# Building Group Key Establishment on Group Theory: A Modular Approach

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## Abstract

**:**

## 1. Introduction

- Instead of the random oracle model, we use the common reference string model. An (expected) price we pay for this, is the need of a decisional assumption instead of a computational one that is used in [21].
- Instead of setting out for a group key establishment directly, we suggest a construction for the two-party case and thereafter apply a protocol compiler of Abdalla et al. [22].

## 2. Preliminaries: Security Model and Protocol Goals

- Two values ${v}_{0}$, ${v}_{1}.$ These will be the input for a pseudorandom function at the time of computing the session identifier and session key;
- The information necessary to implement a non-interactive and non-malleable commitment scheme (see Section 3.1 for further details);
- Two elements, chosen independently and uniformly at random, each taken from a family of universal hash functions (one as needed for the compiler in [22] and one for our two-party solution as detailed in Section 3.1).

#### 2.1. Communication Model and Adversarial Capabilities

#### 2.1.1. Protocol Instances

- ${\mathsf{used}}_{i}^{{s}_{i}}$
- will indicate whether this instance is or has been used for a protocol run. The ${\mathsf{used}}_{i}^{{s}_{i}}$ flag is set through a protocol message received by the corresponding instance due to a call to the $\mathsf{Send}$ oracle (see below);
- ${\mathsf{state}}_{i}^{{s}_{i}}$
- stores the state information needed during the protocol execution;
- ${\mathsf{term}}_{i}^{{s}_{i}}$
- indicates if the execution has terminated;
- ${\mathsf{sid}}_{i}^{{s}_{i}}$
- denotes a session identifier (which may be public) which may be later use as identifier for the session key ${\mathsf{sk}}_{i}^{{s}_{i}}$ (in particular, the adversary is thus allowed to learn session identifiers);
- ${\mathsf{pid}}_{i}^{{s}_{i}}$
- stores the user identities that ${\Pi}_{i}^{{s}_{i}}$ aims at establishing a key with. This set includes ${U}_{i}$ himself;
- ${\mathsf{acc}}_{i}^{{s}_{i}}$
- indicates that the protocol instance completed a protocol successfully. That is, whether the involved user accepted the session key or not;
- ${\mathsf{sk}}_{i}^{{s}_{i}}$
- stores a distinguished null value in the beginning. After a session key is accepted by ${\Pi}_{i}^{{s}_{i}},$ this session key replaces the null value.

#### 2.1.2. Communication Network

#### 2.1.3. Adversarial Capabilities

- Send(U
_{i}, s_{i}, M) - This oracle sends a message M to instance ${\Pi}_{i}^{{s}_{i}}$ and returns the message generated by this instance. In case the instance ${\Pi}_{i}^{{s}_{i}}$ is previously unused and the message $M\subseteq \mathcal{P}$ contains a set of user identities, the ${\mathsf{used}}_{i}^{{s}_{i}}$-flag is set, ${\mathsf{pid}}_{i}^{{s}_{i}}$ initialized with ${\mathsf{pid}}_{i}^{{s}_{i}}:=\left\{{U}_{i}\right\}\cup M$. ${\Pi}_{i}^{{s}_{i}}$ initiates the protocol with the first message which is returned.
- Reveal(U
_{i}, s_{i}) - This outputs the computed key of the instance stored in ${\mathsf{sk}}_{i}^{{s}_{i}}$.
- Test(U
_{i}, s_{i}) - If the corresponding session key is defined (i. e., ${\mathsf{acc}}_{i}^{{s}_{i}}=\mathsf{true}$ and ${\mathsf{sk}}_{i}^{{s}_{i}}\ne $ null) and instance ${\Pi}_{i}^{{s}_{i}}$ is fresh (see Definition 4), $\mathcal{A}$ can execute this oracle query at any time when being activated. Then, if $b=0$ the session key ${\mathsf{sk}}_{i}^{{s}_{i}}$ is returned, while if $b=1$ a uniformly chosen random session key is returned. An arbitrary number of $\mathsf{Test}$ queries is allowed for the adversary $\mathcal{A}$, but once the $\mathsf{Test}$ oracle returned a value for an instance ${\Pi}_{i}^{{s}_{i}}$, the same value will be returned for all instances partnered with ${\Pi}_{i}^{{s}_{i}}$ (see Definition 3).
- Corrupt(U
_{i}) - This oracle models forward secrecy, as this query will output the secret signing key of user ${U}_{i}$.

#### 2.2. Goals of a Key Establishment Protocol: Correctness, Integrity, and Security

**Definition**

**1**

**.**A group key establishment protocol $\mathsf{P}$ is correct , if in the presence of a passive adversary $\mathcal{A}$ the following holds: for all $i,j$ with both ${\mathsf{sid}}_{i}^{{s}_{i}}={\mathsf{sid}}_{j}^{{s}_{j}}$ and ${\mathsf{acc}}_{i}^{{s}_{i}}={\mathsf{acc}}_{j}^{{s}_{j}}=\mathsf{true}$, we have ${\mathsf{sk}}_{i}^{{s}_{i}}={\mathsf{sk}}_{j}^{{s}_{j}}\ne $ null and ${\mathsf{pid}}_{i}^{{s}_{i}}={\mathsf{pid}}_{j}^{{s}_{j}}$.

**Definition**

**2**

**.**A correct group key establishment protocol fulfills key integrity, if all instances of users that have accepted with the same session identifier ${\mathsf{sid}}_{j}^{{s}_{j}}$ hold with overwhelming probability identical session keys ${\mathsf{sk}}_{j}^{{s}_{j}}$ and identical partner identifiers ${\mathsf{pid}}_{j}^{{s}_{j}}$.

**Definition**

**3**

**.**Instances ${\Pi}_{i}^{{s}_{i}}$ and ${\Pi}_{j}^{{s}_{j}}$ are partnered if ${\mathsf{pid}}_{i}^{{s}_{i}}={\mathsf{pid}}_{j}^{{s}_{j}}$, ${\mathsf{sid}}_{i}^{{s}_{i}}={\mathsf{sid}}_{j}^{{s}_{j}}$, and ${\mathsf{acc}}_{i}^{{s}_{i}}={\mathsf{acc}}_{j}^{{s}_{j}}=\mathsf{true}$.

**Definition**

**4**

**.**An instance ${\Pi}_{i}^{{s}_{i}}$ is called fresh provided that none of the following condition holds:

- For some${U}_{j}\in {\mathsf{pid}}_{i}^{{s}_{i}}$a query$\mathsf{Corrupt}\left({U}_{j}\right)$was executed before a query of the form$\mathsf{Send}({U}_{k},{s}_{k},\ast )$has taken place where${U}_{k}\in {\mathsf{pid}}_{i}^{{s}_{i}}$.
- The adversary queried$\mathsf{Reveal}({U}_{j},{s}_{j})$with${\Pi}_{i}^{{s}_{i}}$and${\Pi}_{j}^{{s}_{j}}$being partnered.

**Definition**

**5.**

## 3. Building on a Group-Theoretic Assumption

#### 3.1. A Two-Party Solution

- A
**non-interactive non-malleable commitment scheme $\mathcal{C}$**, satisfying the following requirements:- –
- It is perfectly binding in the sense that every commitment can be decommitted to at most one value.
- –
- It is non-malleable for multiple commitments. This means that an adversary who knows commitments to a polynomial sized set of values $\nu $, will not be able to output commitments to a polynomial sized set of values $\beta $ related to $\nu $ in a meaningful way. It is well-known that in the CRS model such a commitment scheme can be implemented by means of any IND-CCA2 secure public key encryption scheme, for instance.

- A
**family of universal hash functions $\mathcal{U}\mathcal{H}$**mapping triples consisting of two elements from G and a ${\mathsf{pid}}_{i}^{{s}_{i}}$-value onto a superpolynomial sized set ${\{0,1\}}^{L}$. A universal hash function $\mathrm{UH}$ will be selected by the CRS from this family. - A
**collision-resistant pseudorandom function family $\mathcal{F}={\left\{{F}^{\ell}\right\}}_{\ell \in \mathbb{N}}$**(see Katz and Shin [28]). We assume ${F}^{\ell}={\left\{{F}_{\eta}^{\ell}\right\}}_{\eta \in {\{0,1\}}^{L}}$ to be indexed by ${\{0,1\}}^{L}$ and further denote by ${v}_{0}={v}_{0}\left(\ell \right)$ a publicly known value such that no ppt adversary can find two different indices $\lambda \ne {\lambda}^{\prime}\in {\{0,1\}}^{L}$ such that ${F}_{\lambda}\left({v}_{0}\right)={F}_{{\lambda}^{\prime}}\left({v}_{0}\right)$. We further use another public value ${v}_{1},$ fulfilling the same requirement as ${v}_{0}$ for deriving the session key (this can also be included in the CRS—see [28] for more details).

- $\mathsf{DomPar}$, the domain parameter generation algorithm, is a (stateless) ppt algorithm that, upon input of the security parameter ${1}^{\ell}$, outputs a finite sequence S of elements in G. The subgroup of G spanned by S, $\langle S\rangle $, will be publicly known. Note that, for the special case of applying our framework to a DDH-assumption, S specifies a public generator of a cyclic group.
- $\mathsf{SamAut}$, the automorphism group sampling algorithm, is a (stateless) ppt algorithm that, upon input of the security parameter ${1}^{\ell}$ and a sequence S output by $\mathsf{DomPar}$, returns a description of an automorphism $\varphi $ on the subgroup $\langle S\rangle $, so that both $\varphi $ and ${\varphi}^{-1}$ can be efficiently evaluated. For example, for a cyclic group, $\varphi $ could be given as an exponent, or for an inner automorphism the conjugating group element could be specified.
- $\mathsf{SamSub}$, the subgroup sampling algorithm, is a (stateless) ppt that, upon input of the security parameter ${1}^{\ell}$ and a sequence S output by $\mathsf{DomPar}$, returns a word $x\left(S\right)$ representing an element $x\in \langle S\rangle $. Intuitively, $\mathsf{SamSub}$ chooses a random $x\in \langle S\rangle $, so that it is hard to recognize x if we know elements of x’s orbit under $\mathrm{Aut}(\langle S\rangle )$. Thus, our protocol requires an explicit representation of x in terms of the generators S.

**Definition**

**6**

**.**Suppose that we have fixed a quadruple $(G,\mathsf{DomPar},\mathsf{SamAut},\mathsf{SamSub})$. Then the decision automorphism application (DAA) assumption states that for all ppt algorithms $\mathcal{A}$ the advantage function ${\mathsf{Adv}}_{\mathcal{A}}^{\mathsf{DAA}}={\mathsf{Adv}}_{\mathcal{A}}^{\mathsf{DAA}}\left(\ell \right):=$

**Example**

**1**

**.**Let G be a finite cyclic group and $S:=\langle g\rangle $ a prime order subgroup with generator g of order q. If we let $\mathsf{SubSam}$ choose uniformly at random an exponent $x\in \{1,\dots ,q-1\}$ and $\mathsf{SamAut}$ uniformly at a random exponent $\varphi \in \{1,\dots ,q-1\}$, then the DAA problem just described can be recognized as polynomial-time equivalent to a decision Diffie–Hellman (DDH) problem:

**“DDH solution**⇒**DAA solution”:**- When facing, the DAA problem, we obtain as input a tuple$(g,{g}^{y},{({g}^{{\varphi}_{i}},{g}^{x{\varphi}_{i}})}_{i=1,2})$where either$y=x$, or y has been chosen uniformly at random from$\{1,\dots ,q-1\}$—independently of x and the${\varphi}_{i}$s. Given a DDH oracle, we just query it with$(g,{g}^{y},{g}^{{\varphi}_{1}},{g}^{x{\varphi}_{1}})$to see with non-negligible success probability which is the case.
**“DDH solution**⇒**DAA solution”:**- When facing the DDH problem, we obtain as input a tuple$(g,{g}^{{\varphi}_{1}},{g}^{x},{g}^{y})$, where either$y={\varphi}_{1}x\phantom{\rule{0.277778em}{0ex}}\mathrm{mod}\phantom{\rule{0.277778em}{0ex}}q$, or y has been chosen uniformly at random from$\{1,\dots ,q-1\}$—independently of x and${\varphi}_{1}$. Choosing another random${\varphi}_{2}\in \{1,\dots ,q-1\}$, we can compute the input$$({g}^{{\varphi}_{1}},{g}^{y},((\underset{=({g}^{{\varphi}_{1}}{)}^{{\varphi}_{1}^{-1}}}{\underbrace{g}},\underset{={\left({g}^{{\varphi}_{1}x}\right)}^{{\varphi}_{1}^{-1}}}{\underbrace{{g}^{x}}}),(\underset{={\left({g}^{{\varphi}_{1}}\right)}^{{\varphi}_{1}^{-1}{\varphi}_{2}}}{\underbrace{{g}^{{\varphi}_{2}}}},\underset{={\left({g}^{{\varphi}_{1}x}\right)}^{{\varphi}_{1}^{-1}{\varphi}_{2}}}{\underbrace{{\left({g}^{x}\right)}^{{\varphi}_{2}}}}))$$

**Proposition**

**1**

**.**Assume that for each ppt time algorithm $\mathcal{A}$, its advantage ${\mathsf{Adv}}_{\mathcal{A}}^{\mathsf{Sig}}$ of achieving an existential forgery under the adaptive chosen-message attack for the underlying signature scheme, and ${\mathsf{Adv}}_{\mathcal{A}}^{\mathsf{DAA}},$ its advantage of solving DAA, can be bounded by a negligible function (in ℓ). Then the protocol in Figure 1 is a correct and secure two-party key establishment protocol fulfilling key integrity.

#### 3.2. Security Analysis for the Two-Party Case: Proof of Proposition 1

**Game 0.**All oracles are simulated as defined in the model. Thus, $\mathsf{Adv}(\mathcal{A},{G}_{0})$ is exactly ${\mathsf{Adv}}_{\mathcal{A}}$ and $\mathsf{Succ}(\mathcal{A},{G}_{0})$ is the probability of violating the security of our key exchange protocol.

**Game 1.**In this game, the simulator keeps a list with entries $(i,M,{\sigma}_{M})$ for every message M and corresponding signature ${\sigma}_{M}$ he has produced and returned to the adversary $\mathcal{A}$ in a Round 2 message following a $\mathsf{Send}$ query.

**Lemma**

**1.**

**Proof.**

**Game 2.**Now the simulation of the $\mathsf{Test}$ oracle is modified, so that, on input of a fresh instance, it will always output an element selected uniformly at random in the key space. Thus, $\mathsf{Adv}(\mathcal{A},{G}_{2})=0.$

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 2.**The protocol compiler from [22].

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**MDPI and ACS Style**

Bohli, J.-M.; González Vasco, M.I.; Steinwandt, R.
Building Group Key Establishment on Group Theory: A Modular Approach. *Symmetry* **2020**, *12*, 197.
https://doi.org/10.3390/sym12020197

**AMA Style**

Bohli J-M, González Vasco MI, Steinwandt R.
Building Group Key Establishment on Group Theory: A Modular Approach. *Symmetry*. 2020; 12(2):197.
https://doi.org/10.3390/sym12020197

**Chicago/Turabian Style**

Bohli, Jens-Matthias, María I. González Vasco, and Rainer Steinwandt.
2020. "Building Group Key Establishment on Group Theory: A Modular Approach" *Symmetry* 12, no. 2: 197.
https://doi.org/10.3390/sym12020197