Algebraic Analysis of LEX

LEX is a stream cipher that progressed to Phase 3 of the eSTREAM stream cipher project. In this paper, we show that the security of LEX against algebraic attacks relies on a small equation system not being solvable faster than exhaustive search. We use the byte leakage in LEX to construct a system of 21 equa- tions in 17 variables. This is very close to the require- ment for an efficient attack, i.e. a system containing 16 variables. The system requires only 36 bytes of keystream, which is very low.

Investigating functional thinking in the elementary classroom : foundations of early algebraic reasoning

This paper examines the development of student functional thinking during a teaching experiment that was conducted in two classrooms with a total of 45 children whose average age was nine years and six months. The teaching comprised four lessons taught by a researcher, with a second researcher and classroom teacher acting as participant observers. These lessons were designed to enable students to build mental representations in order to explore the use of function tables by focusing on the relationship between input and output numbers with the intention of extracting the algebraic nature of the arithmetic involved. All lessons were videotaped. The results indicate that elementary students are not only capable of developing functional thinking but also of communicating their thinking both verbally and symbolically.

Algebraic analysis of small scale LEX-BES

This paper examines the algebraic cryptanalysis of small scale variants of the LEX-BES. LEX-BES is a stream cipher based on the Advanced Encryption Standard (AES) block cipher. LEX is a generic method proposed for constructing a stream cipher from a block cipher, initially introduced by Biryukov at eSTREAM, the ECRYPT Stream Cipher project in 2005. The Big Encryption System (BES) is a block cipher introduced at CRYPTO 2002 which facilitates the algebraic analysis of the AES block cipher. In this paper, experiments were conducted to find solution of the equation system describing small scale LEX-BES using Gröbner Basis computations. This follows a similar approach to the work by Cid, Murphy and Robshaw at FSE 2005 that investigated algebraic cryptanalysis on small scale variants of the BES. The difference between LEX-BES and BES is that due to the way the keystream is extracted, the number of unknowns in LEX-BES equations is fewer than the number in BES. As far as the author knows, this attempt is the first at creating solvable equation systems for stream ciphers based on the LEX method using Gröbner Basis computations.

An analysis of the RC4 family of stream ciphers against algebraic attacks

To date, most applications of algebraic analysis and attacks on stream ciphers are on those based on lin- ear feedback shift registers (LFSRs). In this paper, we extend algebraic analysis to non-LFSR based stream ciphers. Specifically, we perform an algebraic analysis on the RC4 family of stream ciphers, an example of stream ciphers based on dynamic tables, and inves- tigate its implications to potential algebraic attacks on the cipher. This is, to our knowledge, the first pa- per that evaluates the security of RC4 against alge- braic attacks through providing a full set of equations that describe the complex word manipulations in the system. For an arbitrary word size, we derive alge- braic representations for the three main operations used in RC4, namely state extraction, word addition and state permutation. Equations relating the inter- nal states and keystream of RC4 are then obtained from each component of the cipher based on these al- gebraic representations, and analysed in terms of their contributions to the security of RC4 against algebraic attacks. Interestingly, it is shown that each of the three main operations contained in the components has its own unique algebraic properties, and when their respective equations are combined, the resulting system becomes infeasible to solve. This results in a high level of security being achieved by RC4 against algebraic attacks. On the other hand, the removal of an operation from the cipher could compromise this security. Experiments on reduced versions of RC4 have been performed, which confirms the validity of our algebraic analysis and the conclusion that the full RC4 stream cipher seems to be immune to algebraic attacks at present.

Improved algebraic cryptanalysis of QUAD, Bivium and Trivium via graph partitioning on equation systems

We present a novel approach for preprocessing systems of polynomial equations via graph partitioning. The variable-sharing graph of a system of polynomial equations is defined. If such graph is disconnected, then the corresponding system of equations can be split into smaller ones that can be solved individually. This can provide a tremendous speed-up in computing the solution to the system, but is unlikely to occur either randomly or in applications. However, by deleting certain vertices on the graph, the variable-sharing graph could be disconnected in a balanced fashion, and in turn the system of polynomial equations would be separated into smaller systems of near-equal sizes. In graph theory terms, this process is equivalent to finding balanced vertex partitions with minimum-weight vertex separators. The techniques of finding these vertex partitions are discussed, and experiments are performed to evaluate its practicality for general graphs and systems of polynomial equations. Applications of this approach in algebraic cryptanalysis on symmetric ciphers are presented: For the QUAD family of stream ciphers, we show how a malicious party can manufacture conforming systems that can be easily broken. For the stream ciphers Bivium and Trivium, we nachieve significant speedups in algebraic attacks against them, mainly in a partial key guess scenario. In each of these cases, the systems of polynomial equations involved are well-suited to our graph partitioning method. These results may open a new avenue for evaluating the security of symmetric ciphers against algebraic attacks.

Performance analysis of adaptive lattice filters for FM signals and alpha-stable processes

The performance of an adaptive filter may be studied through the behaviour of the optimal and adaptive coefficients in a given environment. This thesis investigates the performance of finite impulse response adaptive lattice filters for two classes of input signals: (a) frequency modulated signals with polynomial phases of order p in complex Gaussian white noise (as nonstationary signals), and (b) the impulsive autoregressive processes with alpha-stable distributions (as non-Gaussian signals). Initially, an overview is given for linear prediction and adaptive filtering. The convergence and tracking properties of the stochastic gradient algorithms are discussed for stationary and nonstationary input signals. It is explained that the stochastic gradient lattice algorithm has many advantages over the least-mean square algorithm. Some of these advantages are having a modular structure, easy-guaranteed stability, less sensitivity to the eigenvalue spread of the input autocorrelation matrix, and easy quantization of filter coefficients (normally called reflection coefficients). We then characterize the performance of the stochastic gradient lattice algorithm for the frequency modulated signals through the optimal and adaptive lattice reflection coefficients. This is a difficult task due to the nonlinear dependence of the adaptive reflection coefficients on the preceding stages and the input signal. To ease the derivations, we assume that reflection coefficients of each stage are independent of the inputs to that stage. Then the optimal lattice filter is derived for the frequency modulated signals. This is performed by computing the optimal values of residual errors, reflection coefficients, and recovery errors. Next, we show the tracking behaviour of adaptive reflection coefficients for frequency modulated signals. This is carried out by computing the tracking model of these coefficients for the stochastic gradient lattice algorithm in average. The second-order convergence of the adaptive coefficients is investigated by modeling the theoretical asymptotic variance of the gradient noise at each stage. The accuracy of the analytical results is verified by computer simulations. Using the previous analytical results, we show a new property, the polynomial order reducing property of adaptive lattice filters. This property may be used to reduce the order of the polynomial phase of input frequency modulated signals. Considering two examples, we show how this property may be used in processing frequency modulated signals. In the first example, a detection procedure in carried out on a frequency modulated signal with a second-order polynomial phase in complex Gaussian white noise. We showed that using this technique a better probability of detection is obtained for the reduced-order phase signals compared to that of the traditional energy detector. Also, it is empirically shown that the distribution of the gradient noise in the first adaptive reflection coefficients approximates the Gaussian law. In the second example, the instantaneous frequency of the same observed signal is estimated. We show that by using this technique a lower mean square error is achieved for the estimated frequencies at high signal-to-noise ratios in comparison to that of the adaptive line enhancer. The performance of adaptive lattice filters is then investigated for the second type of input signals, i.e., impulsive autoregressive processes with alpha-stable distributions . The concept of alpha-stable distributions is first introduced. We discuss that the stochastic gradient algorithm which performs desirable results for finite variance input signals (like frequency modulated signals in noise) does not perform a fast convergence for infinite variance stable processes (due to using the minimum mean-square error criterion). To deal with such problems, the concept of minimum dispersion criterion, fractional lower order moments, and recently-developed algorithms for stable processes are introduced. We then study the possibility of using the lattice structure for impulsive stable processes. Accordingly, two new algorithms including the least-mean P-norm lattice algorithm and its normalized version are proposed for lattice filters based on the fractional lower order moments. Simulation results show that using the proposed algorithms, faster convergence speeds are achieved for parameters estimation of autoregressive stable processes with low to moderate degrees of impulsiveness in comparison to many other algorithms. Also, we discuss the effect of impulsiveness of stable processes on generating some misalignment between the estimated parameters and the true values. Due to the infinite variance of stable processes, the performance of the proposed algorithms is only investigated using extensive computer simulations.

Algebraic attacks on clock-controlled stream ciphers

Stream ciphers are encryption algorithms used for ensuring the privacy of digital telecommunications. They have been widely used for encrypting military communications, satellite communications, pay TV encryption and for voice encryption of both fixed lined and wireless networks. The current multi year European project eSTREAM, which aims to select stream ciphers suitable for widespread adoptation, reflects the importance of this area of research. Stream ciphers consist of a keystream generator and an output function. Keystream generators produce a sequence that appears to be random, which is combined with the plaintext message using the output function. Most commonly, the output function is binary addition modulo two. Cryptanalysis of these ciphers focuses largely on analysis of the keystream generators and of relationships between the generator and the keystream it produces. Linear feedback shift registers are widely used components in building keystream generators, as the sequences they produce are well understood. Many types of attack have been proposed for breaking various LFSR based stream ciphers. A recent attack type is known as an algebraic attack. Algebraic attacks transform the problem of recovering the key into a problem of solving multivariate system of equations, which eventually recover the internal state bits or the key bits. This type of attack has been shown to be effective on a number of regularly clocked LFSR based stream ciphers. In this thesis, algebraic attacks are extended to a number of well known stream ciphers where at least one LFSR in the system is irregularly clocked. Applying algebriac attacks to these ciphers has only been discussed previously in the open literature for LILI-128. In this thesis, algebraic attacks are first applied to keystream generators using stop-and go clocking. Four ciphers belonging to this group are investigated: the Beth-Piper stop-and-go generator, the alternating step generator, the Gollmann cascade generator and the eSTREAM candidate: the Pomaranch cipher. It is shown that algebraic attacks are very effective on the first three of these ciphers. Although no effective algebraic attack was found for Pomaranch, the algebraic analysis lead to some interesting findings including weaknesses that may be exploited in future attacks. Algebraic attacks are then applied to keystream generators using (p; q) clocking. Two well known examples of such ciphers, the step1/step2 generator and the self decimated generator are investigated. Algebraic attacks are shown to be very powerful attack in recovering the internal state of these generators. A more complex clocking mechanism than either stop-and-go or the (p; q) clocking keystream generators is known as mutual clock control. In mutual clock control generators, the LFSRs control the clocking of each other. Four well known stream ciphers belonging to this group are investigated with respect to algebraic attacks: the Bilateral-stop-and-go generator, A5/1 stream cipher, Alpha 1 stream cipher, and the more recent eSTREAM proposal, the MICKEY stream ciphers. Some theoretical results with regards to the complexity of algebraic attacks on these ciphers are presented. The algebraic analysis of these ciphers showed that generally, it is hard to generate the system of equations required for an algebraic attack on these ciphers. As the algebraic attack could not be applied directly on these ciphers, a different approach was used, namely guessing some bits of the internal state, in order to reduce the degree of the equations. Finally, an algebraic attack on Alpha 1 that requires only 128 bits of keystream to recover the 128 internal state bits is presented. An essential process associated with stream cipher proposals is key initialization. Many recently proposed stream ciphers use an algorithm to initialize the large internal state with a smaller key and possibly publicly known initialization vectors. The effect of key initialization on the performance of algebraic attacks is also investigated in this thesis. The relationships between the two have not been investigated before in the open literature. The investigation is conducted on Trivium and Grain-128, two eSTREAM ciphers. It is shown that the key initialization process has an effect on the success of algebraic attacks, unlike other conventional attacks. In particular, the key initialization process allows an attacker to firstly generate a small number of equations of low degree and then perform an algebraic attack using multiple keystreams. The effect of the number of iterations performed during key initialization is investigated. It is shown that both the number of iterations and the maximum number of initialization vectors to be used with one key should be carefully chosen. Some experimental results on Trivium and Grain-128 are then presented. Finally, the security with respect to algebraic attacks of the well known LILI family of stream ciphers, including the unbroken LILI-II, is investigated. These are irregularly clock- controlled nonlinear filtered generators. While the structure is defined for the LILI family, a particular paramater choice defines a specific instance. Two well known such instances are LILI-128 and LILI-II. The security of these and other instances is investigated to identify which instances are vulnerable to algebraic attacks. The feasibility of recovering the key bits using algebraic attacks is then investigated for both LILI- 128 and LILI-II. Algebraic attacks which recover the internal state with less effort than exhaustive key search are possible for LILI-128 but not for LILI-II. Given the internal state at some point in time, the feasibility of recovering the key bits is also investigated, showing that the parameters used in the key initialization process, if poorly chosen, can lead to a key recovery using algebraic attacks.

Algebraic analysis of small Scale LEX-BES

This work examines the algebraic cryptanalysis of small scale variants of the LEX-BES. LEX-BES is a stream cipher based on the Advanced Encryption Standard (AES) block cipher. LEX is a generic method proposed for constructing a stream cipher from a block cipher, initially introduced by Biryukov at eSTREAM, the ECRYPT Stream Cipher project in 2005. The Big Encryption System (BES) is a block cipher introduced at CRYPTO 2002 which facilitates the algebraic analysis of the AES block cipher. In this article, experiments were conducted to find solutions of equation systems describing small scale LEX-BES using Gröbner Basis computations. This follows a similar approach to the work by Cid, Murphy and Robshaw at FSE 2005 that investigated algebraic cryptanalysis on small scale variants of the BES. The difference between LEX-BES and BES is that due to the way the keystream is extracted, the number of unknowns in LEX-BES equations is fewer than the number in BES. As far as the authors know, this attempt is the first at creating solvable equation systems for stream ciphers based on the LEX method using Gröbner Basis computations.

Correlations between designability and various structural characteristics of protein lattice models

Using six kinds of lattice types (4×4 ,5×5 , and6×6 square lattices;3×3×3 cubic lattice; and2+3+4+3+2 and4+5+6+5+4 triangular lattices), three different size alphabets (HP ,HNUP , and 20 letters), and two energy functions, the designability of proteinstructures is calculated based on random samplings of structures and common biased sampling (CBS) of proteinsequence space. Then three quantities stability (average energy gap),foldability, and partnum of the structure, which are defined to elucidate the designability, are calculated. The authors find that whatever the type of lattice, alphabet size, and energy function used, there will be an emergence of highly designable (preferred) structure. For all cases considered, the local interactions reduce degeneracy and make the designability higher. The designability is sensitive to the lattice type, alphabet size, energy function, and sampling method of the sequence space. Compared with the random sampling method, both the CBS and the Metropolis Monte Carlo sampling methods make the designability higher. The correlation coefficients between the designability, stability, and foldability are mostly larger than 0.5, which demonstrate that they have strong correlation relationship. But the correlation relationship between the designability and the partnum is not so strong because the partnum is independent of the energy. The results are useful in practical use of the designability principle, such as to predict the proteintertiary structure.

A different algebraic analysis of the ZUC stream cipher

Existing algebraic analyses of the ZUC cipher indicate that the cipher should be secure against algebraic attacks. In this paper, we present an alternative algebraic analysis method for the ZUC stream cipher, where a combiner is used to represent the nonlinear function and to derive equations representing the cipher. Using this approach, the initial states of ZUC can be recovered from 2^97 observed words of keystream, with a complexity of 2^282 operations. This method is more successful when applied to a modified version of ZUC, where the number of output words per clock is increased. If the cipher outputs 120 bits of keystream per clock, the attack can succeed with 219 observed keystream bits and 2^47 operations. Therefore, the security of ZUC against algebraic attack could be significantly reduced if its throughput was to be increased for efficiency.