Background Previous work Preliminaries Main results On the 2-Adic Complexity of A Class of Binary Sequences of Period 4 p with Optimal Autocorrelation Magnitude Minghui Yang joint work with Lulu Zhang and Keqin Feng State Key Laboratory of Information Security Institute of Information Engineering Chinese Academy of Sciences 2020-6-3 1 / 25
Background Previous work Preliminaries Main results Outline Background 1 Previous work 2 Preliminaries 3 Main results 4 2 / 25
Background Previous work Preliminaries Main results Outline Background 1 Previous work 2 Preliminaries 3 Main results 4 3 / 25
Background Previous work Preliminaries Main results The encryption process of the stream cipher Figure: The key stream sequence plays the role of masking the plaintext sequence. The key stream sequence should have good pseudorandomness in order to ensure that the attacker cannot recover the plaintext sequence even if they obtain the ciphertext sequence. 4 / 25
Background Previous work Preliminaries Main results Both correlation attacks and algebraic attacks cause serious threat to stream cipher system based on the linear feedback shift register. 5 / 25
Background Previous work Preliminaries Main results Both correlation attacks and algebraic attacks cause serious threat to stream cipher system based on the linear feedback shift register. It is main trend that using nonlinear feedback shift register sequences with good pseudo randomness as driving sequences in stream cipher design. 5 / 25
Background Previous work Preliminaries Main results Both correlation attacks and algebraic attacks cause serious threat to stream cipher system based on the linear feedback shift register. It is main trend that using nonlinear feedback shift register sequences with good pseudo randomness as driving sequences in stream cipher design. As a class of nonlinear sequence generator, the feedback shift register with carry has received lots of attention. 5 / 25
Background Previous work Preliminaries Main results Binary sequences with pseudo randomness: long period, low autocorrelation, large linear complexity, etc. the Rational Approximation Algorithm: the 2-adic complexity of a safe sequence should exceed half of its period. It is interesting to investigate the 2-adic complexity of sequences with optimal autocorrelation and large linear complexity. 6 / 25
Background Previous work Preliminaries Main results The 2-adic complexity of binary sequences S = ( s 0 , s 1 , . . . , s N − 1 ) with period N can be computed by 2 N − 1 gcd(2 N − 1 , S (2)) , where S (2) = s 0 + s 1 2 + · · · + s N − 1 2 N − 1 . log 2 The sequence s is called an optimal autocorrelation sequence if for any τ � = 0, the autocorrelation function C s ( τ ) satisfies (1) C s ( τ ) = − 1 for N ≡ 3 (mod 4); or (2) C s ( τ ) ∈ { 1 , − 3 } for N ≡ 1 (mod 4); or (3) C s ( τ ) ∈ { 2 , − 2 } for N ≡ 2 (mod 4); or (4) C s ( τ ) ∈ { 0 , − 4 } for N ≡ 0 (mod 4). 7 / 25
Background Previous work Preliminaries Main results Outline Background 1 Previous work 2 Preliminaries 3 Main results 4 8 / 25
Background Previous work Preliminaries Main results The known results about the 2-adic complexity of sequences with optimal autocorrelation m sequence Type (1) the maxi- property of m mum sequence known sequences Type (1-2) the maxi- determinant of mum a matrix all sequences Type (1) the maxi- autocorrelation mum function DHM sequence Type (3) close to the “Gauss sum- maximum s”, “Gauss period” a kind of sequences Type (4) the maxi- direct compu- mum tation 9 / 25
Background Previous work Preliminaries Main results The corresponding references T. Tian, W. F. Qi, 2-Adic complexity of binary m-sequences, IEEE Trans. Inf. Theory, 56(1): 450-454, 2010. H. Xiong, L. Qu, C. Li, A new method to compute 2-adic complexity of binary sequences, IEEE Trans. Inf. Theory, 60(4): 2399-2406, 2014. H. Hu, Comments on a new method to compute the 2-adic complexity of binary sequences, IEEE Trans. Inform. Theory, 60(4): 5803-5804, 2014. L. Zhang, J. Zhang, M. Yang, K. Feng, On the 2-adic complexity of the Ding-Helleseth-Martinsen binary sequences, IEEE Trans. Inf. Theory, DOI 10.1109/TIT.2020.2964171, 2020. Xiong, L. Qu, and C. Li, 2-Adic complexity of binary sequnces with interleaved structure, Finite Fields Appl., 33: 14-28, 2015. 10 / 25
Background Previous work Preliminaries Main results Outline Background 1 Previous work 2 Preliminaries 3 Main results 4 11 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude For a sequence with period N ≡ 0 (mod 4), if C s ( τ ) ∈ { 0 , ± 4 } when τ ranges from 1 to N − 1, then s is referred to as a sequence with optimal autocorrelation magnitude. W. Su, Y. Yang, C. Fan, New optimal binary sequences with period 4 p via interleaving Ding-Helleseth-Lam sequences, Des. Codes Cryptogr., 86(6): 1329 õ 1338, 2018. 12 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude The linear complexity of such sequences is close to its period. C. Fan, The linear complexity of a class of binary sequences with optimal autocorrelation, Des. Codes Cryptogr., 86(10): 2441 õ 2450, 2018. b = (0 , 1 , 0 , 1). The 2-adic complexity of such sequence S with period 4 p is proven to be no less than half of its period by using the method of autocorrelation function proposed by Hu. Conjecture: gcd( S (2) , 2 2 p + 1) = 5, where S (2) = s 0 + · · · + s 2 p − 1 2 2 p − 1 . Y. Sun, T. Yan, Z. Chen, The 2-adic complexity of a class of binary sequences with optimal autocorrelation magnitude, The 10th conference on sequences and their applications, 2018. 13 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude Let s ( t ) = ( s ( t ) 0 , s ( t ) 1 , . . . , s ( t ) N − 1 ) be a binary sequence of period N , where 0 ≤ t ≤ M − 1. 14 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude Let s ( t ) = ( s ( t ) 0 , s ( t ) 1 , . . . , s ( t ) N − 1 ) be a binary sequence of period N , where 0 ≤ t ≤ M − 1. An N × M matrix is obtained from these M binary sequences and given by s (0) s (1) s ( M − 1) · · · 0 0 0 s (0) s (1) s ( M − 1) · · · 1 1 1 U = . . . . ... . . . . . . s (0) s (1) s ( M − 1) · · · N − 1 N − 1 N − 1 14 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude An interleaved sequence u = ( u h ) of period MN is obtained by concatenating the successive rows of the matrix and defined by u iM + j = U i , j , 0 ≤ i < N , 0 ≤ j < M . 15 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude An interleaved sequence u = ( u h ) of period MN is obtained by concatenating the successive rows of the matrix and defined by u iM + j = U i , j , 0 ≤ i < N , 0 ≤ j < M . The sequence u is denoted by u = I ( s (0) , s (1) , ..., s ( M − 1)) for simplicity. 15 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude Let g be a primitive root of p . Define D j = { g j +4 i : 0 ≤ i ≤ p − 1 − 1 } for 0 ≤ j ≤ 3. 4 16 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude Let g be a primitive root of p . Define D j = { g j +4 i : 0 ≤ i ≤ p − 1 − 1 } for 0 ≤ j ≤ 3. 4 Let u = ( u 0 , u 1 , . . . , u N − 1 ) be a binary sequence of period N . The set B u = { t ∈ Z N : u t = 1 } is called the support of u . Let s 1 , s 2 , s 3 be the Ding-Hellseth-Lam sequences of period p with supports D 0 ∪ D 1 , D 0 ∪ D 3 , D 1 ∪ D 2 , respectively, where p = 4 f + 1 = x 2 + 4 y 2 is a prime number, f is odd and y = ± 1. 16 / 25
Background Previous work Preliminaries Main results The known results about interleaved sequences with optimal magnitude Lemma (Su et al.) Assume that p = 4 f + 1 = x 2 + 4 y 2 is a prime number, where f is odd and y = ± 1. Let b = ( b (0) , b (1) , b (0) , b (1)) be a binary sequence. Then the binary sequence of period 4 p constructed by S = I ( s 3 + b (0) , L d ( s 2 ) + b (1) , L 2 d ( s 1 ) + b (0) , L 3 d ( s 1 ) + b (1)) is optimal with respect to the autocorrelation magnitude. 17 / 25
Background Previous work Preliminaries Main results Outline Background 1 Previous work 2 Preliminaries 3 Main results 4 18 / 25
Background Previous work Preliminaries Main results Main results Theorem For the sequence S with b = (0 , 1 , 0 , 1), we have gcd( S (2) , 2 2 p + 1) = 5 . 19 / 25
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