The Numbers of De Bruijn Sequences in Extremal Weight Classes Ming - PowerPoint PPT Presentation

The Numbers of De Bruijn Sequences in Extremal Weight Classes Ming Li, Yupeng Jiang, Dongdai Lin State Key Laboratory of Information Security, Institute of Information Engineering Chinese Academy of Sciences, Beijing, China May 26, 2020 1 / 16

Introduction - Feedback shift registers feedback function . . . x n − 1 x 0 x 1 Figure: feedback shift register Output sequence: s 0 , s 1 , . . . s p − 1 , . . . , with p ≤ 2 n . (non-singular) Feedback function: x n = x 0 + f ( x 1 , x 2 , . . . , x n − 1 ). Censor function: f ( x 1 , x 2 , . . . , x n − 1 ). Related concepts: LFSR, NFSR, cycle structure 2 / 16

Introduction - Definition of de Bruijn sequences: Definition An n -th order De Bruijn sequence is a sequences of period 2 n such that each n -tuple appears exactly once in one period. Example The sequence, (0000111101001011), is a 4-th order De Bruijn sequence. The sequence, (0000001111011100110110100111010100011001011000101011111100001001), is a 6-th order De Bruijn sequence. 3 / 16

Introduction - Basic results about de Bruijn sequences 1 The number of n -th order de Bruijn sequences is 2 2 n − 1 − n . 2 The linear span of an n -th order de Bruijn sequence satisfies, 2 n − 1 + n ≤ LC ( s ) ≤ 2 n − 1. 3 The k -error linear span of an n -th order de Bruijn sequence satisfies, 2 n − 1 + 1 ≤ LC k ( s ) < 2 n , when k ≤ ⌈ 2 n − 1 n ⌉ . 4 · · · De Bruijn sequences have many applications in communication systems, coding theory and cryptography. 4 / 16

Introduction - Weight classes of de Bruijn sequences Fredricksen showed that the weights of de Bruijn sequences are odd numbers and in the range Z ( n ) ≤ wt ( g ) ≤ 2 n − 1 − Z ∗ ( n ) + 1 , where Z ( n ) = 1 Z ∗ ( n ) = 1 n n � � d , φ ( n )2 and φ ( n )2 d 2 n n d | n d | n d odd are the number of cycles in the n -th order pure circulating register and complementing circulating register respectively. H. Fredricksen, “A survey of full length nonlinear shift register cycle algorithms”, SIAM Rev., vol.24, no. 2, pp. 195-221, Apr. 1982. 5 / 16

Introduction - Related conjectures Conjectures : 1 S max ( n ) divides S min ( n ). 2 S min ( n ) divides S min ( n + 1). 3 S max ( n ) divides S max ( n + 1). 4 For a prime p and order n , p divides S min ( n ) for all p < n . 5 For a prime p and order n , p divides S max ( n ) for all 2 p < n . 6 If 2 α || S max ( n ), then 2 α | η ( w , n ) for any w and n . H. Fredricksen, “A survey of full length nonlinear shift register cycle algorithms”, SIAM Rev., vol.24, no. 2, pp. 195-221, Apr. 1982. G. L. Mayhew, “Extreme weight classes of de Bruijn sequences,”Discrete Mathematics, 2002. 6 / 16

Introduction - Distribution of de Bruijn sequences Define S ( n ) to be the set of functions that generate a de Bruijn sequence of order n . Define S ( f ; k ) to be the set of g ∈ S ( n ) such that the weight of g + f is k . Moreover, let N ( f ; k ) = | S ( f ; k ) | . Lemma Let l : F n − 1 → F 2 be a linear function. Then 2 � (1 + y ) wt ( c ) − (1 − y ) wt ( c ) � G ( l ; y ) = 2 − n � . (1) c � = 0 ∈ C ( l ) D. Coppersmith, R. Rhoades, and J. Vanderkam, “Counting De Bruijn sequences as perturbations of linear recursions”, arXiv, 2017. 7 / 16

Weight class distributions of de Bruijn sequences In the case of l = 0, we get, G (0; y ) = 2 − n · [(1 + y ) w − (1 − y ) w ] e ( w , n ) , � 1 ≤ w ≤ n where e ( w , n ) is the number of cycles of weight w in the pure circulating register. Lemma � � 1 l / s � � e ( w , n ) = l µ ( s ) . w / s l | n s | gcd( w , l ) 8 / 16

The extremal weight classes By expanding (1 + y ) w − (1 − y ) w , we get that,  y 3 + . . . + 2 � w � w � w y w 2 � y + 2 � � If w is odd (1 + y ) w − (1 − y ) w =  1 3 w � w y 3 + . . . + 2 � w � w y w − 1 2 � y + 2 � � If w is even .  1 3 w − 1 Theorem Let N min ( n ) and N max ( n ) be the numbers of de Bruijn sequences in the minimal and maximal weight classes, respectively. Then, 1. N min ( n ) = 2 Z ( n ) − n − 1 · � w e ( w , n ) . 1 ≤ w ≤ n − 1 2. N max ( n ) = 2 Z ( n ) − n − 1 · � w e ( w , n ) . 1 ≤ w ≤ n − 1 w even 9 / 16

De Bruijn sequences in the extremal weight classes Order S max S min 1 1 1 2 1 1 3 2 2 2 2 × 3 2 2 4 2 6 × 3 2 2 6 5 2 14 × 3 3 × 5 2 14 6 2 26 × 3 2 26 × 3 6 × 5 3 7 2 50 × 3 3 2 50 × 3 11 × 5 7 × 7 8 2 95 × 3 9 2 95 × 3 18 × 5 14 × 7 4 9 2 177 × 3 20 2 177 × 3 36 × 5 25 × 7 12 10 2 329 × 3 42 × 5 2 329 × 3 67 × 5 43 × 7 30 11 2 632 × 3 75 × 5 5 2 632 × 3 133 × 5 72 × 7 66 × 11 12 2 1187 × 3 133 × 5 22 2 1187 × 3 265 × 5 121 × 7 132 × 11 6 13 2 2257 × 3 219 × 5 70 2 2257 × 3 536 × 5 216 × 7 245 × 11 26 × 13 14 2 4251 × 3 363 × 5 200 × 7 2 4251 × 3 1061 × 5 400 × 7 430 × 11 91 × 13 7 15 2 7978 × 3 612 × 5 497 × 7 7 2 7978 × 3 2086 × 5 778 × 7 723 × 11 273 × 13 35 16 2 14916 × 3 1092 × 5 1144 × 7 40 2 14916 × 3 4000 × 5 1516 × 7 1184 × 11 728 × 13 140 17 2 27952 × 3 2061 × 5 2424 × 7 168 2 27952 × 3 7563 × 5 2960 × 7 1940 × 11 1768 × 13 476 × 17 18 2 52338 × 3 4082 × 5 4862 × 7 612 2 52338 × 3 14061 × 5 5678 × 7 3264 × 11 3978 × 13 1428 × 17 9 19 2 98535 × 3 8258 × 5 9225 × 7 1932 2 98535 × 3 25903 × 5 10800 × 7 5820 × 11 8398 × 13 3876 × 17 57 × 19 20 10 / 16

Weight class distribution for de Bruijn sequences of order 7 Number of sequences Weight class 2 26 × 3 6 × 5 3 19 2 30 × 3 6 × 5 3 21 2 26 × 3 5 × 5 2 × 7 × 13 × 19 23 2 30 × 3 3 × 5 2 × 59 × 71 25 2 28 × 3 4 × 5 × 80513 27 2 30 × 3 2 × 5 × 7 × 52973 29 2 28 × 17 × 1567 × 3769 31 2 30 × 5 × 181 × 31307 33 2 27 × 461 × 421607 35 2 30 × 3 × 5 2 × 11 × 19301 37 2 27 × 3 2 × 5 2 × 283949 39 2 30 × 3 × 11 × 19 × 4871 41 2 28 × 5 × 13 × 67 × 811 43 2 30 × 41 × 4637 45 2 28 × 5 2 × 11 × 433 47 2 30 × 5 × 653 49 2 26 × 7 3 × 11 51 2 31 × 5 53 2 26 × 3 55 11 / 16

Weight class distribution for de Bruijn sequences of order 8 Number of sequences Number of sequences Weight class Weight class 2 50 × 3 11 × 5 7 × 7 2 51 × 21758744604075000469 35 75 2 50 × 3 10 × 5 7 × 7 2 × 17 2 51 × 3 × 61 × 5747701 × 10537594667 37 77 2 50 × 3 9 × 5 6 × 7 2 × 4799 2 51 × 13 × 47 × 30711169 × 263121107 39 79 2 50 × 3 9 × 5 6 × 11 × 254281 2 51 × 7 × 37 × 1879 × 1561463 × 2527559 41 81 2 50 × 3 8 × 5 5 × 31 × 11436329 2 52 × 5 2 × 7 × 1860740956442243 43 83 2 50 × 3 7 × 5 5 × 13 × 367 × 1425883 2 52 × 541 × 8151161 × 21768917 45 85 2 50 × 3 6 × 5 5 × 61 × 1685504311 2 52 × 7 × 800959 × 4374440849 47 87 2 50 × 3 5 × 5 5 × 197 × 12569 × 507809 2 52 × 5 × 7 × 73 × 2117079729071 49 89 2 52 × 3 4 × 5 4 × 103 × 1493 × 1583 × 65141 2 52 × 7 × 11 × 1048391 × 12694057 51 91 2 52 × 3 2 × 5 5 × 7 × 43 × 109 × 139 × 1063 × 16573 2 52 × 7 × 761 × 1613 × 2003 × 9631 53 93 2 52 × 3 × 5 2 × 17 × 43 × 98168800363397 2 52 × 7 × 23 × 141135121727 55 95 2 52 × 5 2 × 7 2 × 29 × 373 × 825689158081 2 52 × 5 × 13 × 40239850067 57 97 2 52 × 3 2 × 5 × 13 × 129287 × 252672123113 2 50 × 3 × 333168905291 59 99 2 52 × 3 2 × 5 × 197 × 367 × 653 × 13571044169 2 50 × 5 2 × 37 × 84395329 61 101 2 52 × 3 × 13 × 193 × 8461 × 592416520393 2 50 × 17 × 157 × 1830833 63 103 2 52 × 3 × 157 × 487 × 154459 × 1212559573 2 50 × 3 × 19 × 4183841 65 105 2 51 × 7 × 71 × 570013 × 301237031203 2 50 × 8716843 67 107 2 51 × 5 × 23 × 644071742408998313 2 50 × 3 2 × 7 × 11 × 17 × 19 69 109 2 51 × 331 × 20735557 × 8194340813 2 50 × 3 3 × 7 × 19 71 111 2 51 × 3 × 486569 × 25611783385373 2 50 × 3 3 73 113 12 / 16

Proof of Conjecture 1 Conjecture 1: S max ( n ) divides S min ( n ) . S min ( n ) = 2 − n · � (2 w ) e ( w , n ) . 1 ≤ w ≤ n S max ( n ) = 2 − n · 2 e ( w , n ) · � � (2 w ) e ( w , n ) . 1 ≤ w ≤ n 1 ≤ w ≤ n w even w odd Conjecture 1 is proved by noting that, S min ( n ) � w e ( w , n ) . S max ( n ) = 1 ≤ w ≤ n w odd 13 / 16

Proofs of Conjectures 4 and 5 Conjecture 4: For a prime p and order n , p divides S min ( n ) for all p < n . Conjecture 5: For a prime p and order n , p divides S max ( n ) for all 2 p < n . Note that: e ( w , n ) = 0 for w = n , and e ( w , n ) ≥ 1 for 1 ≤ w ≤ n − 1. Recall that, S min ( n ) = 2 − n · � (2 w ) e ( w , n ) , 1 ≤ w ≤ n which is, S min ( n ) = 2 − n · 2 e (1 , n ) · 4 e (2 , n ) . . . (2 n ) e ( n , n ) Therefore we have, P ( S min ( n )) = P (( n − 1)!) = { p is a prime | 1 < p < n } . Similarly, P ( S max ( n )) = P ( ⌊ ( n − 1) / 2 ⌋ !) = { p is a prime | 1 < p < n / 2 } . 14 / 16

Proof of Conjecture 6 Conjecture 6: If 2 α || S max ( n ), then 2 α | η ( w , n ) for any w and n . Recall that, G (0; y ) = 2 − n · [(1 + y ) w − (1 − y ) w ] e ( w , n ) , � (2) 1 ≤ w ≤ n where e ( w , n ) is the number of cycles of weight w in the pure circulating register.  y 3 + . . . + 2 � w � � w � � w � y w 2 y + 2 If w is odd  1 3 w p w ( y ) = (3) � w y 3 + . . . + 2 � w � � w � � y w − 1 2 y + 2 If w is even .  1 3 w − 1 Let w = 2 l u , where u is an odd number. Let 1 ≤ t ≤ w be an odd number. We just need to show that 2 l | � w � . This is true because t is odd and, t � � � � w = w w − 1 . t t t − 1 15 / 16

Thank you for your attention! 16 / 16