Introduction A General Method for Binary Sequence Implementation Efficient Algorithm for the Linear Complexity of Sequences and Some Related Consequences Johan Chrisnata ISIT 2020 Joint work with: Yeow Meng Chee Tuvi Etzion Han Mao Kiah Johan Chrisnata Linear Complexity of Sequences 1 / 29
Introduction A General Method for Binary Sequence Implementation Outline 1 Introduction Motivation Problem Notations 2 A General Method for Binary Sequence Powers of Primitive Polynomial General Idea 3 Implementation For period p · 2 n , where 2 is a generator in F p Johan Chrisnata Linear Complexity of Sequences 2 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Outline 1 Introduction Motivation Problem Notations 2 A General Method for Binary Sequence Powers of Primitive Polynomial General Idea 3 Implementation For period p · 2 n , where 2 is a generator in F p Johan Chrisnata Linear Complexity of Sequences 3 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Motivation Binary sequences with good pseudorandomness and complexity properties are widely used as keystreams in cryptographic applications. Linear complexity c ( s ) is one of the measure. Sequences of low linear complexity are fully determined via a solution of c ( s ) linear equations if 2 c ( s ) consecutive terms of the sequence are known. Johan Chrisnata Linear Complexity of Sequences 4 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Motivation Binary sequences with good pseudorandomness and complexity properties are widely used as keystreams in cryptographic applications. Linear complexity c ( s ) is one of the measure. Sequences of low linear complexity are fully determined via a solution of c ( s ) linear equations if 2 c ( s ) consecutive terms of the sequence are known. Johan Chrisnata Linear Complexity of Sequences 4 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Motivation Binary sequences with good pseudorandomness and complexity properties are widely used as keystreams in cryptographic applications. Linear complexity c ( s ) is one of the measure. Sequences of low linear complexity are fully determined via a solution of c ( s ) linear equations if 2 c ( s ) consecutive terms of the sequence are known. Johan Chrisnata Linear Complexity of Sequences 4 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Berlekamp-Massey Algorithm The linear complexity of a sequence s of length N over a finite field F q can be determined with the well-known Berlekamp-Massey algorithm (1968,1969) in O ( N 2 ) symbol field operations. Improved to O ( N (log N ) 2 log log N ) by R. E. Blahut(1985) and also S. R. Blackburn(1997). However, in many applications, only periodic sequences are considered. For example, when the period is a power of two. Specific algorithm to find the linear complexity of such sequences, can be done in linear time. Johan Chrisnata Linear Complexity of Sequences 5 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Berlekamp-Massey Algorithm The linear complexity of a sequence s of length N over a finite field F q can be determined with the well-known Berlekamp-Massey algorithm (1968,1969) in O ( N 2 ) symbol field operations. Improved to O ( N (log N ) 2 log log N ) by R. E. Blahut(1985) and also S. R. Blackburn(1997). However, in many applications, only periodic sequences are considered. For example, when the period is a power of two. Specific algorithm to find the linear complexity of such sequences, can be done in linear time. Johan Chrisnata Linear Complexity of Sequences 5 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Berlekamp-Massey Algorithm The linear complexity of a sequence s of length N over a finite field F q can be determined with the well-known Berlekamp-Massey algorithm (1968,1969) in O ( N 2 ) symbol field operations. Improved to O ( N (log N ) 2 log log N ) by R. E. Blahut(1985) and also S. R. Blackburn(1997). However, in many applications, only periodic sequences are considered. For example, when the period is a power of two. Specific algorithm to find the linear complexity of such sequences, can be done in linear time. Johan Chrisnata Linear Complexity of Sequences 5 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Berlekamp-Massey Algorithm The linear complexity of a sequence s of length N over a finite field F q can be determined with the well-known Berlekamp-Massey algorithm (1968,1969) in O ( N 2 ) symbol field operations. Improved to O ( N (log N ) 2 log log N ) by R. E. Blahut(1985) and also S. R. Blackburn(1997). However, in many applications, only periodic sequences are considered. For example, when the period is a power of two. Specific algorithm to find the linear complexity of such sequences, can be done in linear time. Johan Chrisnata Linear Complexity of Sequences 5 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Past Result Length/Period Field Author 2 n Games and Chan F 2 (1983) p n F p t C. Ding (1991) and K. Imamura and T. Mo- riuchi (1993) p t , 2 p t where p F q , where q is a primitive Xiao et al. root modulo p 2 is odd (2000,2002) F p t ,where 2 n | p t − 1 and ℓ · 2 n Chen(2005) g . c . d . ( ℓ, p t − 1) = 1 F p t , where ℓ | p t − 1 and ℓ · n Chen(2006) g . c . d . ( n , p t − 1) = 1 ℓ · 2 n W. Meidl (2008) F 2 Johan Chrisnata Linear Complexity of Sequences 6 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Past Result Meidl(2008) proposed the most efficient algorithm for computing the linear complexities of binary sequences of period N = ℓ · 2 n , which is of interest for large N , a small odd integer ℓ such that the smallest k for which ℓ divides 2 k − 1 is not large. Johan Chrisnata Linear Complexity of Sequences 7 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Contribution Direct generalization of Games and Chan’s algorithm, namely the Powers of Primitive Polynomial (PPP) algorithm for an important class of sequences. Using PPP algorithm, we present a general idea to find linear complexity of binary sequence of any length N . The algorithm can handle efficiently sequences of even period as in Meidl’s paper(2008) and also binary sequences of some odd period. The minimal polynomial which generates the sequence is also computed in the algorithm, a feature that does not exist in the algorithm of Meidl. Johan Chrisnata Linear Complexity of Sequences 8 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Contribution Direct generalization of Games and Chan’s algorithm, namely the Powers of Primitive Polynomial (PPP) algorithm for an important class of sequences. Using PPP algorithm, we present a general idea to find linear complexity of binary sequence of any length N . The algorithm can handle efficiently sequences of even period as in Meidl’s paper(2008) and also binary sequences of some odd period. The minimal polynomial which generates the sequence is also computed in the algorithm, a feature that does not exist in the algorithm of Meidl. Johan Chrisnata Linear Complexity of Sequences 8 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Contribution Direct generalization of Games and Chan’s algorithm, namely the Powers of Primitive Polynomial (PPP) algorithm for an important class of sequences. Using PPP algorithm, we present a general idea to find linear complexity of binary sequence of any length N . The algorithm can handle efficiently sequences of even period as in Meidl’s paper(2008) and also binary sequences of some odd period. The minimal polynomial which generates the sequence is also computed in the algorithm, a feature that does not exist in the algorithm of Meidl. Johan Chrisnata Linear Complexity of Sequences 8 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Contribution Direct generalization of Games and Chan’s algorithm, namely the Powers of Primitive Polynomial (PPP) algorithm for an important class of sequences. Using PPP algorithm, we present a general idea to find linear complexity of binary sequence of any length N . The algorithm can handle efficiently sequences of even period as in Meidl’s paper(2008) and also binary sequences of some odd period. The minimal polynomial which generates the sequence is also computed in the algorithm, a feature that does not exist in the algorithm of Meidl. Johan Chrisnata Linear Complexity of Sequences 8 / 29
Introduction Motivation A General Method for Binary Sequence Problem Implementation Notations Contribution For the cases mentioned, the algorithm requires β N bit operations to compute the linear complexity of a binary sequence s of length N , where the constant β is relatively small. Johan Chrisnata Linear Complexity of Sequences 9 / 29
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