Groups With Two Generators Having Unsolvable Word Problem And Presentations of Mihailova Subgroups

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Groups With Two Generators Having Unsolvable Word Problem

And Presentations of Mihailova Subgroups

Xiaofeng Wang, Chen Xu, Guo Li

∗

, and Hanling Lin

School of Mathematics, Shenzhen University Shenzhen 518060, P. R. China

Abstract. A presentation of a group with two generators having unsolvable word problem and an explicit countable presentation of Mihailova subgroup of F2×F2 with finite number of generators are given. Where Mihailova subgroup ofF2×F2enjoys the unsolvable subgroup membership problem. One then can use the presentation to create entities’ private keys in a public key cryptsystem.

Key words: word problem, subgroup membership problem, Mihailova subgroup, insolvability

1 Introduction

In 1997, Shor published in [37] his distinguished quantum computational algorithms and he pointed out in this paper that with his algorithms the factorization of an integer and the computation of discrete logarithm are computable in a polynomial time. Therefore, the most used public key cryptsystems (such as RSA, ECC, et cetera) are really in jeopardy since people believe that quantum computers or quantum computation systems are in fact not far from the reality. Therefore, one of the most urgent task for the cryptologists is to find new public key cryptsystems which are much more safe and free of the quantum computational attack.

In the last decade, due to the works done by Anshelet al[1], and Koet al[24], the decision problems from combinatorial group theory (i.e. the conjugacy search problem, the decomposition problem, the root extraction search problem, and the subgroup membership problem) have been intensively employed as the core for the establishment of alleged secure and effective cryptographic primitives. In particular, due to having very complicated structures, very nice geometrical interpretations, exponential growth, and unique normal form for all words representing any fixed element, the non-commutative braid groups Bn have

been used as the platforms of setting up cryptographic schemes [2, 3, 25, 8, 13, 27, 38, 40, 41, 42, 44, 45] with the hope that the corresponding protocols have high level security. Unfortunately, it was shown that some of these primitives are feasible to the attackers, for examples see [9, 14, 15, 16, 17, 19, 21, 22, 23, 26, 28, 29, 30, 31, 34, 36].

Clearly, one of the resolutions is to find a group with word problem solvable in polynomial time and with some decision problem very hard decidable. Followed then, taking the group as the platform one can try to set up public key cryptographic primitives with safety guaranteed by the hard decision problem.

Collins [12] proved that there are subgroups of a braid groupBnwithn≥6 which are isomorphic to

the groupF2×F2, where the groupF2is the free group of rank 2. Then, as Shpilrain and Ushakov have pointed out in [39] that there are some Mihailova subgroups [32] in a braid groupBnwithn≥6 such that

the subgroup membership problem of these subgroups is unsolvable. Therefore, one of the key points of this undecidability is able to be applied to propose new cryptsystems is to give an explicit presentation of Mihailova subgroups ofBn.

∗

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2 Subgroups with unsolvable GWP

2.1 Some decision problems of groups

In this section, for the use in the sequel, we present some decision problems of groups. A presentation of a groupGis as the following

P =hx1, x2, x3,· · · |R1, R2, R3,· · · i

where the set X = {x1, x2, x3,· · · } is called an alphabet and the Rj’s are words on X ∪X−1 with

X−1 ₌_{x−1 1 , x

−1 2 , x

−1

3 ,· · · }. The group presented by P denoted G(P) is the quotient group of the free

group onX by the normal closure of the set{R1, R2, R3,· · · }in the free group. Usually it is not necessary to distinguish so carefully between a group and its presentation and we often write simply

G=hx1, x2, x3,· · · |R1, R2, R3,· · · i

to mean theGis the group defined by the given presentation, and we call that the elements in X are generators ofG, the words Rj’s are defining relators. When the setsX is finite we then say thatG is

finitely generated, and when both setsX and {R1, R2, R3,· · · } are finite we then say that Gis finitely

presented. Sometimes, one may uses so-calleddefining relationsof the formRl=Rr(which is equivalent

to being a relator of the formRlR−r1) to replace relators in a presentation of a groupG.

Suppose thatG is a finitely presented group defined by a presentation as above. We present some decision problems inG.

Word problem (WP):

Given any two wordswanduonX∪X−1_{, decide if}_w₌_u_in_G_.

Generalized word problem or subgroup membership problem (GWP):

Given a subgroupH of Ggenerated by elementsa1, a2,· · ·, al, and an element g ofG, decide ifg is

an element ofH, or equivalently ifg can be written as a word on the set

{a1, a2,· · · , al} ∪ {a−11, a− 1

2 ,· · ·, a− 1

l }

2.2 Presentation of groups with two generators having unsolvable WP

Novikov [35] and Boone [4] independently proved that there is a finitely presented group having un-solvable word problem. In 1969, Borisov [6] gave an elegant simplification on Boone’s approach. Then in 1986, applying to a C´ejtin’s [7] semigroup presentation with Borisov’s method, Collins [11] set up a simple finite group presentation having unsolvable word problem with 10 generators and 27 relations as the following.

Presentation A Generators:

c1, c2, c3, c4, c5, c6, c7, c8, c9, c10

Relations:

c₁−1c10₇ c1=c7, c₂−1c10₇ c2=c7, c−₃1c10₇ c3=c7, c−₄1c10₇ c4=c7, c−₅1c10₇ c5=c7,

c₁−1c8c1=c10₈ , c₂−1c8c2=c10₈ , c−₃1c8c3=c10₈ , c−₄1c8c4=c₈10, c−₅1c8c5=c10₈ ,

c−₉1c1c9=c1, c−₉1c2c9=c2, c₉−1c3c9=c3, c−₉1c4c9=c4, c−₉1c5c9=c5,

c₁₀−1c7c10=c7, c10−1c8c10=c8, c−61c

−3

1 c10c31c6=c−13c10c31

c−₉1c7c1c3c8c9=c7c3c1c8, c−91c 2

7c1c4c28c9=c27c4c1c28, c

−1 9 c

3

7c2c3c38c9=c37c3c2c38,

c−₉1c4₇c2c4c4₈c9=c4₇c4c2c4₈, c−₉1c5₇c3c5c₈5c9=c5₇c5c3c1c5₈, c−₉1c6₇c4c5c6₈c9=c6₇c5c4c2c6₈,

c−₉1c7₇c3c4c3c7₈c9=c7₇c3c4c3c5c7₈, c−₉1c8₇c3c3₁c8₈c9=c8₇c3₁c8₈, c−₉1c9₇c4c3₁c9₈c9=c9₇c3₁c9₈

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presentation.

Presentation B

hJ, t;v=t−1ut, t−1v−1uvt=c1u−1vu, t−1v−2uv2t=c2u−2vu2, t−1v−3uv3t=c3u−3vu3

t−1v−4uv4t=c4u−4vu4, t−1v−5uv5t=c5u−5vu5, t−1v−6uv6t=c6u−6vu6, t−1v−7uv7t=c7u−7vu7

t−1v−8uv8t=c8u−8vu8, t−1v−9uv9t=c9u−9vu9, t−1v−10uv10t=c10u−10vu10i

whereJ =C∗ hu, viis the free product of the groupC and the free group generated by lettersuandv. By Lemma 2.1 of [33] we then have the following.

Proposition 2.1 The word problem for the group defined byPresentation Bis unsolvable.

Now we apply Tietze transformations [43] as pointed in [20] to replace all the occurrences of v by

t−1_ut_{and replace all occurrences of}_c

i by

ci=t−1t−1u−itut−1uittu−it−1u−1tui, i= 1,2,· · ·,10

in the relations in Presentation B, and then eliminate all generators (by using Tieze transformations agian)ci,i= 1,2,· · · ,10 andv to get a finite presentation with only two generators as follows.

Presentation C

Two generators: u, t

27 relations:

R1: (t−2u−7tut−1u7t2u−7t−1u−1tu7)10t−2u−1tut−1ut2u−1t−1u−1tu

=t−2u−1tut−1ut2u−1t−1u−1tut−2u−7tut−1u7t2u−7t−1u−1tu7

R2: (t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎10_t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_tu2

=t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_tu2_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7

R3: (t−2u−7tut−1u7t2u−7t−1u−1tu7)10t−2u−3tut−1u3t2u−3t−1u−1tu3

=t−2u−5tut−1u5t2u−5t−1u−1tu5t−2u−7tut−1u7t2u−7t−1u−1tu7

R6: t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu

=t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎10

R7: t−2u−8tut−1u8t2u−8t−1u−1tu8t−2u−2tut−1u2t2u−2t−1u−1tu2

=t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_tu2₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎10

=t−2u−4tut−1u4t2u−4t−1u−1tu4(t−2u−8tut−1u8t2u−8t−1u−1tu8)10

R10: t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5

R11: t−2u−9tut−1u9t2u−9t−1u−1tu9t−2u−1tut−1ut2u−1t−1u−1tu

=t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tut−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9

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=t−2u−5tut−1u5t2u−5t−1u−1tu5t−2u−9tut−1u9t2u−9t−1u−1tu9

R18: (t−2u−1tut−1ut2u−1t−1u−1tu)−3t−2u−10tut−1u10t2u−10t−1u−1tu10(t−2u−1tut−1ut2u−1t−1u−1tu)3 t−2_u−6_tut−1_u6_t2_u−6_t−1_u−1_tu6

=t−2_u−6_tut−1_u6_t2_u−6_t−1_u−1_tu6₍_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎−3_t−2_u−10_tut−1_u10_t2_u−10_t−1_u−1_tu10

(t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3

R19: t−2u−7tut−1u7t2u−7t−1u−1tu7t−2u−1tut−1ut2u−1t−1u−1tut−2u−3tut−1u3t2u−3t−1u−1tu3 t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9

=t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3 t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tut−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8

R20: (t−2u−7tut−1u7t2u−7t−1u−1tu7)2t−2u−1tut−1ut2u−1t−1u−1tut−2u−4tut−1u4t2u−4t−1u−1tu4

(t−2u−8tut−1u8t2u−8t−1u−1tu8)2t−2u−9tut−1u9t2u−9t−1u−1tu9

=t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎2_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4 t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎2

R21: (t−2u−7tut−1u7t2u−7t−1u−1tu7)3t−2u−2tut−1u2t2u−2t−1u−1tu2t−2u−3tut−1u3t2u−3t−1u−1tu3

=t−2u−9tut−1u9t2u−9t−1u−1tu9(t−2u−7tut−1u7t2u−7t−1u−1tu7)3t−2u−3tut−1u3t2u−3t−1u−1tu3 t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_tu2₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎3

R22: (t−2u−7tut−1u7t2u−7t−1u−1tu7)4t−2u−2tut−1u2t2u−2t−1u−1tu2t−2u−4tut−1u4t2u−4t−1u−1tu4

=t−2u−9tut−1u9t2u−9t−1u−1tu9(t−2u−7tut−1u7t2u−7t−1u−1tu7)4t−2u−4tut−1u4t2u−4t−1u−1tu4 t−2u−2tut−1u2t2u−2t−1u−1tu2(t−2u−8tut−1u8t2u−8t−1u−1tu8)4

R23: (t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎5_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5

(t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎5_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9

=t−2u−9tut−1u9t2u−9t−1u−1tu9(t−2u−7tut−1u7t2u−7t−1u−1tu7)5t−2u−5tut−1u5t2u−5t−1u−1tu5 t−2u−3tut−1u3t2u−3t−1u−1tu3t−2u−1tut−1ut2u−1t−1u−1tu(t−2u−8tut−1u8t2u−8t−1u−1tu8)5

R24: (t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎6_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5

=t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎6_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5 t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4_t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_tu2₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎6

R25: (t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎7_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4 t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎7_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9

=t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎7_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3 t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5

(t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎7

R26: (t−2u−7tut−1u7t2u−7t−1u−1tu7)8t−2u−3tut−1u3t2u−3t−1u−1tu3(t−2u−1tut−1ut2u−1t−1u−1tu)3

=t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎8₍_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3

(t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎8

R27: (t−2u−7tut−1u7t2u−7t−1u−1tu7)9t−2u−4tut−1u4t2u−4t−1u−1tu4(t−2u−1tut−1ut2u−1t−1u−1tu)3

=t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎9₍_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3

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By removing a number of inverse pairs on the relations in the above presentation we then have the following presentation.

Presentation C0

Two generators: u, t

27 relations:

R0₁: u−6_tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎9_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_t

=tut−1_ut2_u−1_t−1_u−1_tut−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu6

R02: u−5tut−1u7t2u−7t−1u−1tu7(t−2u−7tut−1u7t2u−7t−1u−1tu7)9t−2u−2tut−1u2t2u−2t−1u−1t

=tut−1u2t2u−2t−1u−1tu2t−2u−7tut−1u7t2u−7t−1u−1tu5

R0₃: u−4_tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎9_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_t

R0₄: u−3_tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎9_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_t

=tut−1_u4_t2_u−4_t−1_u−1_tu4_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu3

R0₅: u−2_tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎9_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_t

R06: u−7tut−1u8t2u−8t−1u−1tu8t−2u−1tut−1ut2u−1t−1u−1t

=tut−1ut2u−1t−1u−1tu(t−2u−8tut−1u8t2u−8t−1u−1tu8)9t−2u−8tut−1u8t2u−8t−1u−1tu7

R0₇: u−6_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_t

=tut−1u2t2u−2t−1u−1tu2(t−2u−8tut−1u8t2u−8t−1u−1tu8)9t−2u−8tut−1u8t2u−8t−1u−1tu6

R0₈: u−5_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_t

=tut−1_u3_t2_u−3_t−1_u−1_tu3₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎9_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu5

R0₉: u−4_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_t

=tut−1_u4_t2_u−4_t−1_u−1_tu4₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎9_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu4

R010: u−3tut−1u8t2u−8t−1u−1tu8t−2u−5tut−1u5t2u−5t−1u−1t

=tut−1u5t2u−5t−1u−1tu5(t−2u−8tut−1u8t2u−8t−1u−1tu8)9t−2u−8tut−1u8t2u−8t−1u−1tu3

R0₁₁: u−8_tut−1_u9_t2_u−9_t−1_u−1_tu9_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_t

=tut−1ut2u−1t−1u−1tut−2u−9tut−1u9t2u−9t−1u−1tu8

R0₁₂: u−7_tut−1_u9_t2_u−9_t−1_u−1_tu9_t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_t

R0₁₃: u−6_tut−1_u9_t2_u−9_t−1_u−1_tu9_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_t

R014: u−5tut−1u9t2u−9t−1u−1tu9t−2u−4tut−1u4t2u−4t−1u−1t

R0₁₅: u−4tut−1u9t2u−9t−1u−1tu9t−2u−5tut−1u5t2u−5t−1u−1t

R0₁₆: tut−1_u7_t2_u−7_t−1_u−1_tu7_t−2_u−10_tut−1_u10_t2_u−10_t−1_u−1_tu3

=u−3_tut−1_u10_t2_u−10_t−1_u−1_tu10_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_t

R0₁₇: tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−10_tut−1_u10_t2_u−10_t−1_u−1_tu2

=u−2_tut−1_u10_t2_u−10_t−1_u−1_tu10_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_t

R018: (t−2u−1tut−1ut2u−1t−1u−1tu)−2u−1t−1utut−2u−1tu−1t−1u−9tut−1u10t2u−10t−1u−1tu10

(t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3_t−2_u−6_tut−1_u6_t2_u−6_t−1_u−1_tu5

=t−2_u−6_tut−1_u6_t2_u−6_t−1_u−1_tu5_t−1_utut−2_u−1_tu−1_t−1_ut2_u−1_t−1_utut−2_u−1_tu−1_t−1_ut2 u−1_t−1_utut−2_u−1_tu−1_t−1_u−9_tut−1_u10_t2_u−10_t−1_u−1_tu10

(t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎2_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_t

(6)

=u−2_tut−1_u9_t2_u−9_t−1_u−1_tu9_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3 t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tut−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_t

R020: tut−1u7t2u−7t−1u−1tu7t−2u−7tut−1u7t2u−7t−1u−1tu7t−2u−1tut−1ut2u−1t−1u−1tu

t−2u−4tut−1u4t2u−4t−1u−1tu4(t−2u−8tut−1u8t2u−8t−1u−1tu8)2t−2u−9tut−1u9t2u−9t−1u−1tu

=u−2tut−1u9t2u−9t−1u−1tu9(t−2u−7tut−1u7t2u−7t−1u−1tu7)2t−2u−4tut−1u4t2u−4t−1u−1tu4 t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tut−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_t

R0₂₁: tut−1u7t2u−7t−1u−1tu7(t−2u−7tut−1u7t2u−7t−1u−1tu7)2t−2u−2tut−1u2t2u−2t−1u−1tu2 t−2u−3tut−1u3t2u−3t−1u−1tu3(t−2u−8tut−1u8t2u−8t−1u−1tu8)3t−2u−9tut−1u9t2u−9t−1u−1tu

=u−2tut−1u9t2u−9t−1u−1tu9(t−2u−7tut−1u7t2u−7t−1u−1tu7)3t−2u−3tut−1u3t2u−3t−1u−1tu3 t−2u−2tut−1u2t2u−2t−1u−1tu2(t−2u−8tut−1u8t2u−8t−1u−1tu8)2t−2u−8tut−1u8t2u−8t−1u−1t

R0₂₂: tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎3_t−2_u−2_tut−1_u2_t2_u−2_t−1_u−1_tu2 t−2u−4tut−1u4t2u−4t−1u−1tu4(t−2u−8tut−1u8t2u−8t−1u−1tu8)4t−2u−9tut−1u9t2u−9t−1u−1tu

=u−2tut−1u9t2u−9t−1u−1tu9(t−2u−7tut−1u7t2u−7t−1u−1tu7)4t−2u−4tut−1u4t2u−4t−1u−1tu4 t−2u−2tut−1u2t2u−2t−1u−1tu2(t−2u−8tut−1u8t2u−8t−1u−1tu8)3t−2u−8tut−1u8t2u−8t−1u−1t

R0₂₃: tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎4_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3 t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎5_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu

=u−2_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎5_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5 t−2u−3tut−1u3t2u−3t−1u−1tu3t−2u−1tut−1ut2u−1t−1u−1tu

(t−2u−8tut−1u8t2u−8t−1u−1tu8)4t−2u−8tut−1u8t2u−8t−1u−1t

R0₂₄: tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎5_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4 t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎6_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu

=u−2_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎6_t−2_u−5_tut−1_u5_t2_u−5_t−1_u−1_tu5 t−2u−4tut−1u4t2u−4t−1u−1tu4t−2u−2tut−1u2t2u−2t−1u−1tu2

R0₂₅: tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎6_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3 t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3

(t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎7_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu

=u−2_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎7_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3 t−2u−4tut−1u4t2u−4t−1u−1tu4t−2u−3tut−1u3t2u−3t−1u−1tu3t−2u−5tut−1u5t2u−5t−1u−1tu5

R0₂₆: tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎7_t−2_u−3_tut−1_u3_t2_u−3_t−1_u−1_tu3

(t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎8_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu

=u−2_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎8₍_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3

(t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎7_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_t

R0₂₇: tut−1_u7_t2_u−7_t−1_u−1_tu7₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎8_t−2_u−4_tut−1_u4_t2_u−4_t−1_u−1_tu4

(t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3₍_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎9_t−2_u−9_tut−1_u9_t2_u−9_t−1_u−1_tu

=u−2_tut−1_u9_t2_u−9_t−1_u−1_tu9₍_t−2_u−7_tut−1_u7_t2_u−7_t−1_u−1_tu7₎9₍_t−2_u−1_tut−1_ut2_u−1_t−1_u−1_tu₎3

(t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_tu8₎8_t−2_u−8_tut−1_u8_t2_u−8_t−1_u−1_t

We denote by H the group defined by Presentation C0. SinceH is isomorphic to the group defined by Presentation B [43], we have the following theorem.

Theorem 2.2 The word problem for the groupH defined byPresentation C0 _{is unsolvable.}

3 Presentations of Mihailova subgroups of

F

2

×

F

2

LetH be a group defined by a presentationP=hx1, x2,· · ·, xk|R1, R2,· · · , Rmiwith integerk≥2,

and letFk be the free group on{x1, x2,· · ·, xk}. Then, in her influent paper [32], Mihailova associated

toH theMihailova subgroupM(H) of the direct product ofFk×Fk defined by

M(H) ={(w1, w2)|w1=w2inH}

(7)

Theorem 3.1 (Mihailova) The membership problem forM(H)inFk×Fk is solvable if and only if the

word problem forH is solvable.

Thus, taking H the group defined by Presentation C0 _{generated by two elements, the Mihailova}

subgroupM(H) ofF2×F2has a unsolvable membership problem, namely there is no algorithm to decide

if any elementxofF2×F2 written as a word on the generators ofF2×F2 is an element ofM(H).

By a result of Grunewald’s [18], ifH enjoys a unsolvable word problem then the Mihailova subgroup

M(H) can not be finitely presented. However, Bogopolski and Venturawe [5] have gaven an explicit countable presentation with finite generators and countably infinite relators for Mihailova subgroupM(H) ofFk×Fk (k≥2) provided that the groupH can be defined by a finite, concise and Peiffer aspherical

presentation as the following theorem.

Theorem 3.2 (Bogopolski and Venturawe) Let Fk be the free group on {x1,· · · , xk}, and let H =

hx1,· · · , xk|R1,· · ·, Rmibe a finite, concise and Peiffer aspherical presentation. Then Mihailova’s group

M(H)≤Fk×Fk admits the following presentation

hd1,· · ·, dk, t1,· · ·, tm|[tj, d−1t−i 1rid],[ti,root(ri)],1≤i, j≤m, d∈Dki

whereDkis the free group with basis{d1,· · ·, dk},ridenotes the word inDkobtained fromRiby replacing

eachxltodl (1≤l≤k), root(ri) denotes the unique elementsi∈Dk such thatriis a positive power of

sibutsiitself is not a proper power, and the elementsdi andtjcorrespond, respectively, to the elements

(xi, xi) and (1, Rj) ofM(H) (1≤i≤k, 1≤j≤m).

To apply the above theorem on Presentation C0 _{we must show that Presentation C}0 _{is concise and}

Peiffer aspherical.

A group presentation P = hx1, x2,· · · , xk|R1, R2,· · ·, Rmi is called concise if every relation Ri is

non-trivial and reduced, and every two relatorsRi, Rj, i6=j, are not conjugate to each other, or to the

inverse of each other. A direct check shows that Presentation C0 is concise.

One can refer [5] for the definition of being Peiffer aspherical. Theorems 3.1 and 4.2, and Lemma 5.1 in [10] imply that respectively, the Peiffer asphericity is preserved under HNN-extensions, under free products, and under Tietze transformations. Therefore, it is sufficient to show that Presentation C (as well as Presentation C0) can be obtained from a free group by performing a number of HNN-extensions, free products, and Tietze transformations.

First, Presentation A can be gained from a free group by three consecutive HNN-extensions as follows. The first HNN-extension is performed by taking the free groupAgenerated byc7, c8as the associated subgroup, andc1, c2, c3, c4, c5, c10as the stable letters to get an HNN-extensionJ1defined by the following presentation.

P1=hA, c1, c2, c3, c4, c5, c10|

c−₁1c10₇ c1=c7, c−₂1c10₇ c2=c7, c−₃1c10₇ c3=c7, c−₄1c10₇ c4=c7, c₅−1c10₇ c5=c7, c−₁₀1c7c10=c7,

c−₁1c8c1=c108 , c− 1

2 c8c2=c 10 8 , c−

1

3 c8c3=c 10 8 , c−

1

4 c8c4=c 10 8 , c−

1

5 c8c5=c 10 8 , c−

1

10c8c10=c8i

Then, by taking the subgroupK1 ofJ1 generated by the the following subset

{c1, c2, c3, c4, c5, c7c1c3c8, c7c3c1c8, c72c1c4c28, c 2

7c4c1c28, c 3

7c2c3c38, c 3

7c3c2c38, c 4

7c2c4c48, c 4

7c4c2c48, c 5 7c3c5c58,

c57c5c3c1c58, c67c4c5c68, c67c5c4c2c68, c77c3c4c3c78, c77c3c4c3c5c78, c87c3c31c88, c87c31c88, c79c4c31c98, c97c31c98}

as the associated subgroup and c9 as the stable letter we have the HNN-extension J2 defined by the

following presentation.

P2=hJ1, c9|c−₉1c1c9=c1, c−₉1c2c9=c2, c₉−1c3c9=c3, c−₉1c4c9=c4, c−₉1c5c9=c5,

c−₉1c7c1c3c8c9=c7c3c1c8, c−₉1c27c1c4c 2 8c9=c

2 7c4c1c

2 8, c−

1 9 c

3 7c2c3c

3 8c9=c

3 7c3c2c

3 8,

c−₉1c4₇c2c4c4₈c9=c4₇c4c2c4₈, c−₉1c5₇c3c5c₈5c9=c5₇c5c3c1c5₈, c−₉1c6₇c4c5c6₈c9=c6₇c5c4c2c6₈,

c−₉1c77c3c4c3c78c9=c77c3c4c3c5c78, c− 1 9 c

8

7c3c31c88c9=c87c31c88, c− 1 9 c

9

(8)

The third one HNN-extension is then clearly performed by taking the subgroupK2ofJ2generated by

the the elementc−₁3c10c31as the associated subgroup andc6the stable letter to obtain the HNN-extension Cdefined by Presentation A.

Now, it is obvious that the group defined by Presentation B is also an HNN-extension by taking the subgroupJ as the associated subgroup andt as the stble letter, whereJ is the free product of groupC

and the free group generated by two lettersuandv.

Finally and clearly, we already have known that Presentation C (as well as Presentation C0) is obtained by performing a number of Tietze transformations from Presentation B. Thus, by the results in [10] we have the following theorem.

Theorem 3.3 Presentation C and Presentation C0_{are Peiffer aspherical, and Presentation C}0_{is concise.}

Since the groupH defined by Presentation C0 is generated by two elementsu, t, we now can apply Theorem 3.2 to present an explicit countable presentation for Mihailova subgroupMF2×F2(H) ofF2×F2

withF2 generated by{u, t}. To do so we need some notations as follows. For eachi= 1,2,· · ·,27, if a relationR0_i in Presentation C0 is of the form

R0_i:R(_il)(u, t) =R_i(r)(u, t)

with bothR(_il)(u, t) andR_i(r)(u, t) being words on{u, t, u−1_{, t}−1_} _{then we denote}

Si = (R

(r)

i (u, t))−

1_R(l)

i (u, t)

Clearly, one can check that root(Si) =Si,i= 1,2,· · ·,27. Thus, we then have

ri= (R

(r)

i ((u, u),(t, t)))

−1_R(l)

i ((u, u),(t, t)), i= 1,2,· · ·,27

whereriis as defined as in the the presentation given in Theorem 3.2.

Finally, by Theorem 3.2 we then have an explicit countable presentation with 56 generators for Mi-hailova subgroupMF2×F2(H) ofF2×F2 as the following.

Presentation D

29 generators:

(u, u),(t, t),(1, Si), i= 1,2,· · · ,27

Countable number of relators:

S_i−1(δ−1S_k−1r_k−1δ)−1Si(δ−1Sk−1r

−1

k δ), S

−1

i r

−1

i Siri, i, k= 1,2,· · · ,27

whereδ∈F2×F2.

Now, since the word problem of the group G defined by Presentation C is unsolvable, Mihailova’s theorem (Theorem 3.1) implies the following conclusion.

Theorem 3.4 The membership problem for Mihailova subgroupMF2×F2(H)of F2×F2 is unsolvable.

Finally, for being used with applications, we give the descriptions of eachSi’s in the generators (1, Si),

i= 1,2,· · ·,27 in Presentation D as follows where for the simplicity we replace all occurrences of (u, u) byδu and all occurrences of (t, t) byδt.

S1: (δtδuδt−1δuδ2tδ−u1δ−

1

t δu−1δtδuδ−t2δu−7δtδuδ−t1δ7uδt2δu−7δ−

1

t δ−u1δtδ6u)−1

δ−6

u δtδuδt−1δu7δ2tδ−u7δ

−1

t δu−1δtδu7(δ

−2

t δu−7δtδuδt−1δ7uδt2δu−7δ

−1

t δ−u1δtδ7u)9

δ−_t2δ−1

u δtδuδt−1δuδt2δu−1δ

−1

t δ−u1δt

S2: (δtδuδt−1δ

2

uδ

2

tδ−

2

u δ

−1

t δ−

1

u δtδu2δ

−2

t δ−

7

u δtδuδ−t1δ

7

uδ

2

tδ−

7

u δ

−1

t δ−

1

u δtδ5u)−

1

δ−u5δtδuδt−1δu7δ2tδ−u7δ−

1

t δu−1δtδu7(δ−

2

t δu−7δtδuδt−1δ7uδt2δu−7δ−

1

t δ−u1δtδ7u)9

δt−2δ−u2δtδuδt−1δ2uδt2δu−2δ−

1

(9)

S3: (δtδuδt−1δu3δ2tδ−u3δ−

1

t δu−1δtδu3δ−

2

1

δ−4

−1

t δu−1δtδu7(δ

−2

−1

t δ−u1δtδ7u)9

δ−_t2δ−3

u δtδuδt−1δ3uδt2δu−3δ

−1

t δ−u1δt

S4: (δtδuδt−1δu4δ

2

tδ−

4

u δ

−1

t δu−1δtδu4δ

−2

t δu−7δtδuδ−t1δ7uδ

2

tδ−

7

u δ

−1

t δ−u1δtδ3u)−

1

δ−u3δtδuδt−1δu7δ2tδ−u7δ−

1

2

1

t δ−u1δtδ7u)9

1

t δ−u1δt

S5: (δtδuδt−1δu5δ2tδ−u5δ

−1

t δu−1δtδu5δ

−2

−1

δ−2

−1

t δu−1δtδu7(δ

−2

−1

t δ−u1δtδ7u)9

δ−_t2δ−5

−1

t δ−u1δt

S6: (δtδuδt−1δuδ2tδ−u1δ

−1

t δu−1δtδu(δ−t2δ−u8δtδuδt−1δu8δt2δu−8δ

−1

t δ−u1δtδ8u)9

δ−_t2δ−8

−1

δ−7

−1

t δu−1δtδu8δ

−2

t δu−1δtδuδt−1δuδ2tδu−1δ

−1

t δ−u1δt

S7: (δtδ2uδ−

1

t δu2δ2tδ−u2δ−

1

2

t δ−u8δtδuδt−1δ8uδt2δu−8δ−

1

t δ−u1δtδ8u)9

1

δ−6

u δtδuδt−1δu8δ2tδ−u8δ−

1

2

t δu−1δtδuδt−1δuδ2tδu−1δ−

1

t δ−u1δt

S8: (δtδ3uδ

−1

t δu3δ2tδ−u3δ

−1

t δu−1δtδu3(δ

−2

t δ−u8δtδuδt−1δ8uδt2δu−8δ

−1

t δ−u1δtδ8u)9

δ−_t2δ−8

−1

δ−_u5δtδuδt−1δ

8

uδ

2

tδ−

8

u δ

−1

t δ−

1

u δtδu8δ

−2

t δ−

1

u δtδuδ−t1δuδ2tδ−

1

u δ

−1

t δ−

1

u δt

S9: (δtδ4uδ

−1

t δu4δ2tδ−u4δ

−1

t δu−1δtδu(δ−t2δ−u8δtδuδt−1δu8δt2δu−8δ

−1

t δ−u1δtδ8u)9

δ−_t2δ−8

−1

δ−4

−1

t δu−1δtδu8δ

−2

−1

t δ−u1δt

S10: (δtδ5uδ−

1

t δu5δ2tδ−u5δ−

1

t δu−1δtδu(δ−t2δ−u8δtδuδt−1δ8uδt2δu−8δ−

1

t δ−u1δtδ8u)9

δ−_t2δ−8

u δtδuδt−1δ8uδt2δu−8δ−

1

δ−3

1

2

t δu−1δtδuδt−1δuδ2tδu−1δ−

1

t δ−u1δt

S11: (δtδuδt−1δuδ2tδ−u1δ

−1

t δu−1δtδuδ−t2δu−9δtδuδ−t1δu9δt2δu−9δ

−1

δ−_u8δtδuδt−1δ

9

uδ

2

tδ−

9

u δ

−1

t δ−

1

u δtδu9δ

−2

t δ−

1

u δtδuδ−t1δuδ2tδ−

1

u δ

−1

t δ−

1

u δt

−1

t δu−1δtδu2δ

−2

−1

δ−7

−1

t δu−1δtδu9δ

−2

t δu−2δtδuδt−1δu2δ2tδu−2δ

−1

t δ−u1δt

S13: (δtδuδt−1δu3δ2tδ−u3δ−

1

2

1

δ−6

1

2

t δu−3δtδuδt−1δu3δ2tδu−3δ−

1

t δ−u1δt

−1

t δu−1δtδu4δ

−2

−1

δ−5

−1

t δu−1δtδu9δ

−2

−1

t δ−u1δt

−1

t δu−1δtδu5δ

−2

−1

δ−4

−1

t δu−1δtδu9δ

−2

−1

t δ−u1δt

S16: (δ−u3δtδuδt−1δu10δt2δu−10δ−

1

t δ−u1δtδ10u δ−

2

1

t δ−u1δt)−1

δtδuδt−1δu7δ2tδ−u7δ−

1

2

t δu−10δtδuδt−1δ10u δ2tδ−u10δ−

1

t δu−1δtδu3

S17: (δ−2

u δtδuδt−1δu10δt2δu−10δ

−1

t δ−u1δtδ10u δ

−2

−1

t δ−u1δt)−1

δtδuδt−1δu8δ2tδ−u8δ

−1

t δu−1δtδu8δ

−2

t δu−10δtδuδt−1δ10u δ2tδ−u10δ

−1

t δu−1δtδu2

S18: (δ−t2δ−u6δtδuδt−1δ6uδt2δu−6δ−

1

t δ−u1δtδ5uδ−

1

t δuδtδuδt−2δ−u1δtδ−u1δ−

1

t δuδ2t

δ−1

u δ

−1

t δuδtδuδ−t2δu−1δtδu−1δ

−1

t δuδt2δu−1δ

−1

t δuδtδuδ−t2δ−u1δtδ−u1δ

−1

t δu−9δt

δuδt−1δu10δt2δu−10δ

−1

t δ−u1δtδ10u (δ

−2

t δ−u1δtδuδt−1δuδt2δ−u1δ

−1

t δu−1δtδu)2

δ−_t2δ−1

−1

t δ−u1δt)−1

(δ_t−2δ−1

u δtδuδt−1δuδt2δ−u1δ

−1

t δu−1δtδu)−2δ−u1δ

−1

t δuδtδuδt−2δu−1δtδ−u1δ

−1

t δu−9δt

δuδt−1δu10δt2δu−10δ

−1

t δ−u1δtδ10u (δ

−2

t δ−u1δtδuδt−1δuδt2δ−u1δ

−1

t δu−1δtδu)3

δ−_t2δ−6

−1

t δ−u1δtδ5u

S19: (δ−u2δtδuδt−1δu9δt2δ−u9δ−

1

t δu−1δtδ9uδ−

2

t δu−7δtδuδt−1δu7δ2tδu−7δ−

1

t δu−1δtδu7

δ−_t2δ−3

−1

t δ−u1δtδ3uδ

−2

t δu−1δtδuδt−1δuδ2tδ−u1δ

−1

(10)

δ−_t2δ−8

1

t δ−u1δt)−1

−1

t δu−1δtδu7δ

−2

t δu−1δtδuδt−1δuδt2δu−1δ

−1

t δ−u1δtδu

δ−_t2δ−3

−1

t δ−u1δtδ3uδ

−2

t δu−8δtδuδt−1δu8δ2tδ−u8δ

−1

t δu−1δtδu8

δ−_t2δ−9

−1

t δ−u1δtδu

S20: (δ−u2δtδuδt−1δu9δt2δ−u9δ−

1

t δu−1δtδ9u(δ−

2

1

t δ−u1δtδ7u)2

1

2

t δu−1δtδuδt−1δuδ2tδ−u1δ−

1

t δu−1δtδu

δ−_t2δ−8

1

2

t δu−8δtδuδt−1δu8δ2tδ−u8δ−

1

t δu−1δt)−1

1

2

1

t δ−u1δtδ7u

δ−_t2δ−1

−1

t δ−u1δtδuδt−2δu−4δtδuδt−1δu4δ2tδ−u4δ

−1

t δu−1δtδu4

(δ_t−2δ−8

u δtδuδt−1δ8uδt2δ−u8δ

−1

t δu−1δtδ8u)2δ

−2

−1

t δ−u1δtδu

S21: (δ−_u2δtδuδt−1δ

9

uδ

2

tδ−

9

u δ

−1

t δ−

1

u δtδ9u(δ

−2

t δ−

7

u δtδuδ−t1δ

7

uδ

2

tδ−

7

u δ

−1

t δ−

1

u δtδ7u)

3

1

2

1

t δu−1δtδu2

(δt−2δ−u8δtδuδt−1δ8uδt2δ−u8δ−

1

t δu−1δtδ8u)2δ−

2

1

t δ−u1δt)−1

1

2

1

t δu−1δtδ7u)2

δ−_t2δ−2

1

2

1

t δu−1δtδu3

(δ_t−2δ−8

u δtδuδt−1δ8uδt2δ−u8δ−

1

2

1

t δu−1δtδu

S22: (δ−2

u δtδuδt−1δu9δt2δ−u9δ

−1

t δu−1δtδ9u(δ

−2

−1

t δ−u1δtδ7u)4

δ−_t2δ−_u4δtδuδt−1δ

4

uδ

2

tδ−

4

u δ

−1

t δ−

1

u δtδ4uδ

−2

t δ−

2

u δtδuδt−1δ

2

uδ

2

tδ−

2

u δ

−1

t δ−

1

u δtδu2

(δt−2δ−u8δtδuδt−1δ8uδ

2

tδ−

8

u δ

−1

t δu−1δtδ8u)

3_δ−2

t δ−u8δtδuδt−1δ8uδ

2

tδ−

8

u δ

−1

t δu−1δt)−1

1

2

1

t δu−1δtδ7u)3

1

2

1

t δu−1δtδu4

(δ_t−2δ−8

1

2

1

t δu−1δtδu

S23: (δ−2

−1

t δu−1δtδ9u(δ

−2

−1

t δ−u1δtδ7u)5

δ−_t2δ−5

−1

t δ−u1δtδ5uδ

−2

−1

t δu−1δtδu3

δ−_t2δ−_u1δtδuδt−1δuδt2δ−

1

u δ

−1

t δ−

1

u δtδu(δt−2δ−

8

u δtδuδ−t1δ

8

uδ

2

tδ−

8

u δ

−1

t δ−

1

u δtδu8)

4

8

uδ

2

tδ−

8

u δ

−1

t δ−

1

u δt)−1

1

2

1

t δu−1δtδ7u)4

1

2

1

t δu−1δtδu5

1

2

1

t δu−1δtδu

S24: (δ−2

−1

t δu−1δtδ9u(δ

−2

−1

t δ−u1δtδ7u)6

δ−_t2δ−5

−1

t δ−u1δtδ5uδ

−2

−1

t δu−1δtδu4

δ−_t2δ−2

−1

t δ−u1δtδ2u(δ

−2

−1

t δu−1δtδu8)5

8

uδ

2

tδ−

8

u δ

−1

t δ−

1

u δt)−1

δtδuδt−1δu7δ

2

tδ−

7

u δ

−1

t δu−1δtδu7(δ

−2

t δ−u7δtδuδt−1δ7uδ

2

tδ−

7

u δ

−1

t δu−1δtδ7u)

5

1

2

1

t δu−1δtδu5

1

2

1

t δu−1δtδu

S25: (δ−2

−1

t δu−1δtδ9u(δ

−2

−1

t δ−u1δtδ7u)7

δ−_t2δ−3

−1

t δ−u1δtδ3uδ

−2

−1

t δu−1δtδu4

δ−_t2δ−3

−1

t δ−u1δtδ3uδ

−2

−1

t δu−1δtδu5

(δ_t−2δ−_u8δtδuδt−1δ

8

uδ

2

tδ−

8

u δ

−1

t δ−

1

u δtδ8u)

6_δ−2

t δ−

8

u δtδuδt−1δ

8

uδ

2

tδ−

8

u δ

−1

t δ−

1

u δt)−1

δtδuδt−1δ

7

uδ

2

tδ−

7

u δ

−1

t δ−

1

u δtδu7(δ

−2

t δ−

7

u δtδuδt−1δ

7

uδ

2

tδ−

7

u δ

−1

t δ−

1

u δtδ7u)

6

1

2

1

t δu−1δtδu4

1

2

1

t δu−1δtδu5

1

2

1

t δu−1δtδu

S26: (δ−2

−1

t δu−1δtδ9u(δ

−2

−1

t δ−u1δtδ7u)8

(δ_t−2δ−1

−1

t δu−1δtδ2u)3(δ

−2

t δ−u8δtδuδt−1δu8δ2tδ−u8δ

−1

t δu−1δtδu8)7

δ−_t2δ−8

−1

t δ−u1δt)−1

δtδuδt−1δ

7

uδ

2

tδ−

7

u δ

−1

t δ−

1

u δtδu7(δ

−2

t δ−

7

u δtδuδt−1δ

7

uδ

2

tδ−

7

u δ

−1

t δ−

1

u δtδ7u)

7

δ−t2δ−u3δtδuδt−1δ3uδ

2

tδ−

3

u δ

−1

t δ−u1δtδ3u(δ

−2

t δu−1δtδuδt−1δuδ2tδ−

1

u δ

−1

t δu−1δtδu)3

1

2

1

t δu−1δtδu

S27: (δ−2

−1

t δu−1δtδ9u(δ

−2

−1

(11)

(δ_t−2δ−1

1

t δu−1δtδ2u)3(δ−

2

t δ−u8δtδuδt−1δu8δ2tδ−u8δ−

1

t δu−1δtδu8)8

δ−_t2δ−8

−1

t δ−u1δt)−1

−1

t δu−1δtδu7(δ

−2

−1

t δu−1δtδ7u)8

δ−_t2δ−4

−1

t δ−u1δtδ4u(δ

−2

−1

t δu−1δtδu)3

(δ_t−2δ−8

−1

t δu−1δtδ8u)9δ

−2

−1

t δ−u1δtδu

References

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[2] I. Anshel, M. Anshel, B. FISHER, New key agreement protocols in braid group cryptography, in NACCACHE D. (ED.): Topics in CryptologyCCT-RSA, San Francisco, CA, USA, 8C12 April 2001, (LNCS, 2020), 13C27.

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[9] J. H. Cheon and B. Jun, A polynomial time algorithm for the braid Diffie-Hellman conjugacy prob-lem, in BONEH D. (ED.): Advances in Cryptology-CRYPTO 2003, Santa Barbara, CA, USA, 17-21 August 2003, (LNCS, 2729), 212-225.

[10] M. Chiswell, D.J. Collins and J. Huebschmann, Aspherical group presentation, Math. Z., 178 (1981), 1-36.

[11] D. J. Collins, A simple presentation of a group with unsolvable word problem, Illnois J. Math., 30(2)(1986), 230-234.

[12] D. J. Collins, Relations among the squares of the generators of the braid group, Invent. Math., 117(1994), 525-530.

[13] P Dehornoy, Using shifted conjugacy in braid-based cryptography, Arxiv ePrint Archive, Report 0609091v1, http://arxiv.org/abs/cs/0609091.

[14] N. Franco and J. Gonzales-Meneses, Conjugacy problem for braid groups and Garside groups, J. Algebra, 266(2003), 112-132.

[15] D. Garber, S. Kaplan, M. Teicher, B. Tsaban, U. Vishne, Probabilistic solutions of equations in the braid group, Advances in Applied Mathematics 35(2005), 323-334.

[16] D. Garber, S. Kaplan, M. Teicher, B. Tsaban, U. Vishne, Length-based conjugacy search in the Braid group, Contemp. Math., Amer. Math. Soc. 418(2006), 75-87.

[17] V. Gebhardt, A new approach to the conjugacy problem in Garside groups, J. Algebra, 292(1)(2005), 282-302.