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

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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 /

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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

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Introduction - Feedback shift registers

x0 x1 xn−1 . . . feedback function

Figure: feedback shift register

Output sequence: s0, s1, . . . sp−1, . . ., with p ≤ 2n. (non-singular) Feedback function: xn = x0 + f (x1, x2, . . . , xn−1). Censor function: f (x1, x2, . . . , xn−1). Related concepts: LFSR, NFSR, cycle structure

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Introduction - Definition of de Bruijn sequences:

Definition An n-th order De Bruijn sequence is a sequences of period 2n 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

  • rder De Bruijn sequence.

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Introduction - Basic results about de Bruijn sequences

1 The number of n-th order de Bruijn sequences is 22n−1−n. 2 The linear span of an n-th order de Bruijn sequence satisfies, 2n−1 + n ≤ LC(s) ≤ 2n − 1. 3 The k-error linear span of an n-th order de Bruijn sequence satisfies, 2n−1 + 1 ≤ LCk(s) < 2n, when k ≤ ⌈ 2n−1

n ⌉.

4 · · · De Bruijn sequences have many applications in communication systems, coding theory and cryptography.

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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) ≤ 2n−1 − Z ∗(n) + 1, where Z(n) = 1 n

  • d|n

φ(n)2

n d

and Z ∗(n) = 1 2n

  • d|n

d odd

φ(n)2

n d ,

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.

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Introduction - Related conjectures

Conjectures: 1 Smax(n) divides Smin(n). 2 Smin(n) divides Smin(n + 1). 3 Smax(n) divides Smax(n + 1). 4 For a prime p and order n, p divides Smin(n) for all p < n. 5 For a prime p and order n, p divides Smax(n) for all 2p < n. 6 If 2α||Smax(n), then 2α|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.

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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 : Fn−1

2

→ F2 be a linear function. Then G(l; y) = 2−n

  • c=0∈C(l)
  • (1 + y)wt(c) − (1 − y)wt(c)

. (1)

  • D. Coppersmith, R. Rhoades, and J. Vanderkam, “Counting De Bruijn sequences as

perturbations of linear recursions”, arXiv, 2017.

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Weight class distributions of de Bruijn sequences

In the case of l = 0, we get, G(0; y) = 2−n ·

  • 1≤w≤n

[(1 + y)w − (1 − y)w]e(w,n) , where e(w, n) is the number of cycles of weight w in the pure circulating register. Lemma e(w, n) =

  • l|n
  • s| gcd(w,l)

1 l µ(s)

  • l/s

w/s

  • .

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The extremal weight classes

By expanding (1 + y)w − (1 − y)w, we get that, (1 + y)w − (1 − y)w =    2 w

1

  • y + 2

w

3

  • y 3 + . . . + 2

w

w

  • y w

If w is odd 2 w

1

  • y + 2

w

3

  • y 3 + . . . + 2

w

w−1

  • y w−1

If w is even. Theorem Let Nmin(n) and Nmax(n) be the numbers of de Bruijn sequences in the minimal and maximal weight classes, respectively. Then, 1. Nmin(n) = 2Z(n)−n−1 ·

  • 1≤w≤n−1

w e(w,n). 2. Nmax(n) = 2Z(n)−n−1 ·

  • 1≤w≤n−1

w even

w e(w,n).

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De Bruijn sequences in the extremal weight classes

Order Smax Smin 1 1 1 2 1 1 3 2 2 4 22 22 × 3 5 26 26 × 32 6 214 214 × 33 × 5 7 226 × 3 226 × 36 × 53 8 250 × 33 250 × 311 × 57 × 7 9 295 × 39 295 × 318 × 514 × 74 10 2177 × 320 2177 × 336 × 525 × 712 11 2329 × 342 × 5 2329 × 367 × 543 × 730 12 2632 × 375 × 55 2632 × 3133 × 572 × 766 × 11 13 21187 × 3133 × 522 21187 × 3265 × 5121 × 7132 × 116 14 22257 × 3219 × 570 22257 × 3536 × 5216 × 7245 × 1126 × 13 15 24251 × 3363 × 5200 × 7 24251 × 31061 × 5400 × 7430 × 1191 × 137 16 27978 × 3612 × 5497 × 77 27978 × 32086 × 5778 × 7723 × 11273 × 1335 17 214916 × 31092 × 51144 × 740 214916 × 34000 × 51516 × 71184 × 11728 × 13140 18 227952 × 32061 × 52424 × 7168 227952 × 37563 × 52960 × 71940 × 111768 × 13476 × 17 19 252338 × 34082 × 54862 × 7612 252338 × 314061 × 55678 × 73264 × 113978 × 131428 × 179 20 298535 × 38258 × 59225 × 71932 298535 × 325903 × 510800 × 75820 × 118398 × 133876 × 1757 × 19

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Weight class distribution for de Bruijn sequences of order 7

Weight class Number of sequences 19 226 × 36 × 53 21 230 × 36 × 53 23 226 × 35 × 52 × 7 × 13 × 19 25 230 × 33 × 52 × 59 × 71 27 228 × 34 × 5 × 80513 29 230 × 32 × 5 × 7 × 52973 31 228 × 17 × 1567 × 3769 33 230 × 5 × 181 × 31307 35 227 × 461 × 421607 37 230 × 3 × 52 × 11 × 19301 39 227 × 32 × 52 × 283949 41 230 × 3 × 11 × 19 × 4871 43 228 × 5 × 13 × 67 × 811 45 230 × 41 × 4637 47 228 × 52 × 11 × 433 49 230 × 5 × 653 51 226 × 73 × 11 53 231 × 5 55 226 × 3

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Weight class distribution for de Bruijn sequences of order 8

Weight class Number of sequences Weight class Number of sequences 35 250 × 311 × 57 × 7 75 251 × 21758744604075000469 37 250 × 310 × 57 × 72 × 17 77 251 × 3 × 61 × 5747701 × 10537594667 39 250 × 39 × 56 × 72 × 4799 79 251 × 13 × 47 × 30711169 × 263121107 41 250 × 39 × 56 × 11 × 254281 81 251 × 7 × 37 × 1879 × 1561463 × 2527559 43 250 × 38 × 55 × 31 × 11436329 83 252 × 52 × 7 × 1860740956442243 45 250 × 37 × 55 × 13 × 367 × 1425883 85 252 × 541 × 8151161 × 21768917 47 250 × 36 × 55 × 61 × 1685504311 87 252 × 7 × 800959 × 4374440849 49 250 × 35 × 55 × 197 × 12569 × 507809 89 252 × 5 × 7 × 73 × 2117079729071 51 252 × 34 × 54 × 103 × 1493 × 1583 × 65141 91 252 × 7 × 11 × 1048391 × 12694057 53 252 × 32 × 55 × 7 × 43 × 109 × 139 × 1063 × 16573 93 252 × 7 × 761 × 1613 × 2003 × 9631 55 252 × 3 × 52 × 17 × 43 × 98168800363397 95 252 × 7 × 23 × 141135121727 57 252 × 52 × 72 × 29 × 373 × 825689158081 97 252 × 5 × 13 × 40239850067 59 252 × 32 × 5 × 13 × 129287 × 252672123113 99 250 × 3 × 333168905291 61 252 × 32 × 5 × 197 × 367 × 653 × 13571044169 101 250 × 52 × 37 × 84395329 63 252 × 3 × 13 × 193 × 8461 × 592416520393 103 250 × 17 × 157 × 1830833 65 252 × 3 × 157 × 487 × 154459 × 1212559573 105 250 × 3 × 19 × 4183841 67 251 × 7 × 71 × 570013 × 301237031203 107 250 × 8716843 69 251 × 5 × 23 × 644071742408998313 109 250 × 32 × 7 × 11 × 17 × 19 71 251 × 331 × 20735557 × 8194340813 111 250 × 33 × 7 × 19 73 251 × 3 × 486569 × 25611783385373 113 250 × 33

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Proof of Conjecture 1

Conjecture 1: Smax(n) divides Smin(n). Smin(n) = 2−n ·

  • 1≤w≤n

(2w)e(w,n). Smax(n) = 2−n ·

  • 1≤w≤n

w odd

2e(w,n) ·

  • 1≤w≤n

w even

(2w)e(w,n). Conjecture 1 is proved by noting that, Smin(n) Smax(n) =

  • 1≤w≤n

w odd

w e(w,n).

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Proofs of Conjectures 4 and 5

Conjecture 4: For a prime p and order n, p divides Smin(n) for all p < n. Conjecture 5: For a prime p and order n, p divides Smax(n) for all 2p < n. Note that: e(w, n) = 0 for w = n, and e(w, n) ≥ 1 for 1 ≤ w ≤ n − 1. Recall that, Smin(n) = 2−n ·

  • 1≤w≤n

(2w)e(w,n), which is, Smin(n) = 2−n · 2e(1,n) · 4e(2,n) . . . (2n)e(n,n) Therefore we have, P(Smin(n)) = P((n − 1)!) = {p is a prime|1 < p < n}. Similarly, P(Smax(n)) = P(⌊(n − 1)/2⌋!) = {p is a prime|1 < p < n/2}.

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Proof of Conjecture 6

Conjecture 6: If 2α||Smax(n), then 2α|η(w, n) for any w and n. Recall that, G(0; y) = 2−n ·

  • 1≤w≤n

[(1 + y)w − (1 − y)w]e(w,n) , (2) where e(w, n) is the number of cycles of weight w in the pure circulating register. pw(y) =    2 w

1

  • y + 2

w

3

  • y 3 + . . . + 2

w

w

  • y w

If w is odd 2 w

1

  • y + 2

w

3

  • y 3 + . . . + 2

w

w−1

  • y w−1

If w is even. (3) Let w = 2lu, where u is an odd number. Let 1 ≤ t ≤ w be an odd number. We just need to show that 2l| w

t

  • . This is true because t is odd and,
  • w

t

  • = w

t

  • w − 1

t − 1

  • .

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Thank you for your attention!

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