uoagr Abstract Pseudorandom sequences have many applications in cryp tography and spread spectrum communications In this dissertation on one hand we develop tools for assessing the randomness of a sequence and on the other hand we propose new constru ID: 26735 Download Pdf

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uoagr Abstract Pseudorandom sequences have many applications in cryp tography and spread spectrum communications In this dissertation on one hand we develop tools for assessing the randomness of a sequence and on the other hand we propose new constru

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Study on Pseudorandom Sequences with Applications in Cryptography and Telecommunications Rizomiliotis Panagiotis National and Kapodistrian University of Athens, Department of Informatics and Telecommunications, University Campus, 15784 Athens, Greece rizop@di.uoa.gr Abstract. Pseudorandom sequences have many applications in cryp- tography and spread spectrum communications. In this dissertation, on one hand we develop tools for assessing the randomness of a sequence, and on the other hand we propose new constructions of pseudorandom sequences. More precisely, we develop tools

for computing the ﬁrst order approximation of a binary sequence with the minimum linear complexity, we propose two eﬃcient algorithms for computing the second order com- plexity (quadratic span) of a binary sequence, and we consider and solve the problem of computing the maximum nonlinear complexity (span) of a sequence. Finally, we investigate the properties of a family of sequences constructed as the direct sum of two sequences with ideal autocorrela- tion, like the GMW sequences. 1 Introduction Traditionally, pseudorandom sequences have been employed in numerous appli-

cations, for instance in spr ead spectrum, code division multiple access, optical and ultrawideband communication systems, in ranging systems, global position- ing systems, circuit testing and cryptography. In this dissertation we concentrate on spread spectrum communications ([11]) and cryptography ([2], [4], [9]). Depending on the context sequences are r equired to possess certain properties such as long period, balance of symbols, good correlation properties and large linear and nonlinear complexity. When families of pseudorandom sequences are applied in a code division multiple access

(CDMA) system, low crosscorrelation combats interference from other users, whereas low out–of–phase autocorrela- tion facilitates synchronization. Furthermore, large linear complexity protects from jamming. On the other hand, sequences with low correlation values em- ployed in cryptosystems, like stream ciphers, are resistant to correlation attacks, while sequences with large linear span resist register synthesis attacks, like the Berlekamp-Massey Algorithm (BMA) ([8]). An informed overview in designing such sequences is given in [5]. Prof. Nicholas Kaloupsidis was the supervisor of this

thesis.

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In this dissertation, on one hand we develop tools for assessing the random- ness of a sequence, and on the other hand we propose new constructions of pseu- dorandom sequences. More precisely, we develop tools for computing the ﬁrst order approximation of a binary sequence with the minimum linear complexity. Moreover, we propose for the ﬁrst time two eﬃcient algorithms for computing the second order complexity ( quadratic span ) of a binary sequence. In addition, we consider and solve the problem of co mputing the maximum nonlinear com- plexity (

span ) of a sequence (the proposed al gorithm is linear with respect of the length of the sequence). Finally, w e investigate the properties of a family constructed as the direct sum of two sequ ences with ideal autocorrelation, like the GMW sequences. The dissertation is organized as follows. Chapter 1 presents an introduction on sequences and their applications. Chapter 2 establishes the necessary math- ematical background needed in the sequential chapters. In Chapter 3, the linear complexity stability of a binary sequence is investigated. In Chapter 4, we pro- pose two eﬃcient

algorithms for computing the quadratic span of a sequence. The ﬁrst algorithm exploits the properties of the equivalent system of linear equations, while the second is a modiﬁcation of the well known fundamental it- erative algorithm (FIA) ([3]). In Chapter 5, an algorithm for the calculation of the span of a binary sequence is introdu ced, and closed form expressions con- necting the cardinality of the set of bin ary ﬁnite sequences of the same length with their span value are presented. In Chapter 6, a family is constructed as the direct sum of two GMW sequences. This

method is applicable to any pair of sequences with ideal autocorrelation. We conclude by proposing some problems that need further investigation. 2 Background In this section we present the notation we are going to use in this abstract paper, as well as some elementary mathematical background. For more details please refer to the full version of the dissertation. Let be the prime ﬁeld ,andlet be the th dimensional vector space over ([7]). Consider the ﬁnite–length binary sequence ,... ,s An –stage feedback shift register (FSR) ( see Fig. 1) with feedback function generates if and

only if ,s ,...,s )(1) for all 0 . The boolean function canbewrittenintheso–called algebraic normal form (ANF) as follows ,...,x )= ,... ,m A product of termsissaidtobea th order product. The ﬁrst and the second order products are also called linear terms and quadratic terms respectively. The

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nonlinear feedback function Fig. 1. The block diagram of a feedback shift register. order of is deﬁned to be the maximum of the orders of the product terms appearing in the algebraic normal form with nonzero coeﬃcients. Deﬁnition 1. The length of the shortest

FSR having a feedback function of order that generates deﬁnes the th order complexity of sequence ,and is denoted by Deﬁnition 2. The length of the shortest FSR that generates is called mini- mum nonlinear complexity or span of sequence , and is given by SPAN min The second order complexity of a sequence is referred to as the quadratic span of the sequence and it is denoted by QS ). In case the feedback function is linear and (0 ,..., 0) = 0, we deﬁne the linear complexity .Itis clear that QS SPAN Let and be two periodic binary sequences of period . Their periodic

crosscorrelation function is deﬁned as x,y )= =0 1) When, the autocorrelation function is deﬁned. Deﬁnition 3. Two sequences of the same period are said to be cyclically equiv- alent, if they are related by a left (or right) cyclic shift. Otherwise, they are cyclically distinct. 3 First Order Approximation of Binary Sequences Let be a binary sequence of period =2 1and its linear span. The Berlekamp–Massey algorithm needs 2 sequence digits in order to determine and the linear feedback shift register ( LFSR) associated to the least order homogeneous linear recursion ([8]).

Therefore, the linear span is a critical index

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for assessing the strength of a sequence against linear cryptanalytic attacks (such as the Berlekamp–Massey algorithm). However, even a large linear span does not ensure that a sequence is cryptographically secure. Consider the periodic extension of the length vector (0 ... 01). This sequence has linear span but can be approximated by the all–zer o sequence with linear span 0. In many practical applications small deviations from a given sequence can be tolerated if substantial gains in the linear span are achieved. In this context

the approximate realization pro blem becomes relevant. Let =( ,... ,s contain the ﬁrst elements of sequence and be a binary vector of weight 1, the single one being the th digit, where and ,...,N Deﬁne , where + denotes modulo 2 addition. The approximate realization problem is equivalent to the following minimization problem WC min (2) WC ) is known in the literature as the weight complexity of sequence [2]. It is conceivable that the approximation problem is of interest when WC )is less than the linear span of the original sequence. For this reason necessary conditions are

provided. The optimal position opt is referred to as the optimal shift The solution of the approximate realization problem is the main subject of Chapter 3 ([12], [13]). Three methods are developed in order to determine the optimal shift, namely the sequential divisions method, the congruential equa- tions method and the phase synchronization method. The sequential divisions method relies on the repetitive application of the Euclidean algorithm by factors of the minimal polynomial of . The congruential equations method works in the frequency domain and determines t he optimal lag through a set

of linear congruential equations. The solvability of these equations is analyzed and closed form expressions are derived. The phase synchronization method is based on the trace representation and builds upon cyclic equivalence in order to identify opt The three issues of characterization, algorithm implementation and sequence design are tightly related with the approximate realization problem. Charac- terization is concerned with the description of sequences whose ﬁrst–order approximations possess a given linear span. Of particular importance are those sequences for which WC . Such

sequences are called robust because their ﬁrst–order approximations do not modify their complexity performance. Directives for the design of robust sequences are proposed. Algorithm implementation is primarily concerned with the design of eﬃcient algorithms in order to determine WC ), opt (1) and (1) ). An algorithm following a decoding approach is given in [2] for determining the optimal shift but its complexity is high. Furthermore, it does not provide any insight to the design of binary sequences which are robust to such approximation attacks. High level algorithm

organizations for the proposed schemes are presented.

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4 On the Quadratic Span of Binary Sequences In Chapter 4, we investigate the quadratic span of ﬁnite binary sequences ([14], [15]). In [1], the quadratic span of the de Bruijn sequences was studied, and a partial generalization of the Berlekamp–Massey algorithm, based on Gaus- sian elimination, was proposed. Two more eﬃcient algorithm for calculating the quadratic span of a sequence are described in this chapter. The ﬁrst one takes advantage of the special structure of the corresponding linear systems

of equations. Let n,m )=( +1 ,andlet )be the quadratic span of the . In connection with the algebraic normal form we introduce the vector )= ,m ,m (3) which contains the coeﬃcients of the unknown quadratic feedback function From (1), the calculation of a quadratic feedback function that generates a given sequence is equivalent to solving the system of linear equations n,m )= n,m )(4) where n,m )isthe( +1) 2matrix n,m )= +1 +1 Based on the following Theorems, we developed an iterative algorithm that computes the quadratic span of a ﬁnite binary sequence. Theorem 1. Let be the

greatest integer such that rank +1 ,d )) = rank n,d )) Then +1)= if rank +1 ,d )) = rank +1 ,d +1 ,d )) +1 otherwise Theorem 2. Let +1)= )+ ,where δ> . Then it holds +2+ )= +1) , for all [0 , 1] After the computation of the quadratic span, we solve the system of linear equa- tions (4), in order to ﬁnd the feedback function of the corresponding FSR. The second algorithm is a modiﬁed version of the fundamental iterative al- gorithm (FIA). FIA was introduced in [3] for solving the multi-sequence shift register synthesis problem. The goal of the algorithm is to ﬁnd the

smallest initial set of columns, in a given matrix, which are linearly independent.

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5 Results on the Nonlinear Span of Binary Sequences The span was studied by Jansen and Boekee ([6]). We proved similar results using a diﬀerent viewpoint, based on the special properties of the corresponding system of linear equations. An eﬃcient algorithm was also introduced for computing the span and a feedback function that generates the given ﬁnite binary sequence. Finally, the properties of the cardinality of the set of ﬁnite sequences with the same span were

studied. The results of Chapter 5 appear in [14], and [15]. Let sp ) denote the span of and n,m )= +1 .From(1), the calculation of a feedback function that generates a given sequence is equivalent to solving the system of linear equations n,m )= n,m )(5) where n,m )isthe( matrix N,m )= LP n,m NLP n,m denotes the all-one vector of length and the matrix LP n,m ) consists of subsequences of LP n,m )= +1 +1 while NLP n,m ) consists of all termwise product combinations of the columns of LP n,m ). ) contains the coeﬃcients of the unknown feedback function writtenintheANF. Our analysis is

based on the block structure of n,m ). The algorithm is divided in two steps. First we compute the span sp ) of the sequence, and then a feedback function of sp ) variables that produces the given ﬁnite binary sequence. The computation of the span is performed by processing the sequence element by element. The following two Theorems describe the way the value of the span changes. Theorem 3. sp >sp 1) if and only if there is an integer sp 1) , such that sp 1) n,sp 1)) = n,sp 1)) and sp 1) where n,m denotes the th row of n,m Theorem 4. If sp >sp 1) ,then sp )= sp 1) + ,where +1 sp 1) and

+1 is the index of the ﬁrst linear dependent row of n,sp 1))

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In order to compute a boolean feedback function of sp ) variables that generates the sequence, we have to solve the system (1). Due to the special structure of n,m ), the 2 possible diﬀerent rows of the matrix form a base )of GF (2) over GF (2), which can always be written as a lower tri- angular matrix. Thus, using appropriate outputs of the span algorithm, we show that the system (5) can be easily reduced to a low triangular system of rank n,sp ))) equations and variables that can be easily solved by

back substitution. The other 2 sp variables of sp )) that do not appear in the reduced system are set equal to zero. The system of linear equations (5) has 2 sp degrees of freedom. Thus, there are 2 sp functions with sp ) variables that can produce the same sequence . In the case of periodic sequences of period ,itholds Finally, we study the cardinality of n,SP ), the set of binary sequences of length with span SP ,as varies. The main results on the span distribution follow. Let δ> 0. 1. (2 SP SP δ, SP (2 SP SP,SP 2. δ, n, ,for even. 6 Construction of Sequences with four-valued

Autocorrelation from GMW Sequences One of the most important families of pseudorandom sequences are Gordon, Mills, Welch (GMW) sequences ([10]). The GMW sequences and their gener- alization called cascaded GMW sequences have been extensively studied in the literature. In Chapter 6, we describe the construction of a large class of balanced bi- nary sequences with four–valued autocorrelation function. Binary sequences with good autocorrelation properties play an important role in communication sys- tems employing phase–reversal modulation techniques. The construction is based on the modulo 2

addition of two GMW sequences with relatively prime periods. The resulting sequences have period equal to the product of the periods. Addi- tionally, other characteristics of the class members, such as the linear span and the periodic crosscorrelation, are investigated ([18]). Deﬁnition 4 ([10]). Let n,k be two integers such that lk ,and be an integer in the range relatively prime to . Consider the binary sequence given by =tr tr ti (6) where is a primitive element of ,and is an integer in the range relatively prime to .Then, is a GMW sequence. The above deﬁnition implies that

GMW sequences are periodic with least period =2 1. Some of the properties of a GMW sequence are the following [10]:

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i. The sequence has the ideal autocorrelation property )= 1if 0(mod /N otherwise (7) ii. Sequence is balanced. iii. The total number of cyclically distinct GMW sequences of period =2 is GMW (2 1) (2 1) (8) where the summation is over all divisors of and ) denotes the Euler’s totient function. We present a new approach for the calculation of the periodic crosscorrelation function of two GMW sequences whose lea st periods are diﬀerent. In accordance with the

above analysis we assume that these sequences, say and ,aregivenby =tr tr (9) and =tr tr (10) where the ﬁeld elements and are primitive elements of and re- spectively. Let us assume that the integer (resp. ) is relatively prime to =2 1(resp. =2 1). Then, sequence (resp. ) has least period (resp. ). Let us denote by the greatest common divisor of and and let =lcm( ,N )= /d We prove that their crosscorrelation function becomes x,y )= =0 mod ) (11) where the sequence and correspond to the autocorrelation of a special dec- imation of and respectively. Of special interest is the case

=1,wherewe get x,y )=1 /N for all integers Next, we examine the properties of binary sequences constructed by the mod- ulo 2 addition of two GMW sequences whose least periods are diﬀerent. Deﬁne the sequence which is given by ,wherethe sequences and are deﬁned in (9) and (10) respectively. We proved the following theorem

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Theorem 5. The spectrum of the autocorrelation function of sequence deﬁned as described above, is given by )= if 0(mod /N if 0(mod /N if 0(mod (0) otherwise Of special interest is the case where the component GMW sequences have

relative prime periods, i.e. = 1. We introduce the sets and which contain all cyclically distinct GMW sequences with periods =2 1and =2 respectively, and the set and for all with 0 i< min ,N }} where gcd( ,N ) = 1. Recall that GMW for =1 2. Clearly, || | min ,N Corollary 1. The spectrum of the autocorrelation function of sequence is four–valued and is given by )= if 0(mod /N if 0(mod /N if 0(mod /N otherwise where . Moreover, the value 1 occurs one time, /N occurs times, /N occurs times and /N occurs +1 times. The linear span of a sequence depends on its component sequences and as the

following Lemma indicates. Lemma 1. Let .Then, (12) Finally we compute the crosscorrelation function of two members of the family . The above results can be easily extended in the case of any family of sequences with ideal autocorrelation. References 1. A. H. Chan and R. A. Games, “On the quadratic spans of de Bruijn sequences, IEEE Trans. Inform. Theory , vol. IT–36, pp. 822–829, Jul. 1990. 2. T. W. Cusick, C. Ding, and A. Renvall, Stream Ciphers and Number Theory North–Holland Mathematical Library. Elsevier Science, 1998.

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3. G.-L. Feng and K. K. Tzeng, “A generalization of

the Berlekamp–Massey al- gorithm for multisequence shift–register synthesis with applications to decoding cyclic codes, IEEE Trans. Inform. Theory , vol. IT–37, pp. 1274–1287, Sep. 1991. 4. S. W. Golomb, Shift Register Sequences . Holden–Day, San Francisco, 1967. 5. T. Helleseth and V. J. Kumar, “Sequences with low correlation, correlation func- tions of geometric sequences, in Handbook of Coding Theory ,V.Plessand C. Huﬀman, Eds. Amsterdam, The Netherlands: Elsevier, 1998. 6. C. J. Jansen and D. E. Boekee, “The shortest feedback shift register that can gen- erate a given sequence,” in

Proc. Advances in Cryptology–CRYPTO ’89 , pp. 90 99, 1990. 7. R. Lidl and H. Niederreiter, Finite Fields ,vol.20of Encyclopedia of Mathematics and its Applications . Cambridge University Press, 1996, 2nd ed. 8. J. L. Massey, “Shift register synthesis and BCH decoding, IEEE Trans. Inform. Theory , vol. IT–15, pp. 122–127, Jan. 1969. 9. A. J. Menezes, P. C. Van Oorschot, and S. A. Vanstone, Handbook of Applied Cryptography . CRC Press, 1996. 10. R. A. Scholtz and L. R. Welch, “GMW sequences, IEEE Transactions on Infor- mation Theory , vol. IT–30, pp. 548–553, May 1984. 11. M. K. Simon, J. K.

Omura, R. A. Scholtz, and B. K. Levitt, Spread Spectrum Communications Handbook . McGraw–Hill, 1994, Revised ed. 12. N. Kolokotronis, P. Rizomiliotis, and N. K alouptsidis, “First–order optimal ap- proximation of binary sequences,” in Proc. Conference on Sequences and Their Applications , T. Helleseth, P. V. Kumar, and K. Yang (Eds). Springer-Verlag. Series in Discrete Mathematics and Theoretical Computer Science, pp. 242-256, May 2001. 13. N. Kolokotronis, P. Rizomiliotis, and N. Kalouptsidis, “Minimum linear span approximation of binary sequences, IEEE Transactions on Information Theory vol.

IT–48, pp. 2758–2764, Oct. 2002. 14. P. Rizomiliotis, N. Kolokotronis, and N. Kalouptsidis, “On the Quadratic Span of Binary Sequences,” in Proc. IEEE Inter. Symp. on Inform. Theory , pp. 377, 2003. 15. P. Rizomiliotis, N. Kolokotronis, and N. Kalouptsidis, “On the Quadratic Span of Binary Sequences, IEEE Transactions on Information Theory , vol. IT–51, pp. 1840–1848, May 2005. 16. P. Rizomiliotis, and N. Kalouptsidis, “Result on the nonlinear span of binary sequences,” in Proc. IEEE Inter. Symp. on Inform. Theory , pp. 124, 2004. 17. P. Rizomiliotis, and N. Kalouptsidis, “Results on the

nonlinear span of binary sequences, IEEE Transactions on Information Theory , vol. IT–51, pp. 1555 1563, April 2005. 18. P. Rizomiliotis, N. Kolokotronis, and N. Kalouptsidis, “Construction of sequences with four–valued autocorrelation from GMW sequences,” in Proc. IEEE Inter. Symp. on Inform. Theory , pp. 183, 2002.

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