Key-agreement based on automaton groups

TL;DR

This paper explores the use of various automaton groups, including Grigorchuk and Basilica groups, as platforms for key-agreement schemes, leveraging their algebraic properties for cryptographic applications.

Contribution

It introduces novel automaton groups as potential cryptographic platforms for the Anshl-Anshel-Goldfeld metascheme, expanding the scope of automaton groups in cryptography.

Findings

Automaton groups like Grigorchuk and Basilica are viable for key-agreement schemes.

The groups exhibit properties suitable for cryptographic protocols.

The subgroup of the affine group has an unsolvable conjugacy problem, enhancing security.

Abstract

We suggest several automaton groups as key-agreement platforms for Anshl-Anshel-Goldfeld metascheme, they include Grigorchuk and universal Grigorchuk groups, Hanoi 3-Towers group, Basilica group and a subgroup of the affine group with the unsolvable conjugacy problem

Equations

$$ba{b}^{-1}=b{a}_{p_1}{b}^{-1}\cdots b{a}_{p_s}{b}^{-1},$$

$$a^{-1}ba=a^{-1}{b}_{q_1}a\cdots a^{-1}{b}_{q_t}a.$$

$$\pi (a,0)=1,\pi (a,1)=0,\pi (b,x)=\pi (c,x)=\pi (d,x)=x;\tau (a,x)=\tau (e,x)=e,$$

$$\tau (b,0)=\tau (c,0)=a,\tau (d,0)=e,\tau (b,1)=c,\tau (c,1)=d,\tau (d,1)=b$$

$$\pi (e,x)=\pi (a,x)=x,\pi (b,0)=1,\pi (b,1)=0;$$

$$\tau (e,x)=\tau (a,0)=\tau (b,0)=e,\tau (a,1)=b,\tau (b,1)=a$$

$$\pi (e,(x,y))=\pi (b,(x,y))=\pi (c,(x,y))=\pi (d,(x,y))=(x,y),$$

$$\pi (a,(0,y))=(1,y),\pi (a,(1,y))=(0,y);$$

$$\tau (e,(x,y))=\tau (a,(x,y))=\tau (b,(0,2))=\tau (c,(0,1))=\tau (d,(0,0))=e,$$

$$\tau (b,(0,0))=\tau (b,(0,1))=\tau (c,(0,2))=\tau (d,(0,1))=\tau (d,(0,2))=a,$$

$$\tau (b,(1,y))=b,\tau (c,(1,y))=c,\tau (d,(1,y))=d$$

$$a_{ij}(iw)=jw,a_{ij}(jw)=iw,a_{ij}(xw)=x{a}_{ij}(w),x\notin \{i,j\}.$$

Full text

Introduction

Typically abelian groups are involved in cryptography, say in RSA and Diffie-Hellman schemes (see e. g. [20], [21] and the references there). But they are vulnerable with respect to quantum machines. Thus, for post-quantum cryptography one tries to use non-abelian groups (some attempts one can find in e. g. [15], [16], [17] and in the references there). In this paper we suggest several groups $G$ as candidates for platforms for Anshel-Anshel-Goldfeld key-agreement metascheme [1] (section 1).

To break Anshel-Anshel-Goldfeld scheme over a group $G$ an adversary needs to solve a system of simultaneous conjugacies of the form $xu_{i}x^{-1}=v_{i},\,1\leq i\leq m$ for given $u_{i},v_{i}\in G,\,1\leq i\leq m,\,a_{1},\dots,a_{n}\in G$ and unknown $x\in\langle a_{1},\dots,a_{n}\rangle$. On the other hand, to perform a communication between Alice and Bob via a public channel, the word problem in $G$ should have a small (say, polynomial) complexity. We suggest some automaton groups [9], [2], [10] (see section 2) as $G$ for which the word problem is known to have the polynomial complexity The conjugacy problem for automaton groups was studied in [6], [12]. Observe that automaton groups are convenient for algorithmic representation.

In section 2.1 we consider Grigorchuk group [9]. Note that in [18], [24] the algorithms (without complexity analysis) for the conjugacy problem in Grigorchuk group were proposed, later in [19] a polynomial complexity algorithm for the conjugacy problem in Grigorchuk group was exhibited. But the problem of simultaneous conjugacies looks difficult in Grigorchuk group. Also mention that there was an attempt to use Grigorchuk group in cryptography in a different way [7] which was later broken [23]. In section 2.2 we discuss the Basilica group [13] which is defined by an automaton with 3 states. In section 2.3 we consider the universal Grigorchuk group [9], [3]. In section 2.4 we discuss the group of Hanoi Towers on 3 pegs [11]. Finally, in section 2.5 we consider a subgroup of the affine group $Aff_{4}(\mathbb{Z})$ with the unsolvable conjugacy problem.

1. Anshel-Anshel-Goldfeld key-agreement metascheme

We recall the key-agreement scheme from [1] (cf. [15] where its extension to multiparty communications is exhibited, also [22] ). Let $G$ be a group and $a_{1},\dots,a_{n},\,b_{1},\dots,b_{m}\in G$ be some publically given elements. Alice chooses her private element $a=a_{p_{1}}\cdots a_{p_{s}}\in\langle a_{1},\dots,a_{n}\rangle$, while Bob chooses his private element $b=b_{q_{1}}\cdots b_{q_{t}}\in\langle b_{1},\dots,b_{m}\rangle$. Alice transmits (via a public channel) elements $a^{-1}b_{i}a,\,1\leq i\leq m$, while Bob transmits $ba_{j}b^{-1},\,1\leq j\leq n$. After that Alice computes

[TABLE]

while Bob computes

[TABLE]

Finally, the commutator $a^{-1}(bab^{-1})=(a^{-1}ba)b^{-1}$ computed by both Alice and Bob, is treated as their common secret key.

So, an adversary has to find $A\in\langle a_{1},\dots,a_{n}\rangle,\,B\in\langle b_{1},\dots,b_{m}\rangle$ such that $A^{-1}b_{i}A=a^{-1}b_{i}a,\,1\leq i\leq m$ and $Ba_{j}B^{-1}=ba_{j}b^{-1},\,1\leq j\leq n$ (note that the right-hand sides of the latter equalities are known). Then one can verify that $a^{-1}bab^{-1}=A^{-1}BAB^{-1}$. We emphasize that an adversary has to search a solution $A$ of the problem $A^{-1}b_{i}A=a^{-1}b_{i}a,\,1\leq i\leq m$ in the subgroup $\langle a_{1},\dots,a_{n}\rangle$ which makes the task even harder than the customary simultaneous conjugacy problem. Thus, our goal is to exhibit groups with the polynomial complexity of the word problem and difficult problem of solving systems of conjugacies.

We produce several candidates for such groups among automaton groups (see e. g. [2], [9], [10]).

2. Automaton groups

Denote by $X=\{0,\dots,k-1\}$ an alphabet and by $S$ a finite set that we will call a set of the states. An automaton of Mealy type on $X$ with a set $S$ of states is defined by a transition function $\tau:S\times X\to S$ and an output function $\pi:S\times X\to X$. If for each $s\in S$ the function $\pi(s,\cdot)\in Sym(k)$ is a permutation then the automaton is called invertible.

Denote by $T$ a rooted $k$-regular tree and by $T_{0},\dots,T_{k-1}$, respectively, the rooted subtrees of $T$ with their roots in the children of the root of $T$. The paths (without back tracking) in $T$ starting at its root correspond to the words in the alphabet $X$. Denote by $X^{l}$ the set of the words of the length $l$ over $X$, by $X^{*}$ the set of all the words, and by $X^{\infty}$ the set of all the infinite words over $X$. Each state $s\in S$ provides an action on $T$ being its automorphism: $s$ acts by a permutation $\pi(s,\cdot)$ on the roots of subtrees $T_{0},\dots,T_{k-1}$, and in its turn $s$ acts recursively as $\tau(s,i)$ on the subtree $T_{i},0\leq i<k$.

Thus, for an invertible automaton $A=(S,X,\tau,\pi)$ this defines a group $G(A)$ of automatically defined automorphisms of $T$ with the operation of composition. The group $G(A)$ (see e. g, [2], [9], [10]) is generated by the words over $S\cup S^{-1}$ where for the state corresponding to $s^{-1}$ the permutation $\pi(s^{-1},\cdot)=(\pi(s,\cdot))^{-1}$ and $\tau(s^{-1},i)=(\tau(s,i))^{-1}$. We refer to the length $|g|$ of an element $g\in G(A)$ as its length in the generators $S\cup S^{-1}$ (clearly, the length depends on a representation in the generators, we’ll be interested in upper bounds on the length, so no misunderstanding would emerge). For an element $g\in G(A)$ we define its portait (see e. g. [2], [9]) of a depth $d$ as the collection of the following data: a permutation of the action of $g$ (denoted by $g_{x}$ on $X^{d}$ and for every word $x=x_{1}\cdots x_{d}\in X^{d}$ the action of $g$ on the subtree $T_{x}$ of $T$ with the root $x$ (being an element of $G(A)$, these elements are called sections).

In all the examples of automaton groups $G(A)$ considered below (except for the last one) two elements $g_{1},g_{2}\in G(A)$ are equal iff their portraits of the depth $\log(|g_{1}|+|g_{2}|)$ coincide. Moreover, the sections of all the words of this length over $X$ have constant size $O(1)$ (we’ll refer to it as the portrait property). This is due to the contracting property established for the groups $G(A)$ considered below (except for the last one): there exist $\lambda<1,c,l$ such that $|g_{x}|<\lambda|g|+c$ for all $g\in G(A),\,x\in X^{l}$. The contracting property immediately allows one to solve the word problem in $G(A)$ within the polynomial complexity. On the other hand, it seems that the problem of solving a system of simultaneous conjugacies is difficult in all the automaton groups under consideration, the key-agreement scheme from section 1 based on any of these groups looks hard to be broken.

Thus, one can compute the portrait within the polynomial complexity, and the portrait (or its binary encoding) will be used as a common secret key by Alice and Bob.

2.1. Grigorchuk group

Grigorchuk group $G$ (see e. g. [9]) can be defined by an automaton with 5 states $a,b,c,d,e$ acting on $X^{*}=\{0,1\}^{*}$ as follows:

[TABLE]

[TABLE]

for any $x\in X$. In particular, $a^{2}=b^{2}=c^{2}=d^{2}=bcd=e$ (where $e$ denotes the identity). Note that $G$ is not finitely presented. Observe that the complexity upper bound for the word problem for $G$ is $O(n\log n)$ [9]. It is known (see e. g. [9]) that the portrait property (see section 2) holds for $G$.

In [19] an algorithm is designed to test whether for given $u,v\in G$ there exists $x\in G$ such that $xux^{-1}=v$. In fact, one can extend this algorithm to produce such $x$, provided it does exist. On the other hand, it seems to be a difficult problem to test whether there exists $x\in G$ such that $xu_{i}x^{-1}=v_{i},\,1\leq i\leq m$ for given $u_{i},v_{i}\in G,\,1\leq i\leq m$ (and so more, to find such $x$).

One could also use the generalizations $G_{\omega}$ [8], [9] of $G$ where $\omega\in\{0,1,2\}^{\infty}$. Observe that the word problem in $G_{\omega}$ has a complexity upper bound polynomial in the complexity of computing a prefix of $\omega$ of a logarithmic length, while for a generic $\omega$ already the single conjugacy equation problem is more difficult than the similar problem in $G$ [8], [9].

2.2. Basilica group

Consider an automaton group $B$ (sometimes called the Basilica group) defined by the following automaton with 3 states $a,b,e$ (again, $e$ is the identity of $B$) over the alphabet $X=\{0,1\}$ [13], [14]:

[TABLE]

[TABLE]

for any $x\in X$.

It is proved in [13] that the group $B$ also satisfies the portrait property. Note that for $B$ only an exponential complexity algorithm is known for the problem of a single conjugacy equation.

2.3. Universal Grigorchuk group

One can represent each group $G_{\omega}=F_{4}/N_{\omega}$ where $N_{\omega}$ is a normal subgroup of 4-free group $F_{4}$ (with the generators $a,b,c,d$). Denote $N=\bigcap_{\omega}N_{\omega}$ where the intersection ranges over all the infinite words $\omega\in\{0,1,2\}^{\infty}$. The universal group is defined $U=F_{4}/N$ [3]. Similar to $G$ (see section 2.1) $a^{2}=b^{2}=c^{2}=d^{2}=bcd=e$ (and again, $U$ is not finitely presented).

One can represent $U$ as an automaton group [3] defined by an automaton with 5 states $a,b,c,d,e$ (again, $e$ is the identity of $U$) over an alphabet $X=\{0,1\}\times\{0,1,2\}$ of size 6 as follows:

[TABLE]

[TABLE]

[TABLE]

[TABLE]

[TABLE]

for any $x\in\{0,1\},\,y\in\{0,1,2\}$.

Similar to the group $G$ (cf. section 2.1) the group $U$ also satisfies the portait property [9], [3].

Apparently, the simultaneous conjugacy problem for $U$ (cf. section 1) is not easier than the same problem for $G$, for $G_{\omega}$ and for $B$.

2.4. Hanoi 3-Towers group

We describe Hanoi Towers group $H^{(3)}$ on 3 pegs as an automaton group [11], [5]. The alphabet $X=\{0,1,2\}$ consists of 3 letters which corresponds to the pegs. Actully, one can generalize to the group $H^{(k)}$ of Hanoi Towers on $k\geq 3$ pegs, then $|X|=k$ [11], [5]. A word $x_{1}\cdots x_{n}\in X^{n}$ has a meaning that the disc $i$ is placed on $x_{i}$-th peg. According to the rules of the game in each peg the discs of sizes $1,2,\dots$ are placed in the decreasing order of their sizes from the bottom to the top.

The automaton of $H^{(3)}$ contains 3 states: $a_{01},a_{02},a_{12}$. For any word $w\in X^{n}$ we have

[TABLE]

This means that $a_{ij}$ takes the disc from the top of either peg $i$ or $j$ being minimal among these two and puts it on another peg among $i$ and $j$. Clearly, $a_{01}^{2}=a_{02}^{2}=a_{12}^{2}=e$ (again, $H^{(k)}$ is not finitely presented).

In [5] the portrait property is proved for $H^{(3)}$. Note that the complexity bound $\exp(O(\log^{k-2}n))$ [5] for the word problem in the group $H^{(k)}$ is not polynomial for $k\geq 4$.

2.5. A group with the unsolvable problem of conjugacy

In Proposition 7.5 [4] a group $F^{\prime}\subset GL_{4}(\mathbb{Z})$ is constructed with generators $M_{1},\dots,M_{s}\in GL_{4}(\mathbb{Z})$ having unsolvable orbit problem, i. e. whether for a pair of vectors $u,v\in\mathbb{Z}^{4}$ there exists $f\in F^{\prime}$ such that $fu=v$. In [25] it is proved that the semidirect product $G^{\prime}=\mathbb{Z}^{4}\rtimes F^{\prime}\subset Aff_{4}(\mathbb{Z})$ has the unsolvable conjugacy problem. Moreover, in Proposition 1.5 [25] this construction is modified to make a group $F\subset GL_{6}(\mathbb{Z})$ free, also having the unsolvable orbit problem and $G=\mathbb{Z}^{6}\rtimes F\subset Aff_{6}(\mathbb{Z})$ having the unsolvable conjugacy problem.

On the other hand, the word problem in $G^{\prime}$ (as well as in $G$) can be solved within the polynomial complexity. Indeed, an element of $G^{\prime}$ one can represent as a composition of affine transformations in $Aff_{4}(\mathbb{Z})$ of $\mathbb{Z}^{4}$ of the form $v\to u+M_{i}v,\,1\leq i\leq s$ for vectors $u\in\mathbb{Z}^{4}$. One can explicitly compute such a composition.

Note that in [25] $G$ is represented as an automaton group. It looks reasonable to use both $G$ and $G^{\prime}$ as platforms for Anshel-Anshel-Goldfeld scheme (see section 1).

Acknowledgements. The frst author graciously acknowledges support from the Simons Foundation through Collaboration Grant 527814, is partially supported by the mega-grant of the Russian Federation Government (N14.W03.31.0030) and is grateful to Max-Planck Institut fuer Mathematik, Bonn during staying in which the paper was conceived. The second author is grateful to the grant RSF 16-11-10075 and to MCCME for inspiring atmosphere.