that produced when the same state previously
occurred.
If the period of the keystream is too short then di erent parts of the
plaintext will be encrypted in an identical way and this constitutes a severe
weakness. Knowledge of the plaintext allows recovery of the corresponding
portion of the keystream and the cryptanalyst can then use the fact that
this portion of keystream is used elsewhere in the encryption to successfully
decrypt the ciphertext. Additionally if only the ciphertext parts are re
ceived then they can be combined to give a stream of data that equals the
combination of two plaintext messages and is independent of the key. The
underlying statistics of the plaintext source might then be used to derive
both the plaintexts and the keystream.
During the Second World War the Lorenz SZ 42 cipher machine was
used to encrypt messages sent from German High Command to various Army
Commands. According to Good 57 who worked on the cryptanalysis of
this cipher one of the biggest single advances came when a radio operator
used the same keystream twice to encrypt two di erent messages. This
allowed cryptanalysts to construct a machine which mimicked the action of
the SZ 42 and paved the way for the subsequent successful cryptanalysis of
this cipher.
The question of how large a period is required for a sequence is open
to debate and depends on the application in mind. We note however that
with a stream cipher encrypting at a speed of 1 Mbyte sec a sequence with
period 232 will repeat itself after only 29 seconds or 8 5 minutes. This would
not generally be considered adequate.
When the modes of use for the DES block cipher were rst published 95
some exibility was provided in one of the parameters for the OFB mode
this uses the block cipher as a keystream generator. An important parameter
was allowed to vary from 8 bits through to 64. It was soon discovered that for
all choices except 64 the period of the generated keystream was very likely
to be around 232 which as we have previously noted is inadequate. Instead
the period of the keystream generated when using 64 as the parameter is
around 263 which is more acceptable. Consequently DES should only be
used in the OFB mode with a feedback of 64 bits 29 .
A good assessment of the period of the keystreams generated by a
keystream generator is essential to the design of any stream cipher. Practi
cally the keystream should be long enough to ensure that it is overwhelm
ingly unlikely that the same portion of keystream is used twice during en
cryption.
On a technical point we note that many of the theoretical results in the
4. Analysis 7
later sections are obtained by considering what is termed a period of the
sequence. Most often the period of some sequence refers to the number of
bits before the sequence recurs. At other times however a period is said to
consist of p successive bits of the sequence where p is the length of the period.
The context provided by the text should avoid any confusion between these
two uses.
4.1.2 Statistical measures
When repeatedly tossing a fair coin one expects to see roughly as many
heads as tails. In a similar fashion many other properties can be formulated
to describe the appearance of a sequence that is purported to be generated
by a totally random source. One of the rst formulations of some basic
ground rules for the appearance of periodic pseudo random sequences was
provided by Golomb 52 and these three rules have come to be known as
Golomb s postulates.
G1 The number of 1 s in every period must di er from the number of 0 s
by no more than one.
G2 In every period half the runs must have length one one quarter must
have length two one eighth must have length three etc. as long as the
number of runs so indicated exceeds one. Moreover for each of these
lengths there must be equally many runs of 1 s and of 0 s.
G3 Suppose we have two copies of the same sequence of period p which
are o set by some amount d. Then for each d 0 d p 1 we can
count the number of agreements between the two sequences Ad and
the number of disagreements Dd. The auto correlation coe cient for
each d is de ned by Ad Dd p and the auto correlation function
takes on several values as d ranges through all permissible values.
For a sequence to satisfy G3 the auto correlation function must be
two valued.
G3 is a technical expression of what Golomb has described as the notion
of independent trials that knowing some previous value in the sequence is
essentially of no help in deducing the current value. Another view of the
auto correlation function is that it is some measure of the ability of being
able to distinguish between a sequence and a copy of the same sequence that
has been started at some other point in the period.
8 Stream Ciphers
A sequence satisfying G1 G3 is often termed a pn sequence where pn
stands for pseudo noise. However it is clear that these rules on their own
are not su cient to capture the full signi cance of a random looking sequence
and a wide range of di erent statistical tests can be applied to a sequence
to assess how well it ts the assertion that it was generated by a perfectly
random source 4 32 67 114 .
We expand a little on quite what we mean when we test a keystream
generator. Suppose a sequence of length p is generated at random and this
nite sequence of p bits is repeated to form a periodic sequence. Such a
sequence is sometimes called a semi in nite sequence. If the p bits were
generated completely at random then any pattern of the p bits would be
equally likely. In particular the sequence consisting of p zero bits which
are then repeated would be as likely to occur as any other. When we
test the generator we test many sequences individually and assess what
proportion of the sequences generated fail the tests we apply. If the failure
rate is comparable to that expected for sequences generated using a perfectly
random source then we pass the generator. Now of course for cryptographic
purposes even sequences generated by a perfectly random source might be
wholly unsuitable for encryption such as the example given above and
so the design of the generator should ensure that catastrophically weak
sequences can never be generated.
4.2 Measures of complexity
As we mentioned previously a great deal of work has centered on providing
an adequate measure of how hard a sequence might be to replicate. The
most popular technique by far is the linear complexity we shall describe
this next. Meanwhile attempts to develop either new techniques or more
general measures of complexityhave also had some success we shall describe
some of these approaches in Section 4.2.2.
4.2.1 Linear complexity
One of the most far reaching papers in stream cipher analysis is due to
Massey 77 . In this paper an algorithm now called the Berlekamp Massey
algorithm is described which identi es the shortest linear recurrence that
can be used to generate a nite binary sequence. Since a linear recurrence
can be implemented using a linear feedback shift register see Section 6.1
this result is often described in terms of the shortest linear feedback shift
register that can be used to generate a sequence.
4. Analysis 9
Every sequence s0s1 of period p satis es a linear recurrence of length
p namely si p si for all i 0. A sequence may additionally satisfy a
shorter recurrence that is each bit of the sequence can be de ned using
some linear expression which involves bits that are less than p bits away.
The length of the shortest recurrence is de ned to be the linear complexity
or linear span of the sequence.
Given a nite sequence the Berlekamp Massey algorithm can be used
to calculate this recurrence over what is mathematically termed a eld an
example of a eld is the set of binary numbers under the operations of
addition and multiplication modulo two hence its applicability to pseudo
random bit generators. An extension of the Berlekamp Massey algorithm is
provided by Reeds and Sloane 103 which acts over more general number
systems those that form what are mathematically called rings.
While the Berlekamp Massey algorithm calculates the linear complexity
of a nite sequence its use can be easily extended to periodic or semi in nite
sequences. It is a very important algorithm since the linear recurrence satis
ed by a sequence with linear complexity k can be e ciently calculated after
observing 2k consecutive bits of the sequence. Since the linear recurrence
also de nes a linear feedback shift register it o ers some indication for how
di cult a sequence might be to replicate. A high linear complexity means
that more of the sequence has to be observed before the recurrence can be
identi ed and that a longer register is required to duplicate the sequence.
While a high linear complexity is a necessary condition the following
example serves to show that it is not a su cient condition on the suitability
of the keystream. Consider the periodic sequence of period p consisting of a
single 1 with the remaining bits set to 0. In this case the linear complexity is
p since no linear relation shorter than si p si for all i 0 will be satis ed
by every bit of the sequence. However it is clear that as a keystream such a
sequence is useless since p 1 bits are zero.
Klapper 63 demonstrates another important consideration for the lin
ear complexity of the keystream namely that the sequence must have a
high linear complexity not only when considered bitwise but also when the
sequence is viewed as numbers which happen to be 0 or 1 over elds of
odd characteristic . In particular Klapper notes that geometric sequences 20
might well be susceptible to this kind of analysis.
Rueppel 114 additionally proposes the use of the linear complexity pro
le in the analysis of stream ciphers. After each bit is added to the keystream
the linear complexity of the sequence seen so far is calculated the value of
the linear complexity can be plotted against the number of bits that have
been examined thereby giving a pro le of the sequence. Rueppel proved
10 Stream Ciphers
several very important theorems concerning the linear complexity pro le
and managed to obtain expressions for the expected behavior of the linear
complexity pro le of a sequence for which each bit is generated at random
114 .
This so called ideal linear complexity pro le has been widely studied 96 .
Rueppel established that the linear complexity pro le for a perfectly random
x
source closely follows the line y a conjecture was posed specifying a
2
class of sequences which possess the ideal linear complexity pro le that is
x
which sequences have a pro le which follows the line y as closely as
2
possible. This conjecture was proved by Dai 28 and a full characterization
of all sequences with an ideal linear complexity pro le was provided by Wang
and Massey 132 .
There are other algorithms for identifying the linear complexity of a
binary sequence some more practical than others. One very interesting
algorithm due to Games and Chan 39 108 is exceptionally elegant but
can be used only on sequences with period 2n for which an entire period of
the sequence is known. While this appears to be an algorithm with limited
general applicability the mathematical foundations were used to prove some
very interesting results 40 38 in the study of what are called de Bruijn
sequences 15 35 . Very recently the validity of this algorithm has been
extended by Blackburn 9 to sequences with any period although an entire
period of the sequence is still required for its application.
4.2.2 Other measures of complexity
As a generalization of the linear complexity a great deal of research was
completed by Jansen 58 who looked closely at algorithms for evaluating
the maximum order complexity of a sequence. This is a generalization of the
linear complexity in that the recurrence that relates bits of the sequence need
no longer be linear. Thus when given a sequence it might be possible to
identify a much shorter recurrence that can be used to recreate the sequence
using a nonlinear feedback shift register instead of one which is entirely
linear.
There are some very obvious relationships between the linear and the
maximum order complexity. For instance the maximum order complexity is
always less than or equal to the linear complexity. However the expected be
havior of the maximum order complexity for a randomly generated sequence
is not easily established 58 . Consequently the practicality of the maximum
order complexity in a statistical test tends to be somewhat limited though
it is of signi cant theoretical interest.
4. Analysis 11
In many ways the maximum order complexity seems to be less easily
related 92 58 to the linear complexity than to another measure of com
plexity called the Ziv Lempel complexity or Lempel Ziv complexity to some
commentators 141 142 . The Ziv Lempel complexity and the maximum
order complexity are in fact based on quite di erent principles since the
Ziv Lempel complexity provides some measure of the rate at which new
patterns are generated within the sequence. The origins of the Ziv Lempel
complexity lie within the eld of data compression and the observation that
a random sequence cannot be signi cantly compressed. Unfortunately the
usefulness of this measure is also currently limited since it is di cult to de
ne a practical test statistic with which to evaluate the performance of a
generator.
The reason that results on the maximum order complexity and the Ziv
Lempel complexity can be closely related is due to the fact that both com
plexities can be computed using what is called a su x tree 32 97 . The
linear complexity cannot be computed in this way and this makes relating
the linear complexity to the Ziv Lempel complexity di cult. Jansen 58
describes a related technique for calculating the maximum order complexity
using directed acyclic word graphs.
The use of the su x tree points out that a measure of complexity can
only be useful if it can be e ciently calculated for the sequence of interest.
This is the major stumbling block with the development of the quadratic
span 21 .
The quadratic span lies between the linear complexity and the maximum
order complexity since it is concerned with using quadratic recurrences to
generate a sequence. At present it is only a measure of theoretical interest
since there is no e cient way to calculate the quadratic span of a sequence.
Consequently there are as yet few results on the calculation of the quadratic
span for a binary sequence 19 62 and even less on the expected values for a
randomly generated sequence. If however these problems can be overcome
then the quadratic span will almost certainly be another useful measure of
complexity.
Finally we mention the 2 adic span 64 53 of a sequence. This fascinat
ing new technique has allowed the cryptanalysis of the previously proposed
summation stream cipher 113 . Perhaps more importantly this work shows
the way toward a rigorous mathematical formalism of another class of shift
registers and while much research has concentrated on the cryptanalytic po
tential of this technique there are also results of interest to the designer
65 66 .
12 Stream Ciphers
4.3 Some theoretical results
The statistical testing of a keystream re ects what some commentators have
described as the system theoretic approach to stream cipher design 126 .
The designer uses the tests that are available and if some statistical weakness
in a class of sequences is discovered a new test is devised and added to the
set. This is very much an ad hoc method of analysis and many prefer to see
some rm theoretical foundations on which the security of stream ciphers
might be based. While a great deal of theory has been established and there
have been several proposals attempting to prove the security provided by
some keystream there still remains a wide gulf between the work of the
researchers and that of the practitioners.
Much of the theoretical work was stimulated by the work of Yao 135 .
Yao succeeded in tying together the two essential concepts we consider a
requisite in a keystream generator those are the ideas of predictability and
random appearance. In short and somewhat casually Yao showed that a
pseudo random generator could be e ciently predicted if and only if the
generator could be e ciently distinguished from a perfectly random source.
Following on this result techniques in the eld of computational com
plexityhave been used to prove results which relate the problem of predicting
the next bit in a pseudo random sequence to the di culty of solving a hard
problem 11 12 123 . We shall look into some of the generators proposed
as a result of this work in Section 7.3.
We have already seen an interesting link provided by the Ziv Lempel
complexity 141 between techniques used in the eld of data compression
and the analysis of pseudo random sequences. There is another technique
due to Maurer called the universal statistical test 83 which is linked to the
eld of data compression. This test was developed with particular interest
in the use of a pseudo random bit generator for obtaining keys for use in a
symmetric key cryptosystem.
Expressing the strength of a cryptosystem against exhaustive search in
terms of the length of the key might be misleading if the keys are not chosen
uniformly. In such a case an opponent can search through the keys that
are more likely to occur giving an improved chance of quickly obtaining
the correct key. Maurer s test evaluates the entropy per output bit of the
generator thereby re ecting the cryptographic strength of a system when
the generator is used to obtain keys in a cryptographic application.
Returning to questions of complexity one of the earliest attempts at as
sessing the complexity of a sequence was provided by Chaitin 18 and Kol
mogorov 70 who attempted to de ne the complexity of a sequence in terms
5. Congruential generators 13
of the size of the smallest Turing machine that could be used to generate the
sequence. The Turing machine is a simple but powerful conceptual comput
ing device which is often used in the theory of computing. While considering
the Turing machines ability to reproduce a sequence leads to a theoretically
interesting measure now called the Turing Kolmogorov Chaitin complexity
it is of little practical signi cance since there is no way to compute it 5 . The
charmingly titled paper On the Complexity of Pseudo Random Sequences
or If You Can Describe a Sequence It Can t be Random 5 provides
a link between the Turing Kolmogorov Chaitin complexity and the linear
complexity. More information on the Turing Kolmogorov Chaitin complex
ity can be found in the work of Chaitin 17 and Martin L of 75 .
5 Congruential generators
Some of the earliest practical systems were intended to act as pseudo random
number generators rather than keystream generators. While the problems
are closely related much of the motivation for pseudo random number gen
eration comes from problems in statistical testing and the cryptographic
value of sequences generated by these techniques can often be questioned.
A congruential generator is often used to generate random numbers and
the next number xi 1 in a sequence of numbers xi is de ned in the following
way
xi 1 axi b mod m
There are many results about the di erent forms of a b and m and their
inter relation to obtain a sequence of pseudo random numbers with large
period 67 .
For cryptographic use the numbers generated should not be predictable
if the modulus m is known then it is easy to solve for a and b given two
consecutive numbers in such a sequence. Knuth considers a variation of
this generator where the modulus m is a power of two 68 but only the high
order bits of the numbers are output this bears a striking similarity to some
work of Dai 27 which is concerned with generating similar sequences using
linear feedback shift registers.
Some results on congruential generators are as follows. Marsaglia 74
questions the claims of su cient random behavior for sequences produced
using linear congruential generators and Reeds 102 Knuth 68 Plumstead
99 Hastad and Shamir 56 and Frieze Kannan and Lagarias 37 have all
cast considerable doubt on the cryptographic value of sequences generated
using the multiplicative congruential generator. A paper by Frieze Hastad
14 Stream Ciphers
Kannan Lagarias and Shamir 36 and one by Boyar Plumstead 13 un
dermine con dence in techniques to use fragments of integers derived from
linear congruences.
While the generation of such sequences can be convenient and there are
analytic results which provide assurances on some of the basic properties
of the sequences generated linear congruential generators cannot be recom
mended for cryptographic use. Surprisingly perhaps a paper by Lagarias
and Reeds 72 implies that there might be little extra cryptographic security
gained by moving to more sophisticated recurrences which involve polyno
mial expressions. Krawczyk 71 has extended both this work and that of
Plumstead to apply a very general analysis to the problem of predicting
sequences generated using di erent forms of polynomial recurrence relation.
6 Shift register based schemes
The vast majority of proposed keystream generators are based in some way
on the use of linear feedback shift registers 4 . There are two primary
reasons for this a class of sequences they generate ideally capture the spirit
of Golomb s Postulates Section 4.1.2 and their behavior is easily analyzed
using algebraic techniques.
6.1 Linear feedback shift registers
Linear feedback shift registers are very familiar to electrical engineers and
coding theorists 10 and they are very suited for high speed implementa
tions since they are easily implemented in both hardware and software. The
two environments generally utilize di erent implementations of the linear
feedback shift register termed the Fibonacci and the Galois registers re
spectively but all theoretical results of major importance are valid for both
types.
A linear feedback shift register consists of a number of stages numbered
say from left to right as 0 n 1 with feedback from each to stage n 1. The
contents of the n stages of a register describe its state . The description of the
action of the register is perhaps easier for the Fibonacci register certainly it
is the most commonly described so we shall consider the Fibonacci register
here. The register is controlled by a clock and at each clocking instance the
contents of stage i are moved to stage i 1. The contents of stage 0 are
output and form part of the sequence while the new contents to stage n 1
which is now conceptually empty are calculated as some linear function
6. Shift register based schemes 15
of the previous contents to stages 0 n 1 the particular function being
dependent on the feedback used.
For completeness we shall brie y describe the Galois register. While
each stage of the Galois register is updated according to the contents of the
stage immediately to its right as in the Fibonacci register the feedback
taps also determine whether the prior contents of stage 0 are exclusive ored
into each stage of the register. Thus in contrast to the Fibonacci register
where feedback is a function of all stages in the register and the result is
stored in the rightmost stage feedback in the Galois register is potentially
applied to every stage of the register though it is a function of only the
leftmost stage.
Despite the implementation di erences between these two forms of the
linear feedback shift register the important thing to note is that for a reg
ister of length n a sequence with maximum period has period 2n 1 since
there are 2n states and the state 0 0 cannot occur and satis es Golomb s
Postulates. Actually this isn t too remarkable Golomb formed the postu
lates with these so called m sequences in mind and every m sequence is a
pn sequence2. What is remarkable is that conditions on the generation of
such m sequences can be easily identi ed and this makes the analysis of
these sequences particularly straightforward.
There clearly has to be a drawback to such sequences that can be easily
and quickly generated and seem to have good properties of random appear
ance. The drawback is that they only have linear complexity n since they
are generated using an n stage linear feedback shift register. Consequently
the Berlekamp Massey algorithm Section 4.2 can be used on 2n successive
bits of the output sequence to deduce the feedback and the initial state of
the register used to generate the sequence.
All shift register based schemes try to exploit the good characteristics
of sequences generated using linear feedback shift registers in such a way
that the new sequences are not susceptible to attacks based on their linear
complexity. Somewhat casually the designers of shift register based schemes
are attempting to introduce su cient nonlinear behavior into the generation
of the sequences to hinder successful cryptanalysis. As Massey is quoted
137 as saying
Linearity is the curse of the cryptographer.
The essential theoretical background to the study of linear feedback shift
2
The question of whether every pn sequence is an m sequence or its binary comple
ment was answered in 1981 when Cheng 22 discovered a pn sequence which could not
be derived from an m sequence.
16 Stream Ciphers
registers and related topics is laid down by the work of Selmer 122 and
Zierler 139 . The work of Ward 133 is often overlooked despite the fact
that many important results were derived a considerable time ago. Much
work has also concentrated on producing a parallel foundation to the theory
of non linear feedback shift registers see for example the work of Ronce
111 but the lack of a convenient and general mechanism for this analysis
is a major handicap.
6.2 Combination and lter generators
When using linear feedback shift registers there are two obvious ways to
generate an alternative output. The rst is to use several registers in parallel
and to combine their output in some hopefully cryptographically secure
way. A generator like this is conventionally called a combination generator.
Another alternative is to generate the output sequence as some nonlinear
function of the state of a single register such a register is termed a lter
generator.
We have described this technique in very general terms and there is little
value in listing speci c choices for the functions used. Clearly bounding the
period and the linear complexity of the sequences generated are important
issues 114 79 117 . In addition unless great care is taken in deciding which
types of function to use these generators may be susceptible to what have
been termed correlation attacks. We will go into more detail on this subject
in Section 6.2.1.
The combination of several sequences may well involve the use of what
is termed the Hadamard product. The product of two sequences is formed
bitwise and we expect to see more 0 s than 1 s in a sequence formed as
the product of two other sequences which had a roughly equal distribu
tion of 0 s and 1 s. While this imbalance in the product sequence tends
to increase the linear complexity considerably the excessive number of 0 s
potentially provides a cryptographic loop hole which the cryptanalyst can
exploit. Much of the theoretical background on the linear complexity of such
product sequences can be found in the work of Zierler 140 and Rueppel and
Sta elbach 116 . Recently Gottfert and Niederreiter 54 proved bounds on
the linear complexity of product sequences.
6.2.1 Correlation attacks
A correlation attack is a widely applicable type of attack which might be
used with success on generators which attempt to combine the output from
6. Shift register based schemes 17
several cryptographically weak keystream generators.
A correlation attack exploits the weakness in some combining function
which allows information about individual input sequences to be observed
in the output sequence. In such a case there is a correlation between the
output sequence and one of the internal sequences. This particular internal
sequence can then be analyzed individually before attention is turned to
one of the other internal sequences. In this way the whole generator can be
deduced this is often called a divide and conquer attack.
Correlation attacks were rst introduced by Siegenthaler 121 119 120 .
Since it is immediately clear that some combining or lter functions are
more susceptible to attack than others the idea of an mth order correlation
immune function was introduced 119 . When at least m 1 internal se
quences must be simultaneously considered in a correlation attack the func
tion is said to be mth order correlation immune. In the same paper 119
Siegenthaler showed that there was an interesting trade o between the lin
ear complexity of the output sequence and the order of correlation immunity
greater correlation immunity meant a reduced linear complexity.
Brynielsson 16 examined how this problem might be adapted to other
non binary elds and researchby Rueppel 113 showed how the use of mem
ory could be used to separate the ideas of correlation immunity and linear
complexity in the binary case. The summation generator 113 see Sec
tion 6.6 introduced the idea of a combining function with memory and
it was established that with this combining function it is possible to attain
maximum order correlation and maximum linear complexity simultaneously.
Meier and Sta elbach 86 have provided more complete details about the
correlation properties of combiners and the role of memory.
Additional work on correlation attacks and some improvements in e
ciency can be found in 23 87 42 90 2 91 45 . Other interesting results
have been established and we shall describe some of them in Section 6.2.3
where the issues addressed are more suitably expressed in terms of Boolean
functions.
6.2.2 Two weak generators
Since they don t fall easily into any other section we will mention here
two early and simple proposals for keystream generators which use multiple
registers and are susceptible to correlation attacks.
The rst is the Ge e generator 41 which was later analyzed by Key 61
and also cryptanalyzed using the linear syndrome algorithm 136 Section
6.2.4 . This generator uses three linear feedback shift registers the third
18 Stream Ciphers
being used to choose whether the bit that is output comes from the rst
register or the second. While such a generator has some nice properties it
is susceptible to a correlation attack. The success of correlation attacks also
defeated the Pless generator 98 which used a widely available logic device
the J K ip op to combine the outputs from eight linear feedback shift
registers.
6.2.3 Boolean functions
It is interesting to observe that with the topic of Boolean functions the design
of stream ciphers and block ciphers are once again related. The interest in
Boolean functions for block ciphers follows from the design of S boxes in
DES like block ciphers 106 . Some of the conditions required for good S
box design are essentially the same as the requirements for good combining
functions.
Meier and Sta elbach 85 consider a measure of the distance of an ar
bitrary Boolean function from the nearest linear function and introduce the
idea of a perfect nonlinear function. A Boolean function taking n inputs
is perfect nonlinear if the output changes with probability 1 2 whenever i
input bits are complemented for 1 i n. It so happens that the notion of
perfect nonlinear functions coincides with the idea of bent functions 112 .
These have already been well researched in other areas of mathematics and
have been connected with functions used in the design of S boxes 118 .
A second issue of interest in the eld of S box design is the so called Strict
Avalanche Criterion SAC 34 73 . A Boolean function f x satis es SAC
if the output changes with probability 1 2 whenever exactly one of the input
bits changes. This property is useful both in the design of S boxes and in
the design of combining functions. So is a generalization of SAC mth order
SAC. A function f x satis es mth order SAC if when any m input bits to
f x are kept constant the output changes with probability 1 2 when one
of the remaining input bits changes.
Preneel et. al. 100 have uni ed these ideas with the concept of the
propagation criterion of degree k. A Boolean function is PC of degree k if the
output changes with probability1 2 whenever i input bits are complemented
for 1 i k. As a consequence we have that the idea of perfect nonlinear
is equivalent to PC of degree n and the property of SAC is equivalent to PC
of degree 1.
Counting and constructing families of Boolean functions which satisfy
various desirable cryptographic properties forms a very active area of re
search. In particular much work is concerned with nding an acceptable
6. Shift register based schemes 19
balance between often con icting requirements.
6.2.4 Three more attacks
In this section we brie y describe three types of attacks that have been
proposed in the literature.
The linear consistency test 137 attempts to e ciently identify some
subset of the key used for encryption. The idea is that a matrix A K1 is
devised for which the entries identify the generator being used. This matrix
is parameterized by some subkey K1 of the complete key K. With some
portion of output sequence b the cryptanalyst attempts to nd some x such
that the matrix equation A K1 x b is consistent.
If a solution is found then it can be shown that provided the portion of
output sequence b is large enough the solution is unique and the correct
subkey K1 has been identi ed. Thus a search need only be performed on
all possible subkeys K1 until a consistent solution is found. In this way an
attack relevant to the entire key K can be mounted.
The second attack uses what is termed the linear syndrome algorithm
136 . This attack is essentially a generalization of the work of Meier and
Sta elbach 85 and relies on being able to write a fragment of captured out
put sequence b as b a x were a is a sequence generated by a known linear
recurrence and x is a sparse unknown sequence where a sparse sequence
consists of more 0 s than 1 s. This algorithm was particularly successful in
the cryptanalysis of both the Ge e generator 41 Section 6.2.2 and the
stop and go generator 6 Section 6.4.1 .
Finally Goli c 43 has proposed the linear cryptanalysis of stream ci
phers. An extension of earlier work 42 and related to other simultaneous
work 44 this technique is potentially applicable to a wide variety of stream
cipher proposals.
6.3 Multiplexers
A multiplexer is a logic device that selects one input from a set of inputs
according to the value of another index input. Sequences based on the
use of multiplexers were initially popular because they are relatively fast
and have some nice provable properties 59 . The keystream generator is
conventionally described using two sequences often m sequences for ease of
analysis and the multiplexer is used to combine these two sequences in a
highly nonlinear way.
At each clocking instance a xed pattern of k bits is taken from the
20 Stream Ciphers
rst sequence. These k bits are viewed as the binary representation of a
number modulo 2k and this number is then mapped using a xed and known
mapping into some other number n. Various conditions are imposed on k
and n to ensure that the mapping is sensibly de ned. The number n is
used to choose some bit from the second sequence which then forms part of
the output sequence. In e ect the keystream generator uses a multiplexer
to select bits from the second sequence according to the values of certain
bits in the rst sequence.
The sequences that result generally have a large period and linear com
plexity 59 . However a technique known as the linear consistency attack
137 has been used to show how the choice of mapping adds little to the
security of the system and it is concluded that the security of multiplexed
sequences might have been previously over estimated. Work by Daemen 26
has highlighted another possible avenue for mounting an attack on multi
plexers and has undermined 25 a European Broadcast Union proposal for
audio video scrambling 31 .
6.4 Clock control
Some of the earlier attempts to introduce non linearityinto the generation of
the keystream use the idea of varying the rate at which a register is clocked.
Recall that in the conventional interpretation of a shift register the register
is clocked regularly and the contents of the stages updated at each clocking
instance. If some arrangement is devised so that the clocking of one register
is in some way dependent on another register then it seems reasonable to
suppose that more complex sequences will be generated.
While it is undoubtedly the case that sequences generated by these tech
niques tend to be more complex than any of the constituent sequences
there is a convenient framework for their analysis and some structure is
inherited in the output sequence. In fact recent theoretical work by Goli c
and O Conner 46 shows that most clock control keystream generators are
at least in theory susceptible to attacks termed embedding and probabilistic
correlation attacks. However these techniques can not in general be readily
extended into practical attacks.
While some of the simpler clock control techniques have not withstood
close analysis more involved designs seem to perform quite well. Baum
and Blackburn 3 discuss a generalization of this technique and a thorough
survey of clock control techniques is provided by Chambers and Gollmann
51 .
6. Shift register based schemes 21
6.4.1 Stop and go with variants
Among the rst investigations were those into what was termed the stop
and go generator 6 . In this simple scenario two registers were connected
so that the second register was clocked if the output of the rst register
was a 1 otherwise the second register repeated its previous output. Some
times this output was then exclusive ored with the output sequence from
a third register. It is not surprising that the repetition of bits in the rst
output sequence roughly half the time leads to poor statistical properties
and unfortunate cryptographic consequences 138 136 .
Alterations can be made to this basic model by making the rst or mo
tor register into one which steps the second register twice when a 1 is output
by the rst register and only once otherwise this arrangement requires that
the second register can run at twice the speed of keystream output but it
certainly has improved statistics. Other generator arrangements including
one by Gunther 55 which has both improved statistics and a constant reg
ister to keystream rate have been proposed and many properties have been
established 127 129 131 .
Perhaps surprisingly it is very straightforward to establish a rm the
oretical basis for the analysis of such sequences. Changing the clocking
pattern of one register merely ensures that the output sequence is some dec
imated or sampled version of the original. The underlying linear algebra can
then be used to establish an expression for the new sequence in terms of
the old and bounds on the period and the linear complexity can usually be
readily established.
Interestingly the operations of decimation that is removing bits from
a sequence and interleaving or interlacing that is combining sequences
together are powerful tools in the investigation of many alternative gener
ators 107 . Often the major di culty is that the bounds we obtain on the
linear complexity of the sequences are upper bounds for practical purposes
we generally wish to obtain a lower bound. Additionally it is often di cult
to establish the conditions that de ne when the upper bound is achieved.
Some form of lower bound can sometimes be obtained by considering the
period of the sequence since a lower bound on the period perhaps obtained
by using combinatorial techniques can usually be translated into a result on
the linear complexity of the sequence. Other more general theoretical results
on both what are termed the regular and irregular decimation of sequences
have been obtained 47 .
After considerable early interest in clock controlled registers during the
mid 1980 s due to the provably high periods and linear complexities of the
22 Stream Ciphers
resultant sequences there has been a slackening of research interest. How
ever work has continued into cascades of registers see Section 6.4.2 which
are often viewed as a generalization of the stop and go type register arrange
ments.
Finally in this section we remark on two other variations. Rueppel 115
obtained bounds on the linear complexity and period of the output from a
single register whose output controls its own clock. While such a register
should not be used as it stands as a keystream generator Rueppel reports
126 that several modi cations have been suggested which might make it
useful in a cascade of registers. Second a generator called the multiple
speed generator 76 which uses two registers clocked at di erent speeds has
many interesting theoretical properties though it is vulnerable to the linear
consistency attack 137 Section 6.2.4 .
6.4.2 Cascades
The main idea behind cascades is to extend the simple stop and go type
arrangements of the previous section into a string of registers for which the
output of the rst is used in some way to control the clock of the second
the output of the second is used for the third and so on. Two major types
of cascades have been studied the rst where each of the registers generates
an m sequence and the second where each of the registers is of length p
where p is prime and there is no feedback from any intermediate stage of
the register. Such registers are called purely cycling registers of length p.
Much of the early theoretical work on cascades took place in tandem with
proposals for clock control 49 51 . The beauty of cascades is that they are
conceptually very simple and they can be used to generate sequences with
vast periods and similarly vast and guaranteed linear complexity 49 . They
also seem to have good statistical qualities 49 48 .
However they are prone to an e ect which has been termed lock in 50 . A
cryptanalyst might try to reconstruct the input to the last register by using
the captured output sequence of the generator and guessing the relevant
but unknown parameters for the last register. Lock in ensures that many
related guesses will su ce to allow the reconstruction of the input sequence
to the last register. In this way a cryptanalyst can unravel a cascade register
by register and the net result of lock in is a reduction in the e ective key
space of the cascade generator. This can be a serious weakness in certain
situations though precautions can be taken to reduce the e ectiveness of a
cryptanalytic attack based on the lock in e ect.
Rather ingeniously Chambers and Gollmann point out that if the cas
6. Shift register based schemes 23
cade is used as an encryption mechanism with plaintext used as input to
the rst stage rather than the all 1 sequence when used as a keystream
generator then the e ect of lock in can be used constructively to regain
synchronization after an error in transmission.
Some very recent cryptanalytic results on cascades are due to Menicocci
89 . It is claimed that there is always some correlation between the output
sequence from the rst register and the output of the cascade and if this
remains signi cant then information about the rst register is leaked in the
output sequence. Menicocci suggests that this might form the basis for an
attack to ensure that this e ect is not exploitable Menicocci suggests that
a cascade should be at least 10 registers long.
6.5 Shrinking and self shrinking generator
These two closely related generators have been proposed recently.
Shrinking generator
The shrinking generator was proposed by Coppersmith Krawczyk and Man
sour 24 . It uses techniques similar to clock control and in fact it has been
pointed out that the generator can be viewed as implementing a form of
variable clock control. One result of this equivalence is that the theoretical
results of Goli c and O Conner 46 Section 6.4 are equally applicable to
the shrinking generator and the self shrinking generator.
In e ect a shrinking generator is implemented by taking two sequences
that are generated in parallel. At any clocking instance a bit is output from
the second sequence if the rst sequence outputs a 1 otherwise nothing is
output.
Like practically all of the previous schemes when the two source se
quences are m sequences bounds on the period and linear complexity of the
resultant sequence can easily be obtained. Also like many of the sequences
built out of m sequence building blocks the statistical appearance of the
sequences is generally good.
It is interesting to apply the techniques of decimation and interleaving
to these output sequences and it can be shown that they can be considered
as the interleaving of many o set copies of some m sequence. While it is
too early to decide whether this is signi cant to a cryptanalyst it is clear
that there is considerable underlying structure in these sequences.
On the practical front this generator is very fast though it su ers from
the problem that the output rate is not regular. A bu ering technique is
24 Stream Ciphers
suggested 24 to get around this problem though it is not clear how great a
problem this irregular rate might be in practice.
Self shrinking generator
The self shrinking generator is a variant of the shrinking generator and was
proposed by Meier and Sta elbach at Eurocrypt 94 88 . Instead of generat
ing the indexing sequence and the sequence to be shrunk from two di erent
registers they are both derived from the same register. While this reduces
the amount of space required for an implementation it does mean that the
sequence is generated at roughly half the speed of the shrinking generator.
There is some duality between the shrinking and the self shrinking gen
erator. It is easy to verify that any shrinking generator can be implemented
using some self shrinking generator and vice versa. However the shrinking
generator equivalents to the self shrinking generators proposed by Meier and
Sta elbach do not possess the same form as those proposed by Coppersmith
et al. 24 and so the previous results on the shrinking register cannot be
carried over.
While there appears to be considerable unexplained behavior in the se
quences produced using the self shrinking generator Meier and Sta elbach
have proved lower bounds on both the period and the linear complexity.
Consequently parameters in an implementation of a self shrinking generator
can be chosen to ensure adequate performance in these regards.
6.6 Summation generator
It is well known 128 that integer addition can be used as a nonlinear com
biner the carry in integer addition is a nonlinear function of the low order
bits of the numbers being added.
Rueppel uses this fact in a generator known as the summation generator.
Here the outputs from several shift registers are combined using a mechanism
involving integer addition. This provides a combining function with good
nonlinearity and high order correlation properties 113 . Importantly it also
provides an example of the role of memory in removing the trade o between
high nonlinearity and the correlation immunity of a function see Section
6.2.1 . Though the work of Meier and Sta elbach 85 on simple summation
generators and that of Klapper and Goresky 64 more generally seems
to have compromised the security o ered by this particular generator it
remains a theoretically interesting technique.
7. Alternative designs 25
7 Alternative designs
It will come as no surprise that there are several important generators and
general techniques which don t really t into the scheme of the report so far.
This penultimate section includes proposals which have not been covered in
this report and considers other very important and widely used techniques.
7.1 RC4
The RC4 stream cipher 104 was designed by Ron Rivest in 1987 for RSA
Data Security Inc. Like its companion block cipher RC2 RC4 is a variable
key size cipher suitable for fast bulk encryption. It is very compact in terms
of code size and it is particularly suitable for byte oriented processors. RC4
can encrypt at speeds of around 1 Mbyte sec on a 33MHz machine and like
RC2 has special status by which the export approval process is considerably
simpli ed 33 .
While RC4 is a con dential and proprietary stream cipher its security
does not depend on the con dentiality of the algorithm. Its design is quite
distinct from the methods we have already seen and uses a random permu
tation during the generation of the keystream. There are no known bad
keys and though there is no proof for the lower bound of the periods of the
sequences generated using RC4 theoretical analysis has established that the
period is overwhelmingly likely to be greater than 10100. A thorough and
extensive analysis into the security of RC4 109 has found no reason to
question the security o ered by the RC4 keystream generator.
7.2 SEAL
SEAL which stands for software optimized encryption algorithm is a re
cently published stream cipher designed by Rogaway and Coppersmith 110 .
SEAL is described as a length increasing pseudo random function and this
can clearly be used as a keystream generator for a stream cipher. This
stream cipher is geared towards 32 bit architectures and encryption requires
about ve machine instructions per byte.
SEAL requires a large amount of pre computation to initialize several
large look up tables which total approximately 3 Kbytes in size. This ini
tialization procedure makes repeated use of the compression function which
lies at the heart of the Secure Hash Algorithm 93 . The algorithm was
optimized with a particular range of popular processors in mind and since
these processors were among those that are more di cult to optimize for it
26 Stream Ciphers
is expected that an implementation will perform well on any modern 32 bit
processor.
Since SEAL is so new there has not been enough time to allow for an
assessment of the security o ered but it marks a welcome new addition to
the di erent design techniques available for stream ciphers.
7.3 Number theoretic techniques
In this section we consider some designs for keystream generators for which
the ability to predict the keystream is in some way related to the solution
of what is considered to be a hard problem.
There are many well known examples of problems which are considered
to be hard perhaps the most commonly cited are inverting the RSA cryp
tosystem 105 establishing quadratic residuosity 69 and solving what is
called the discrete logarithm problem 69 . The aim of the designer is to
ensure that any successful method of predicting the keystream can then be
used to successfully solve some di cult problem. Under the assumption
that this problem is in reality intractable this implies that the keystream
cannot be e ciently predicted. The work of Yao 135 is then cited which
then provides the nal link to show that the keystream cannot be e ciently
distinguished from a perfectly random source.
While these generators have considerable theoretical appeal there are
some considerations we should keep in mind. First the di culty of a prob
lem is usually expressed using techniques in the eld of study known as
complexity theory. Such results are asymptotic in nature that is they de
scribe the di culty of a problem in terms of an increasingly large instance
of the problem. The element of provability for which we are striving is thus
asymptotic in nature and it is lost when we move to a problem instance
of xed size. Nevertheless these techniques do provide us with a scale by
which the security of a system can be quanti ed against a problem that is
known to be di cult to solve in practice. More importantly perhaps the
number theoretic operations that these schemes use tend to be slow. As a
result these keystream generators tend to have poor performance attributes.
We shall merely list here some of the proposals in the literature and provide
some initial references for the interested reader.
Shamir 123 relates the security of a generator to inverting the RSA cryp
tosystem but Blum and Micali highlight some interesting limitations 12 .
Meanwhile Blum and Micali 12 themselves propose a generator whichis re
lated to the problem of e ciently computing the discrete logarithm Kaliski
60 provides similar work on the use of the discrete logarithm problem over
7. Alternative designs 27
elliptic curves. Meanwhile Alexi Chor Goldreich and Schnorr 1 propose
a scheme based on the di culty of inverting the RSA cryptosystem. Blum
Blum and Shub 11 use the problem of deciding quadratic residuosity as the
basis for the security of another keystream generator.
7.4 Other schemes
7.4.1 1 p generator
The 1 p generator has a long history and can be traced back to the work
of Dickson 30 and Knuth 67 . The pseudo random sequence is generated
by expanding the fraction 1 p to some base b where p and b are relatively
prime. While the sequence itself has nice statistical features and certain
conditions on p and b can ensure a provably large period it has been shown
11 that this generator is insecure.
7.4.2 Knapsack generator
The security of this generator is based on what is typically called a hard
problem and might therefore be more consistently presented in Section 7.3.
However it is also a shift register based scheme and this makes its classi
cation somewhat problematical.
The problem on which the generator is based is called the knapsack
problem because an analogy is often drawn between solving this problem
and packing a knapsack with di erent sized items so that the knapsack is
lled exactly. The mathematical exposition of this problem is to nd some
subset of a large set of numbers such that the sum of the subset equals a
speci ed chosen target value.
In the knapsack generator 114 a set of weights are chosen as part of the
key. The state of the register at some time instance is combined with the
set of weights to give a set of integers. These are then added together using
conventional integer arithmetic. Finally bits are chosen from this sum and
output as part of the keystream.
The sequences produced have good period and linear complexity prop
erties. However it seems that the bad name acquired by other speci c
knapsack based systems during the early days of public key cryptography
124 14 makes many people wary of any knapsack based system. There do
not appear to be however any results in the literature on the successful
cryptanalysis of this generator.
28 Stream Ciphers
7.4.3 PKZIP
PKZIP is a widely used compression function that has an option allowing
stream cipher encryption with a variable length key. This cipher however
is not secure and Biham and Kocher 8 have described an attack which will
nd the internal representation of the key in less than one day with a few
hundred bytes of known plaintext.
7.5 Final examples
We brie y present yet more alternative approaches.
7.5.1 Randomized ciphers
We have seen cryptographers attempt to prove security against an unlim
ited adversary or an adversary who is unable to e ciently solve a hard
problem. An interesting new direction is provided by techniques which at
tempt to ensure that the amount of work physically required for successful
cryptanalysis is too demanding. A class of stream ciphers designed with
this goal in mind have been labelled randomized stream ciphers by Rueppel
126 . Two schemes require massive computation or communication over
heads for the legitimate users and can safely be considered impractical but
the third scheme we mention is practical assuming that there is some vast
public source of random bits.
The two less practical ciphers are as follows. Di e s randomized stream
cipher is described by Rueppel 126 . The plaintext is encrypted using one
of 2n randomly generated keystreams which are all sent along with the ci
phertext over the communication channel. The key speci es which of the 2n
sequences the legitimate receiver should pick to use for decryption giving a
considerable advantage over any opponent. Clearly this scheme requires a
considerable communication overhead.
Meanwhile Massey and Ingemarsson 78 have presented the Rip van
Winkle cipher so called because as Massey has said
One can easily guarantee that the enemy cryptanalyst will need
thousands of years to break the cipher if one is willing to wait
millions of years to read the plaintext.
In a third scheme Maurer considers an information theoretic approach
to the abilities of an adversary when the adversary is computationally lim
ited 82 . Note the contrast between this concept and that of Shannon s
information theory 125 where the computational power of the adversary is
8. Conclusions 29
assumed to be unlimited. Maurer s scheme relies on the public availabilityof
a vast amount of random information an example of such a source might be
a satellite which continually beams randomly generated data back to earth.
While randomized ciphers are theoretically interesting it seems that only
Maurer s proposal can be viewed as being near to practical.
7.5.2 Cellular automata
Proposed by Wolfram 134 the cellular automata scheme provides a tech
nique for generating sequences with large periods and good statistical prop
erties. The scheme also marks a departure from shift register based schemes
and as a consequence does not lend itself to the ready analysis applicable to
shift register schemes.
The generator consists of n cells that are arranged in a ring. Each cell
is updated at a given time instance according to some simple but non linear
rule de ned in terms of adjacent cells. The sequence of values of one chosen
cell de nes the keystream sequence.
While the lack of a convenient framework for analysis makes cryptanaly
sis that much harder it also hinders attempts to assess such basic properties
of the system as the period. This is particularly the case when the theoreti
cally interesting model of an in nite array of cells is replaced by the practical
realization described above. Meier and Sta elbach 84 have analyzed this
proposal and shown that the parameters originally proposed for a practical
implementation do not o er adequate security. Daemen 26 has proposed
another cipher based on cellular automata that is resistant to the attack of
Meier and Sta elbach.
8 Conclusions
While there is no single algorithm which acts as a focus for cryptanalysis in
the eld of stream ciphers the impression left by many reviews of stream
cipher techniques is that an overwhelming interest has been paid to shift
register based schemes. This report is clearly no exception. Though a
huge variety of schemes and di erent theoretical techniques are available
the reality is that the open literature is dominated with shift register based
results. As we have seen there is a close interplay between shift registers
and the techniques of linear algebra and this provides much of the emphasis
of research interest.
Despite the wealth of results on both the design and cryptanalysis of
shift register based schemes there are numerous other approaches each with
30 Stream Ciphers
advantages and disadvantages. In the future we might expect to see some of
the alternative approaches to stream cipher design such as those provided
by RC4 and SEAL becoming extremely popular.
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