Cofinitary Permutation Groups

On several problems about automorphisms of the free group of rank two

There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the information that there are objects with the property P, find at least one particular object with the property P. So far, no cryptographic protocol based on a search problem in a non-commutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability “very close” to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a “brute force attack”, the “1” or “0” bits in Bob’s binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her “brute force attack” program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob’s sequence.

There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the information that there are objects with the property P, find at least one particular object with the property P. So far, no cryptographic protocol based on a search problem in a non-commutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. Decision problems are problems of the following nature: given a property P and an object O, find out whether or not the object O has the property P. By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability “very close” to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a “brute force attack”, the “1” or “0” bits in Bob’s binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her “brute force attack” program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob’s sequence.

There are several public key establishment protocols as well as complete public key cryptosystems based on allegedly hard problems from combinatorial (semi)group theory known by now. Most of these problems are search problems, i.e., they are of the following nature: given a property P and the information that there are objects with the property P, find at least one particular object with the property P. So far, no cryptographic protocol based on a search problem in a non-commutative (semi)group has been recognized as secure enough to be a viable alternative to established protocols (such as RSA) based on commutative (semi)groups, although most of these protocols are more efficient than RSA is. In this paper, we suggest to use decision problems from combinatorial group theory as the core of a public key establishment protocol or a public key cryptosystem. By using a popular decision problem, the word problem, we design a cryptosystem with the following features: (1) Bob transmits to Alice an encrypted binary sequence which Alice decrypts correctly with probability very close to 1; (2) the adversary, Eve, who is granted arbitrarily high (but fixed) computational speed, cannot positively identify (at least, in theory), by using a brute force attack, the 1 or 0 bits in Bob's binary sequence. In other words: no matter what computational speed we grant Eve at the outset, there is no guarantee that her brute force attack program will give a conclusive answer (or an answer which is correct with overwhelming probability) about any bit in Bob's sequence.

(1997). An Introduction to Combinatorial Group Theory and the Word Problem. Mathematics Magazine: Vol. 70, No. 1, pp. 3-10.

This book is about relations between three different areas of mathematics and theoretical computer science: combinatorial group theory, cryptography, and complexity theory. It explores how non-commutative (infinite) groups, which are typically studied in combinatorial group theory, can be used in public-key cryptography. It also shows that there is remarkable feedback from cryptography to combinatorial group theory because some of the problems motivated by cryptography appear to be new to group theory, and they open many interesting research avenues within group theory. In particular, a lot of emphasis in the book is put on studying search problems, as compared to decision problems traditionally studied in combinatorial group theory. Then, complexity theory, notably generic-case complexity of algorithms, is employed for cryptanalysis of various cryptographic protocols based on infinite groups, and the ideas and machinery from the theory of generic-case complexity are used to study asymptotically dominant properties of some infinite groups that have been applied in public-key cryptography so far. This book also describes new interesting developments in the algorithmic theory of solvable groups and another spectacular new development related to complexity of group-theoretic problems, which is based on the ideas of compressed words and straight-line programs coming from computer science.

In digital forensics, the issue of data integrity protection for increasingly widespread applied SSD (Solid State Disk, SSD) is to be resolved. Based on Combinatorial Group Theory, mapping data objects in SSD validation process and test object in combination group testing methods, using the non-adaptive mode to the initial calculation, stored procedures, and re-calculate, verify process, and carefully selecting the design parameters to construct the test matrix of Combinatorial Group Testing method in response to different application environments, we design testing group. The algorithm capabilities meet the requirements and that is higher efficiently by repeated tests. It is feasible for SSD integrity check based on the Combinatorial Group Theory.

First we introduce the history of group theory. Group theory has three main historical sources: number theory, the theory of algebraic equations, and geometry. Secondly, we give the main classes of groups: permutation groups, matrix groups, transformation groups, abstract groups and topological and algebraic groups. Finally, we give two different presentations of a group: combinatorial group theory and geometric group theory.

In this chapter, we recall basic definitions from combinatorial group theory. We assume some basic knowledge of group theory; see, for instance, [149]. More background on combinatorial group theory can be found in [119, 158]. Groups will be written multiplicatively throughout this work and the group identity will be denoted with 1.

Let Fn be a free group of rank n ≥ 2. Two elements g, h in Fn are said to be translation equivalent in Fn if the cyclic length of φ(g) equals the cyclic length of φ(h) for every automorphism φ of Fn. Let F(a, b) be the free group generated by {a, b} and let w(a, b) be an arbitrary word in F(a, b). We prove that w(g, h) and w(h, g) are translation equivalent in Fn whenever g, h ∈ Fn are translation equivalent in Fn, and thereby give an affermative solution to problem F38b in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edition. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001), Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.].

Let F 2 be a free group of rank 2. We prove that there is an algorithm that decides whether or not, for two given elements u , v of F 2 , u and v are translation equivalent in F 2 , that is, whether or not u and v have the property that the cyclic length of ( u ) equals the cyclic length of ( v ) for every automorphism of F 2 . This gives an affirmative solution to problem F38a in the online version (http://www.grouptheory.info) of [G. Baumslag, A. G. Myasnikov and V. Shpilrain. Open problems in combinatorial group theory, 2nd edn. In Combinatorial and geometric group theory (New York, 2000/Hoboken, NJ, 2001) , Contemp. Math. 296 (American Mathematical Society, 2002), pp. 1–38.] for the case of F 2 .

The classical method of abelianization is an important tool in combinatorial group theory. It is an algorithm for tbe calculation of the commutator factor group and thus gives a survey of the abelian homomorphic images of a group, since any homomorphism of a group into an abelian group factors through the commutator factor group. A non-commutative analogue of abelianization can be developped if one replaces the abelian groups by the (generalized) dihedral groups; in other words, there exists a method which gives a survey of the (generalized) dihedral homomorphic images of a group. The foundations of this new method will be given in Sections 1 and 2. The most conspicuous difference between abelianization and "dihedralization" is the fact that there is no single substitute for the commutator subgroup; it will be shown that the commutator subgroup has to be replaced by a certain canonical set of normal subgroups. Subsequently, instead of having a single algorithm as in the abelian case, several algorithms have to be carried out. The algorithms, which are described in Section 3, are not more complicated than the corresponding one of abelianization. Notation, preliminaries. X- Y denotes the set difference; if Y consists of a single element y, we write X - y. IX[ denotes the cardinal of the set X. If S is a subset of the group G, (S) denotes the subgroup generated by the set S, and S- 1 denotes the set of all inverse elements of S. If G and H are groups, H < G means that H is a subgroup of G. IfH is a normal subgroup of G and x, y are elements of G such that xyN = yxN, we will occasionally say that x and y commute rood N. An elementary abelian 2-group is a group which contains only elements of order at most two. If G is a group, the subgroup generated by all squares is denoted by G 2. Clearly, G 2 iS the smallest of all normal subgroups which have an elementary abelian 2-group as factor group.

A Schottky extension group is a Kleinian group K containing a Schottky group G of rank g ≥ 2 as a normal subgroup. It is well known that the index of G in K is at most 12(g − 1); if the index is 12(g − 1), then we say that K is a maximal Schottky extension group. A structural description of the maximal Schottky extension groups using 2-dimensional arguments, internal to Riemann surfaces and classical Kleinian groups in spirit, is provided. As a consequence, we re-obtain Zimmermann’s result which states that a maximal Schottky extension group is isomorphic to one of the following groups $$D_{2}*_{{\mathbb Z}_{2}} D_{3}, \; D_{3}*_{{\mathbb Z}_{3}} {\mathcal A}_{4}, \;D_{4}*_{{\mathbb Z}_{4}} {\mathfrak S}_{4}, \; D_{5}*_{{\mathbb Z}_{5}} {\mathcal A}_{5},$$ where D r is the dihedral group of order $${2r, {\mathcal A}_{r}}$$ is the alternating group in r letters and $${{\mathfrak S}_{4}}$$ is the symmetric group in 4 letters. The methods used by Zimmermann are from combinatorial group theory (finite extensions of free groups) and also dimension three, so our arguments are different.

In all of the previous chapters we have dealt with developments which started before the middle of the Twentieth Century, although, in many cases, we also reported on some of their later phases. In the present chapter we are making an exception to this rule which we imposed on our account in order to keep it from becoming too voluminous. Our reason for doing so is the exceptional nature of the impact of mathematical logic on combinatorial group theory. We encounter here the situation where concepts and methods from one mathematical discipline become essential ingredients of the theorems and of the answers to previously open questions of another discipline. Of course, events of this type have happened before, but they deserve to be mentioned with distinction. In a way, the connection between topology and combinatorial group theory provides a similar example. However, there exists a notable difference between these two occurrences. Topology and combinatorial group theory grew up together, but the event we are now going to describe involves the action of a highly developed sophisticated theory on another equally developed discipline.

Having enjoyed a steady diet of geometric group theory for many years, in this brief talk I would like to re-introduce a classical problem of group theory and topology that deserves to be considered by a new generation. (There are others.) In 1941, J. H. C. Whitehead asked whether asphericity is a hereditary property for two-dimensional CW complexes. The problem is still unsolved and I would not be surprised if it outlasts us all. Immediately upon posing the question, Whitehead reduced it to a question in combinatorial group theory. This algebraic question turns out to be much more difficult than the heredity question, so I will not speak about that. Instead, I will highlight some major themes in group theory and topology that directly relate to Whitehead’s problem. More such themes will emerge in the future, probably in other areas and possibly soon. I will point out open questions as appropriate. Finally, as time allows I will try to illustrate why the problem is hard.