Growth functions of braid monoids and generation of random braids

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1 Growth functions of braid monoids and generation of random braids Universidad de Sevilla Joint with Volker Gebhardt

2 Introduction Random braids Most authors working in braid-cryptography or computational braid theory, when they perform computations using random braids, they don t use random braids. They use random words.

3 Introduction Random braids We will focus on the positive braid monoid: Length of a (positive) braid = Word length Lattice structure of B n Unique gcd s (Æ) and lcm s (Ç)

4 Introduction Random braids We want to generate a random positive braid of length k. What if we take the product of k randomly chosen generators? There is only one word representing the braid There are 16 words representing the braid The probability of obtaining is 16 times the probability of obtaining This becomes more dramatic as n and k grow.

5 Generating random braids Lex-representative How do we generate braids with the same probability? Given a braid a, its Lex-representative, w(a ), is the (lexicographically) smallest word representing a. Bij. { Braids of length k } { Lex-representatives of length k }

6 Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B 4 : Root. The only braid of length 0 in B 4.

7 Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B 4 : 3 braids of length 1 in B 4.

8 Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B 4 : 8 braids of length 2 in B 4. The word is not there, as the Lex-representative of is.

9 Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B 4 : 19 braids of length 3 in B 4. There is no subword, neither, nor.

10 Generating random braids Braid tree The situation can be described using a rooted tree. Example, in B 4 : 19 braids of length 3 in B 4. There is no subword, neither, nor.

11 Generating random braids Generation procedure To generate a random braid: 1) Compute the number of leaves of the tree: 19 2) Choose a random number between 1 and 19: 16 3) Find the braid corresponding to the 16th leaf: In polynomial time! Warning: the tree is exponentially big!

12 Counting braids of given length Deligne s theorem x n,k = Number of braids in B n of length k. How to compute this number? Growth function of B n : It is a rational function, 9 recurrence relation for It is the inverse of a polynomial, Deligne (1972): The growth function of B n is the inverse of a polynomial.

13 Counting braids of given length Deligne s theorem x n,k = Number of braids in B n of length k. How to compute this number? Growth function of B n : It is a rational function, 9 recurrence relation for It is the inverse of a polynomial, Deligne (1972): The growth function of B n is the inverse of a polynomial.

14 Counting braids of given length Deligne s theorem Deligne (1972): The growth function of B n is the inverse of a polynomial. Proof: Denote

15 Counting braids of given length Deligne s theorem Deligne (1972): The growth function of B n is the inverse of a polynomial. Proof: Denote

16 Counting braids of given length Deligne s theorem Deligne (1972): The growth function of B n is the inverse of a polynomial. Proof: Inclusion - exclusion principle Each term equals for Recurrence relation! Q.E.D.

17 Counting braids of given length Deligne s theorem Deligne (1972): The growth function of B n is the inverse of a polynomial. Proof: Inclusion - exclusion principle Each term equals for Recurrence relation! Q.E.D.

Counting braids of given length Bronfman s

Bronfman (2001): Recursive formula: As H 0 (t)=

18 Counting braids of given length Bronfman s method Deligne (1972): The growth function of B n is the inverse of a polynomial. Actually: Hard to compute! Bronfman (2001): Recursive formula: As H 0 (t)= 1, one can compute H n (t), and then x n,k. (in polynomial time) We will use another method

19 Counting braids of given length Our method Example: # (Lex-representatives of length k starting with s 5 ) =

20 Counting braids of given length Our method This yields an easy recursive formula: - - =

21 Counting braids of given length Example, in B 4 : Our method =

22 Counting braids of given length Our method Example, in B 4 : =

23 Counting braids of given length Our method Example, in B 4 : =

24 Counting braids of given length Our method Example, in B 4 : =

25 Counting braids of given length Our method Example, in B 4 : =

26 Counting braids of given length Our method Example, in B 4 : = What are we computing?

27 Counting braids of given length Our method The first column of our table contains x n,1, x n,2, x n,3, Computing k rows, we obtain x n,k. In time

28 Consequence A formula for the growth function A new line is obtained from the 6 previous ones by a linear transformation = Multiplying from the left by ( ), one gets x n,k

29 Consequence A formula for the growth function

30 Consequence A formula for the growth function

31 Consequence A formula for the growth function After some easy computations (Bronfman s recursive formula = expansion along the first column)

32 Other Artin-Tits monoids A formula for the growth function Artin-Tits monoid of type B n 4

33 Other Artin-Tits monoids A formula for the growth function Artin-Tits monoid of type D n

34 Generating random braids Generation procedure To generate a random braid: 1) Compute the number of leaves of the tree: 19 2) Choose a random number between 1 and 19: 16 3) Find the braid corresponding to the 16th leaf: Next

35 Finding the rth braid of length k Hanging leaves Consider the graph of height k as before. Given a vertex w, suppose we can compute, in polynomial time, the number of leaves hanging from w:

36 Finding the rth braid of length k The procedure We want to compute the 16th braid: The first letter is. The second letter is. The third letter is.

37 Computing hanging leaves Forbidden prefixes Computing at most (n-2)k times the hanging leaves of a vertex, one computes the r-th braid of length k. How to compute the hanging leaves? w k - w w Lemma: Can put any braid of length k - w, except those starting by some forbidden prefixes.

38 Computing hanging leaves Forbidden prefixes Example: Forbidden prefixes for the word Know how to compute

39 Computing hanging leaves Forbidden prefixes By the inclusion-exclusion principle, we can compute

40 Computing hanging leaves Forbidden prefixes By the inclusion-exclusion principle, we can compute

41 Computing hanging leaves Forbidden prefixes Good news: Every summand is equal to x n,t, for some t k. These are the elements in the first column of our table. Bad news: There are exponentially many summands. But we can collect all summands corresponding to elts. with the same length.

42 Generating random positive braids The result As a consequence: One can compute in ploynomial space and time with respect to n and k. ) Theorem: (GM-Gebhardt, 2011) There is a procedure to generate a random positive braid in B n of length k, whose time and space complexity is a polynomial in n and k. Time?

43 Generating random positive braids Effective computations Time (in ms.) for computing a random positive braid of length k, on n strands k n Linux system with an Intel E bit CPU (core: 3\,GHz, FSB: 1333\,MHz) and a main memory bandwidth of 6.5\,GB/s (X38 chipset, dual channel DDR2 RAM, memory bus: 1066\,MHz) using a development version of Magma V.2.16.

44 Open problems ² Generate group elements instead of monoid elements. ² Use another generating set: permutation braids. ² Generalize to other Artin groups of finite type.

45 Thank you!

MATH 243E Test #3 Solutions

MATH 243E Test #3 Solutions MATH 4E Test # Solutions () Find a recurrence relation for the number of bit strings of length n that contain a pair of consecutive 0s. You do not need to solve this recurrence relation. (Hint: Consider