On homomorphic encryption using abelian groups: Classical security analysis

TL;DR

This paper analyzes the classical and quantum security of a homomorphic encryption scheme based on abelian groups, focusing on the hardness assumptions and potential vulnerabilities in different group settings.

Contribution

It provides a detailed security analysis of the scheme using abelian and solvable groups, clarifying its classical and quantum security properties.

Findings

Classical security of the scheme with abelian groups is examined.

Quantum attacks on instantiations with solvable groups are analyzed.

The scheme's security depends on the hardness of the LHN problem.

Abstract

In [15], Leonardi and Ruiz-Lopez propose an additively homomorphic public key encryption scheme whose security is expected to depend on the hardness of the learning homomorphism with noise problem (LHN). Choosing parameters for their primitive requires choosing three groups $G$, $H$, and $K$. In their paper, Leonardi and Ruiz-Lopez claim that, when $G$, $H$, and $K$ are abelian, then their public key cryptosystem is not quantum secure. In this paper, we study security for finite abelian groups $G$, $H$, and $K$ in the classical case. Moreover, we study quantum attacks on instantiations with solvable groups.

Equations

$$\{(g_1,\phi (g_1)h_1),\dots ,(g_m,\phi (g_m)h_m),\tau \}.$$

$$(g,h^{\prime })=(g_{w_1}\cdots g_{w_{\ell }},\phi (g_{w_1})h_{w_1}\cdots \phi (g_{w_{\ell }})h_{w_{\ell }}).$$

$$(g,h^{\prime }{\tau }^{\beta })=(g_{w_1}\cdots g_{w_{\ell }},\phi (g_{w_1})h_{w_1}\cdots \phi (g_{w_{\ell }})h_{w_{\ell }}{\tau }^{\beta })$$

$$(\tilde{g},{\tilde{h}}^{\prime }{\tau }^{\tilde{\beta }})=(g_{{\tilde{w}}_1}\cdots g_{{\tilde{w}}_{\tilde{\ell }}},\phi (g_{{\tilde{w}}_1})h_{{\tilde{w}}_1}\cdots \phi (g_{{\tilde{w}}_{\tilde{\ell }}})h_{{\tilde{w}}_{\tilde{\ell }}}{\tau }^{\tilde{\beta }})$$

$$\phi (g_{w_1})h_{w_1}\cdots \phi (g_{w_{\ell }})h_{w_{\ell }}{\tau }^{\beta }\phi (g_{{\tilde{w}}_1})h_{{\tilde{w}}_1}\cdots \phi (g_{{\tilde{w}}_{\tilde{\ell }}})h_{{\tilde{w}}_{\tilde{\ell }}}{\tau }^{\tilde{\beta }}$$

$$=$$

$$(g,h^{\prime }{\tau }^{\beta })(\tilde{g},{\tilde{h}}^{\prime }{\tau }^{\tilde{\beta }})=(g\tilde{g},h^{\prime }{\tau }^{\beta }\widetilde{h^{\prime }}{\tau }^{\tilde{\beta }})=(g\tilde{g},h^{\prime }\widetilde{h^{\prime }}{\tau }^{\beta +\tilde{\beta }}).$$

$$E:\{0,1\}\to G\times H,$$

$$\text{polylog}(|G|)+\text{polylog}(|H|)+O\left(\frac{{\log }_2(|G_2|)}{m}{2}^{m/2}\right),$$

$$(g,h)=(g_{w_1}\cdots g_{w_{\ell }},\phi (g_{w_1})\cdots \phi (g_{w_{\ell }})h_{w_1}\cdots h_{w_{\ell }}\cdot {\tau }^{\beta })=(g,\phi (g)h_{w_1}\cdots h_{w_{\ell }}\cdot {\tau }^{\beta }).$$

$$(g,h)=(g_1^{r_1}\cdots g_m^{r_m},\phi (g_1^{r_1})\cdots \phi (g_m^{r_m})h_1^{r_1}\cdots h_m^{r_m}\cdot {\tau }^{\beta })=(g,\phi (g)h_1^{r_1}\cdots h_m^{r_m}\cdot {\tau }^{\beta }).$$

$$(g,h)=(g_{w_1}\cdots g_{w_{\ell }},\phi (g_{w_1})\cdots \phi (g_{w_{\ell }}){\tau }^{\beta })=(g,\phi (g){\tau }^{\beta }).$$

$$\mathrm{Hom}(H,K{)}^{-}=\{\psi \in \mathrm{Hom}(H,K):\mathrm{Z}(H)⊈\ker (\psi )\}.$$

$$(g,h^{\prime })=(g_{w_1}\cdots g_{w_{\ell }},\phi (g_{w_1})h_{w_1}\cdots \phi (g_{w_{\ell }})h_{w_{\ell }})$$

$$|G|=2^{n_G}{q}_G,|H|=2^{n_H}{q}_H,$$

$$(g^q,h^q)=((g_1^q{)}^{r_1}\cdots (g_m^q{)}^{r_m},\phi (g^q)\cdot (h_1^q{)}^{r_1}\cdots (h_m^q{)}^{r_m}\cdot ({\tau }^q{)}^{\beta })=(g^q,(h^{\prime }{)}^q{\tau }^{\beta })\in G_2\times H_2.$$

$$({\alpha }_1,\dots ,{\alpha }_m)\in \mathbb{Z}/|⟨{\gamma }_1⟩|\mathbb{Z}\times \dots \times \mathbb{Z}/|⟨{\gamma }_m⟩|\mathbb{Z}$$

$$x={\gamma }_1^{{\alpha }_1}\cdots {\gamma }_m^{{\alpha }_m}.$$

$$g=g_1^{r_1}\cdots g_m^{r_m}\text{ and }h=\phi (g)h_1^{r_1}\cdots h_m^{r_m}\cdot {\tau }^{\beta }.$$

$${\ell }_i^{|⟨g_i⟩|}=(\phi (g_i)h_i{)}^{|⟨g_i⟩|}=h_i^{|⟨g_i⟩|}.$$

$$x=\prod _{i=1}^m({\ell }_i^{s_i}{)}^{-1}\cdot h=\prod _{i=1}^m{\ell }_i^{r_i-s_i}\cdot {\tau }^{\beta }=\prod _{i=1}^m({\ell }_i^{|⟨g_i⟩|}{)}^{t_i}\cdot {\tau }^{\beta }=:h_M\cdot {\tau }^{\beta },$$

$$O\left(\frac{\log \log |G_2|}{\log \log \log |G_2|}\log |G_2|+\frac{{\log }_2|G_2|}{m}{2}^{m/2}\right)$$

$${\overline{g_1}}^q,\dots ,{\overline{g_m}}^q\text{ and }{h}_1^q,\dots ,h_m^q$$

$${\overline{G}}_2=⟨{\overline{g_1}}^q⟩\oplus \dots \oplus ⟨{\overline{g_m}}^q⟩.$$

$$G=⟨g_1⟩\oplus \dots \oplus ⟨g_m⟩,$$

$$O\left(\frac{\log (e+1)}{\log \log (e+2)}\log |G|+\frac{{\log }_p|G|}{m}{p}^{m/2}\right)$$

$$\frac{(2^{\lambda }-1)(2^{\lambda }-2)(2^{\lambda }-2^2)\cdots (2^{\lambda }-2^{m-1})}{2^{m\lambda }},$$

$$(a_1,\dots ,a_{m+1})⟼g^{a_{m+1}}\cdot g_1^{a_1}\cdots g_m^{a_m}.$$

$$g_1^{r_1+s_1{s}_{m+1}}\cdots g_m^{r_m+s_m{s}_{m+1}}=1.$$

$$r_i\equiv -s_i{s}_{m+1}\mathrm{mod}|⟨g_i⟩|.$$

Full text

00footnotetext: Author list in alphabetical order; see https://www.ams.org/profession/leaders/culture/CultureStatement04.pdf.

Date of this document: 1st March 2024.

Acknowledgements.

We warmly thank the organizers of Women in Numbers Europe 4 for putting together this team. We also wish to thank Chris Leonardi and Andrew Sutherland for helpful conversations around the contents of [20] and [30] respectively. In addition, we thank the three anonymous referees for their comments, which helped improve the exposition of this paper.

The first author was supported by the European Union’s H2020 Programme under grant agreement number ERC-669891. The second author was supported by the Austrian Science Fund, Project P34808. The fifth author was supported by the Magnus Ehrnrooth grant 336005, the Academy of Finland project grant (351271, PI Camilla Hollanti) and by the MATINE project grant, Ministry of Defence Finland (2500M-0147, PI Camilla Hollanti). The last author was partially supported by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) – Project-ID 286237555 – TRR 195 and by the Italian program Rita Levi Montalcini for young researchers, Edition 2020.

Introduction

Homomorphic encryption is a method of encrypting plaintext that allows users to compute directly with the ciphertext. This has many interesting applications, including being able to engage in cloud computing without giving up your data to the owner of the cloud. Scientifically, the premise is easy to describe: Suppose that the plaintext and the ciphertext space both have a ring structure, and that we encrypt plaintext via a map between these spaces. If this map is a ring homomorphism, then this describes a fully homomorphic encryption scheme. Creating such a ring homomorphism that describes secure encryption (requiring, for example, that such a map should be efficiently computable and hard to invert) is, however, much harder than describing its properties.

The closest the scientific community has come to constructing an example of fully homomorphic encryption is using maps based on (variants of) the Learning With Errors (LWE) problem from lattice-based cryptography [25, 22, 14, 9, 6]. However, although practical fully homomorphic encryption can be achieved from LWE-based constructions [7], all known such constructions are not naturally fully homomorphic: Decrypting a message that was encrypted using LWE relies on the ‘error’ that was used in the encryption being small, and adding and especially multiplying encrypted messages together causes the error to grow. Once the error is too large, the data can no longer be decrypted, so methods such as bootstrapping need to be employed to correct this growth (see e.g. [9]). These methods, although somewhat ad hoc, may still be the best solution for providing practical fully homomorphic encryption in the near future.

In this paper, we explore an alternative approach for homomorphic encryption, introduced by Leonardi and Ruiz-Lopez in [20]. The interested reader may find more on homomorphic encryption in [13]. The construction of Leonardi and Ruiz-Lopez, relies on something similar to the Learning Homomorphisms with Noise (LHN) problem introduced by Baumslag, Fazio, Nicolosi, Shpilrain, and Skeith in [2]: Roughly speaking, this is the problem of recovering a group homomorphism from the knowledge of the images of certain elements multiplied by noise. The focus of [20] is on the difficulty of constructing post-quantum secure instantiations of their primitive, but we believe the construction is interesting even in a classical setting. A big advantage of the Leonardi–Ruiz-Lopez approach over a LWE approach is that the noise, which plays the role of the errors in LWE-based homomorphic encryption, does not grow with repeated computation. As such, there is no limitation on the number of additions that can be computed on encrypted data. However, it is not clear if this construction can be extended to multiplicative homomorphic encryption. In general extending additive homomorphic encryption to fully homomomorphic encryption is a hard problem: LWE-based homomorphic encryption is the only known construction. As such, the Leonardi–Ruiz-Lopez approach is akin in some sense to the Benaloh [3] or Paillier [24] cryptosystems. In the nonabelian setting, Leonardi and Ruiz-Lopez’s construction has some hopes of being post-quantum secure unlike the Benaloh or Paillier constructions, but as they explored already in their work this is nontrivial to instantiate, and our work only strengthens this claim as we show that even solvable groups may admit quantum attacks.

Our main contributions address finite groups and include:

Reducing the security of an instantiation of Leonardi and Ruiz-Lopez’s public key homomorphic encryption scheme in which one of the groups is abelian (in particular, the group in which the noise lives) to the discrete logarithm problem in 2-groups (under certain plausible assumptions); this gives a polynomial-time classical attack if the 2-part of the relevant group is cyclic, and a practical classical attack if it is a product of a small number of cyclic groups. See Theorem 1.1 and Section 3.4. 2. 2.

Highlighting an abelian group instantiation of Leonardi and Ruiz-Lopez’s homomorphic encryption scheme where there is no known practical classical attack, namely, the product of many cyclic 2-groups. This may be of interest to the community as a new example of unbounded additively homomorphic encryption. 3. 3.

Highlighting assumptions that need to be made in order to apply any discrete-logarithm derived attack, with a view to constructing (more) examples of groups on which there is no known practical attack on Leonardi and Ruiz-Lopez’s homomorphic encryption scheme. 4. 4.

A description of a quantum attack on an instantiation with solvable groups, under certain assumptions.

Note that all our contributions are studying the hardness of the LHN-PKE problem, given in Definition 1, not the general LHN problem, as this is the relevant underlying “hard problem" in Leonardi–Ruiz-Lopez encryption.

The layout of this paper is as follows: In Section 1, we recap the public key homomorphic encryption scheme proposed by Leonardi and Ruiz-Lopez in [20]. In Section 2, we discuss some simple instantiations: The abelian case and the noiseless case. In this section we also give some basic security requirements (including some recalled from [20]). In Section 3 we describe our reduction from the abelian group instantiation of the Leonardi Ruiz-Lopez primitive to the (extended) discrete logarithm problem, under certain assumptions. In Section 4, we describe some ways of instantiating the primitive with nonabelian groups to which our attack on abelian groups would also apply. In Section 5, we describe our quantum attack on instantiations with solvable groups, under certain assumptions on how such groups would be represented. In Section 6, we outline our plans for future work.

1 Leonardi–Ruiz-Lopez encryption

In this section, we describe the public key additive homomorphic encryption of Leonardi and Ruiz-Lopez [20, Sec. 5.2]; we will refer to this throughout this work as Leonardi–Ruiz-Lopez encryption. Note that this paper shows that the following encryption scheme is not secure (even classically) in most natural abelian instantiations, see Theorem 1.1, and we make no claims on the security of the nonabelian case except for some warnings, see Section 3.4, Section 4, and Section 5. In the course of this work we also discovered the necessity for some extra points in the basic Leonardi–Ruiz-Lopez encryption in order to avoid security problems even in the most general case; our additions are marked with $\,{}^{\textup{new}}$ (both when they are completely new and when they are clarifications). These points are discussed in later subsections.

Fix three finitely generated groups $G,H,K$ and probability distributions $\xi$ on $G$ and $\chi$ on $H$. This data should be chosen in such a way that operations can be performed efficiently in the groups and we can sample from both distributions efficiently. A natural choice could be, for instance, to take $G,H,K$ finite and $\xi,\chi$ to be uniform distributions. The groups $G,H,K$ and the distributions $\xi,\chi$ are public. These groups must also satisfy security assumptions $(\mathrm{S}1)$${}^{\text{new}}$ and $(\mathrm{S}3)$ discussed below. In the following sections we will mostly work with finite groups and we will always make it clear when this is the case.

For the key generation, Alice

• •

chooses efficiently computable secret homomorphisms $\varphi\colon G\to H$ and $\psi\colon H\to K$ such that she can efficiently sample from $\ker(\psi)$ and such that the center $\mathrm{Z}(H)$ of $H$ is not contained in $\ker(\psi)$;

• •

chooses a positive integer $m$;

• •

samples elements $g_{1},\dots,g_{m}\in G$ via $\xi$ and secret elements $h_{1},\dots,h_{m}\in\ker(\psi)$ via $\chi$;

• •

chooses an element $\tau\in\mathrm{Z}(H)\backslash\ker(\psi)$ of order $2$111The element $\tau$ having order $2$ ensures that the encryption is additive as explained in Section 1.1; the construction is also viable without this requirement. that satisfies $(\mathrm{S}4)$${}^{\text{new}}$ described below.

Alice computes the public key as the set

[TABLE]

Note that, whereas the elements $g_{1},\dots,g_{m}$ are public, both $\varphi$ and $h_{1},\dots,h_{m}$ are private (as are also $\psi$ and $\ker(\psi)$).

For encrypting a one-bit message $\beta\in\{0,1\}$, Bob chooses a natural number $\ell$, then samples a word $w=w_{1}\cdots w_{\ell}$222The sample space and method must satisfy security assumption $(\mathrm{S}2)$${}^{\text{new}}$ described below. over the indices $\{1,\dots,m\}$ and using Alice’s public key, he computes

[TABLE]

He then sends $(g,h)=(g,h^{\prime}\tau^{\beta})$ to Alice.

For decrypting $(g,h)$, Alice computes $\nu=\psi(\varphi(g))^{-1}\cdot\psi(h)\in K$ and deduces that the message $\beta$ equals [math] if $\nu$ equals $1_{K}$ (and else, $\beta$ equals $1$).

To see that the decryption indeed produces the correct message $\beta$, recall that $h_{1},\dots,h_{m}$ are contained in $\ker(\psi)$. Hence $\nu=\psi(\tau)^{\beta}$ and, since $\tau$ is not contained in $\ker(\psi)$, the element $\nu$ equals $1$ if and only if $\beta$ equals [math].

For the convenience of the reader, we give a schematic summary of the data described above:

[TABLE]

1.1 Homomorphic properties

The primary selling point of Leonardi–Ruiz-Lopez encryption is that it is unbounded additive homomorphic. We say that an encryption function $E$ from plaintext to ciphertext space is additive if the plaintext space admits an additive operator $+$ and, given encryptions $E(\beta)$ and $E(\tilde{\beta})$ of messages $\beta$ and $\tilde{\beta}$ respectively, one can compute a valid encryption $E(\beta+\tilde{\beta})$ of $\beta+\tilde{\beta}$ (without the knowledge of the plaintext $\beta+\tilde{\beta}$). We say that $E$ is unbounded additive homomorphic if such additions can be performed an unbounded number of times without introducing systematic decryption failures.333 The number of additions is bounded in, for example, LWE-based homomorphic encryption, where the error grows too large.

As seen above, $\tau$ having order two is not necessary for successful decryption; this property is needed to make the encryption additive, as we now recall from [20].

Write the encryptions of $\beta$ and $\tilde{\beta}$ sampled from $\{0,1\}$ as

[TABLE]

and

[TABLE]

respectively. Then, as $\tau$ is central in $H$ and has order 2, we can construct a valid encryption of $\beta+\tilde{\beta}$ via the observation that

[TABLE]

this encryption is given by

[TABLE]

Remark 1

Note that, for Leonardi–Ruiz-Lopez encryption to be fully homomorphic, it would also need to be multiplicative: That is, at the very least, given valid encryptions of $E(\beta)$ and $E(\tilde{\beta})$, we should be able to deduce a valid encryption of $\beta\tilde{\beta}$. Since we only have one group operation on the cipher text space $G\times H$, we cannot expect to find a general strategy that provides fully homomorphic encryption for arbitrary abstract groups $G$, $H$. Our encryption function from plaintext to ciphertext space is

[TABLE]

where the domain can be naturally endowed with a ring structure using addition and multiplication mod 2, but there is no natural way of adding extra data on $G\times H$ that would allow us to deduce a valid encryption of $\beta\tilde{\beta}$. We stress that in the case of LWE-based homomorphic encryption, both plaintext and ciphertext spaces come equipped with a ring structure, so the equivalent of our function $E$ is generally taken to be a ring homomorphism.

The map $E$ being a ring homomorphism is, however, not always strictly necessary to deduce a valid encryption of $\beta\tilde{\beta}$. If in future work we were to succeed in deducing a valid encryption of $\beta\tilde{\beta}$, we expect that this will only apply to a specific instantiation of Leonardi–Ruiz-Lopez encryption, not one for abstract groups, where the groups provide more structure that can be exploited.

1.2 Remarks on Leonardi–Ruiz-Lopez encryption

Some remarks on the construction above:

1. $(\mathrm{R}1)$

The underlying hard problem of this encryption scheme is described as the LHN-PKE problem, so named as it is based on the Learning Homomorphisms with Noise problem (LHN) but is adapted to this Public Key Encryption scheme (PKE).

Definition 1

Let $G,H,K,\xi,$ and $\chi$ be as above. We define the LHN-PKE problem for $G,H,K,\xi,$ and $\chi$ to be: Given $G,H,K,\xi,$ and $\chi$, for any

• •

$\varphi$ sampled uniformly at random from $\operatorname{Hom}(G,H)$,

• •

$\psi$ sampled uniformly at random from the elements of $\operatorname{Hom}(H,K)$ whose kernel does not contain $\mathrm{Z}(H)$,

• •

$g_{1},\ldots,g_{m}$ sampled from $G$ using $\xi$,

• •

$h_{1},\ldots,h_{m}$ sampled from $\ker(\psi)$ using $\chi$,

• •

$\tau$ sampled from the order $2$ elements of $\mathrm{Z}(H)\setminus\ker(\psi)$ using $\chi$,

• •

$\beta$ sampled uniformly at random from $\{0,1\}$,

• •

small $\ell$ and word $w=w_{1}\cdots w_{\ell}$ sampled uniformly at random from $\{1,\cdots,m\}^{\ell}$,

recover $\beta$ from the following information:

• •

$(g_{1},\varphi(g_{1})h_{1}),\ldots,(g_{m},\varphi(g_{m})h_{m})$,

• •

$\tau$,

• •

$g=g_{w_{1}}\cdots g_{w_{\ell}}$,

• •

$h=h^{\prime}\tau^{\beta}=\varphi(g_{w_{1}})h_{w_{1}}\cdots\varphi(g_{w_{\ell}})h_{w_{\ell}}\tau^{\beta}$. 2. $(\mathrm{R}2)$

In order for the encryption and decryption to work, the assumptions that $\tau$ is central or of order $2$ are not necessary. The reason we work under these assumptions is, as explained in Section 1.1, that in this case, the cryptosystem is unbounded additive homomorphic. 3. $(\mathrm{R}3)$

Once Alice has fixed the elements $g_{1},\dots,g_{m}$ and determined the public key, all computations inside $G$ actually take place inside the subgroup $\langle g_{1},\dots,g_{m}\rangle$ that is generated by $g_{1},\ldots,g_{m}$. So for cryptanalysis, we may and will assume that $G$ is generated by $g_{1},\dots,g_{m}$, i.e. that $G=\langle g_{1},\dots,g_{m}\rangle$. 4. $(\mathrm{R}4)$

The work of [20] was inspired by [2], which introduces the Learning Homomorphisms with Noise problem in order to construct a symmetric primitive. However, the noise accumulates in the construction of [2] in a manner akin to the error growth in LWE constructions. Leonardi and Ruiz-Lopez also introduce a symmetric primitive in [20], but we focus on the PKE construction in this work. 5. $(\mathrm{R}5)$

The noise consists of the elements $h_{1},\dots,h_{m}$ that are mixed into the product $h^{\prime}$ in the second component $h$ of the ciphertext. These elements are chosen to be in the kernel of $\psi$ and therefore get erased during decryption. ‘Being contained in the kernel’ of $\psi$ can thus be thought of as an equivalent of ‘the error being small’ in the LWE-based encryption of [25] or ‘the noise being small’ in LHN-based encryption of [2]. The strength of Leonardi–Ruiz-Lopez encryption is that the noise does not accumulate and will not lead to systematic decryption errors, since in the decryption process, we can erase the noise neatly by applying $\psi$.

1.3 Cryptanalysis of the LHN-PKE problem in the abelian case

In this paper we show that, if $G$ and $H$ are finite and abelian, under some reasonable assumptions that we introduce here, the LHN-PKE problem for $G$ and $H$ can be reduced to the extended discrete logarithm problem (cf. Definition 2) and membership problem (cf. Definition 3) in some specific abelian $2$-groups, which leads to a polynomial-time attack in case the 2-part of $G$ is cyclic; the general asymptotic complexity is given in our main theorem below. We also attack the case with finite but nonabelian $G$ and abelian $H$, again under some reasonable assumptions, in Section 3.4.

In each section that follows, we explicitly mention under which assumptions from the following list we are working. For a group $\Gamma$ and $\gamma_{1},\ldots,\gamma_{m}\in\Gamma$, we are interested in the following properties:

1. $(\mathrm{A}1)$

$\Gamma$ is abelian; 2. $(\mathrm{A}2)$

the largest positive odd factor of the order $|\Gamma|$ of $\Gamma$ is known (or easily computable); 3. $(\mathrm{A}3)$

$\gamma_{1},\ldots,\gamma_{m}\in\Gamma$ are such that $\Gamma=\langle\gamma_{1}\rangle\oplus\ldots\oplus\langle\gamma_{m}\rangle$; 4. $(\mathrm{A}4)$

The orders $|\gamma_{1}|,\ldots,|\gamma_{m}|$ of $\gamma_{1},\ldots,\gamma_{m}$ are known (or easily computable).

We discuss these assumptions – why and when they are reasonable and/or needed – in Sections 3.4.3 (for $(\mathrm{A}2)$ and $(\mathrm{A}4)$) and 3.4.4 (for $(\mathrm{A}3)$). We can now state the main theorem of this paper:

Theorem 1.1

*Let $G=\langle g_{1}\rangle\oplus\cdots\oplus\langle g_{m}\rangle$ be a finite group which satisfies $(\mathrm{A}1)$, $(\mathrm{A}2)$, and $(\mathrm{A}3)$.

Let $H$ be a finite group that satisfies $(\mathrm{A}1)$ and $(\mathrm{A}2)$ and for which we can efficiently compute a cyclic decomposition of its Sylow $2$-subgroup $H_{2}$.444Note that if we assumed $(\mathrm{A}3)$ for $H$ together with a choice of generators, computing such a decomposition for $H_{2}$ would be easy, as $H$ satisfies $(\mathrm{A}2)$; that is, this requirement for $H_{2}$ together with $(\mathrm{A}2)$ for $H$, is weaker than $(\mathrm{A}3)$ for $H$. Let $K$ be an arbitrary group. Let $\xi$ and $\chi$ be probability distributions on $G$ and $H$ respectively. Then there is a classical algorithm to solve LHN-PKE for $G,H,K,\xi,$ and $\chi$ with asymptotic complexity*

[TABLE]

where $G_{2}$ denotes the Sylow $2$-subgroup of $G$.

We will give the proof of this theorem in Section 3.3.

Note in particular that if $m=1$, that is if $G$ is cyclic, this attack algorithm is polynomial-time (in fact, as we will see, it is sufficient for $G_{2}$ to be cyclic). In Section 3.4, we discuss how to attack instances with nonabelian $G$, assuming that we can work with the abelianization $G/[G,G]$ of $G$. In Section 3.5.2, we also briefly discuss a polynomial-time quantum algorithm, complementing the existing literature for the torsion-free case [20]. In Section 4, we discuss instantiations in which $H$ is not abelian (i.e. does not satisfy $(\mathrm{A}1)$) but which reduce to the case in which $H$ is abelian. In Section 5, we discuss how one could develop a quantum algorithm for the solvable case.

2 Simple instantiations and security

In this section, we describe some simple instantiations of Leonardi–Ruiz-Lopez encryption. The abelian case is a central focus of this paper as it is much simpler to describe than the general case; the description below is for the reader who wishes only to understand the abelian case. We also describe the noiseless case in order to highlight the role that the noise plays in the encryption. Finally we discuss the requirements on the setup parameters of Leonardi–Ruiz-Lopez encryption in order to achieve security against some naive classical attacks, concluding this section with a list of properties that the groups must have in order to avoid the trivial classical attacks outlined in this section.

2.1 The abelian case

If $H$ is abelian, we can rewrite $(g,h)=(g_{w_{1}}\cdots g_{w_{\ell}},\varphi(g_{w_{1}})h_{w_{1}}\cdots\varphi(g_{w_{\ell}})h_{w_{\ell}}\cdot\tau^{\beta})$ as

[TABLE]

If both $G$ and $H$ are abelian, it makes sense to switch to the following notation: Instead of choosing indices $w_{1},\dots,w_{\ell}$, Bob just chooses non-negative integers $r_{1},\dots,r_{m}$ and encrypts $\beta$ to

[TABLE]

This system has been claimed to not be quantum secure in [20, Section 8.2], cf. also Section 3.5, while we discussed security in the classical sense in Section 1.3. Moreover, in Section 4 we describe instantiations of the LHN-PKE problem in which $G$ and $H$ are nonabelian but the security reduces to the case in which they are.

2.2 The noiseless case

Let us assume that $h_{1}=h_{2}=\dots=h_{m}=1$. Then the public key consists of all pairs $(g_{i},\varphi(g_{i}))$ together with $\tau$ and Bob would encrypt the message $\beta\in\{0,1\}$ to

[TABLE]

Let Eve be an attacker who is aware of the fact that Alice decided to work in a noiseless setting. Then Eve knows all $g_{i}$’s as well as their images $\varphi(g_{i})$ from the public key. If she can write $g$ as a product in $g_{1},\dots,g_{m}$, then she can compute $\varphi(g)$. Knowing $\tau$ from the public key, Eve can then decrypt $(g,h)=(g,\varphi(g)\tau^{\beta})$. Note that even if Eve did not use the same word $w_{1}\cdots w_{\ell}$ as Bob to write $g$ as a product in $g_{1},\dots,g_{m}$, she would nonetheless obtain the correct value of $\varphi(g)$.

Of course, finding such a word might still be a hard problem, even if $m=1$. For example, if $G$ is the multiplicative group of a finite field and $g_{1}$ is a generator, finding a word in $g_{1}$ defining $g$ is the same as solving the discrete logarithm problem, which is known to be hard for classical computers (though there are quantum algorithms to solve it, see [26, 27]). Alice, on the other hand, will probably have a closed form describing $\varphi$ that does not require to write elements as products in $g_{1},\dots,g_{m}$ when she applies $\varphi$ in the decryption process (and similarly for $\psi$). In the $m=1$ example above, she would choose $\varphi$ to take every element to a certain power.

For attacking the cryptosystem in the general case, a possible strategy is to construct attacks that reduce to the noiseless case. We will come back to such attacks in Section 2.3.

2.3 Security

In all that follows, let $\lambda$ be the security parameter.555 Typically, we want any computations undertaken by the user to have complexity that is polynomial in $\lambda$, and an attacker who attempts to decrypt by guessing any unknowns should only succeed with probability at most $2^{-\lambda}$.

2.3.1 Many homomorphisms

First of all, note that an attacker who can guess both $\varphi$ and $\psi$ can decrypt in the same way Alice does. Denote the set of all possible choices for $\psi$ by

[TABLE]

To avert brute force attacks, the groups $G,H,K$ should be chosen in such a way that $\operatorname{Hom}(G,H)$ and $\operatorname{Hom}(H,K)^{-}$ are of size at least $\Theta(2^{\lambda})$,666 We are using Bachmann-Landau notation for complexity, see for example Section 1.2.11.1 of [18].

and $\varphi$ and $\psi$ should be sampled uniformly at random from $\operatorname{Hom}(G,H)$ and $\operatorname{Hom}(H,K)^{-}$ respectively. This ensures that if an attacker guesses $\varphi$ she succeeds with probability $2^{-\lambda}$, and similarly for $\psi$.

2.3.2 Words

As we already saw in the noiseless case, there are links between the LHN-PKE problem and the ability of an attacker to write $g$ as an expression in the generators $g_{1},\ldots,g_{m}$. Assume for instance that Eve wants to decrypt the cyphertext $(g,h)$. She knows that $g$ is a product in $g_{1},\dots,g_{m}$ and recall that $g_{1},\dots,g_{m}$ are public. If she knows the exact expression of $g$ as a product in $g_{1},\dots,g_{m}$ that Bob used in the encryption process, then she can compute $h^{\prime}$ from the public key, erase it from $h=h^{\prime}\tau^{\beta}$, and recover the message $\beta$. It is important to note that other than in the noiseless case, it is in general not enough to find any expression of $g$ as a product in $g_{1},\dots,g_{m}$, because that product will in general not produce the correct term $h^{\prime}$ yielding to a different accumulation of the noise. That is, the attacker needs to recover the correct word $w$, not just any expression of $g$ in $g_{1}\ldots,g_{m}$. To see this, recall that $(g,h)=(g,h^{\prime}\tau^{\beta})$ with

[TABLE]

and assume that the attacker found some word $\tilde{w}_{1}\dots\tilde{w}_{\tilde{\ell}}$ such that $g=g_{\tilde{w}_{1}}\cdots g_{\tilde{w}_{\tilde{\ell}}}$. Using the public elements $(g_{1},\varphi(g_{1})h_{1}),\dots,(g_{m},\varphi(g_{m})h_{m})$, it is then possible to compute an element $\tilde{h}^{\prime}=\varphi(g_{\tilde{w}_{1}})h_{\tilde{w}_{1}}\cdots\varphi(g_{\tilde{w}_{\tilde{\ell}}})h_{\tilde{w}_{\tilde{\ell}}}$ but in general $\tilde{h}^{\prime}\neq h^{\prime}$ and the knowledge of $\tilde{h}^{\prime}$ together with $h$ does not help to compute $\tau^{\beta}$. In the cryptanalysis we carry out in Section 3.2 for finite abelian $G$, $H$, and $K$, we give a reduction of LHN-PKE to the extended discrete logarithm problem for finite $2$-groups; our reduction circumnavigates the issue of finding the correct word.

2.3.3 An attack on instances with few normal subgroups

The idea behind the following attack is to replace Alice’s secret $\psi:H\rightarrow K$ with some new $\bar{\psi}:H\rightarrow L$ erasing the noise without erasing $\tau^{\beta}$ (for instance $L$ could be a quotient of $H$, as described below). A similar attack was also described in Section 7.2 of [20]. Assume that Eve knows a normal subgroup $N$ of $H$ that contains all elements $h_{1},\dots,h_{m}$ but does not contain $\tau$. She can then define $\bar{\psi}\colon H\to H/N$ as the natural projection and by applying $\bar{\psi}$ to all second coordinates of the elements $(g_{i},\varphi(g_{i})h_{i})$ in the public key and to the second coordinate of the encrypted message $(g,h)$ she can switch to the noiseless case; cf. Section 2.2. We deduce in particular, that there should be at least $\Theta(2^{\lambda})$ normal subgroups in $H$, so that if an attacker guesses $\ker(\psi)$ she succeeds with probability $2^{-\lambda}$.

2.3.4 An attack on instances with weak normal subgroups

Now suppose that an attacker can find a normal subgroup $N$ of $H$ that contains $\varphi(g_{i})h_{i}$ for all $i=1,\dots,m$ but does not contain $\tau$ (note that these elements are all public). If efficiently computable, she can then directly apply the projection $H\to H/N$ to the second coordinate in the encrypted message $(g,h)$ and can deduce that $\beta$ equals zero if and only if she obtained the neutral element in $H/N$. To avoid such an attack, after the key generation process, Alice must check (assuming that she is capable of it) whether the normal closure of $\langle\varphi(g_{1})h_{1},\dots,\varphi(g_{m})h_{m}\rangle$ contains $\tau$ and, if it doesn’t, she should choose a different key. For cryptanalysis, we may thus assume that $H$ equals the normal closure of $\langle\varphi(g_{1})h_{1},\dots,\varphi(g_{m})h_{m}\rangle$ (as otherwise, we can just work in this smaller group). In particular, if $H$ is abelian, we may assume that $H=\varphi(G)\ker\psi$.

2.3.5 A summary of the discussed security assumptions

We conclude this section with a list of necessary properties for security deduced from the list of naive attacks above:

1. $(\mathrm{S}1)$

$\operatorname{Hom}(G,H)$ and $\operatorname{Hom}(H,K)^{-}$ are of size exponential in the security parameter; 2. $(\mathrm{S}2)$

finding the precise word $w$ used to express $g$ as a product in the $g_{i}$’s in the encryption phase has complexity that is exponential in the security parameter; 3. $(\mathrm{S}3)$

the number of normal subgroups in H is exponential in the security parameter; 4. $(\mathrm{S}4)$

the normal closure of $\langle\varphi(g_{1})h_{1},\dots,\varphi(g_{m})h_{m}\rangle$ contains $\tau$.

3 Cryptanalysis in the finite abelian case

In this section, we prove Theorem 1.1 via a series of reductions of the LHN-PKE problem. Recall that Theorem 1.1 assumes that $G$ and $H$ are finite and abelian (in addition to a few other assumptions). In Section 3.1, we will show that this can be taken one step further by giving a reduction from $G$ being finite and $H$ being finite and abelian to both $G$ and $H$ being finite and abelian. In Section 3.2.1, we will give a reduction from $G$ and $H$ being finite and abelian to $G$ and $H$ being finite abelian 2-groups (under some reasonable assumptions). In Section 3.2.2, we will give a reduction from LHN-PKE for finite abelian $G$ and $H$ to the eDLP problem777See Definition 2 for a formal definition. for $G$ and the generator membership problem888See Definition 3 for a formal definition. for $H$ (under some reasonable assumptions).

In Section 3.3 we give the proof of Theorem 1.1 and in Section 3.4 we discuss the genericity and limitations of the assumptions made in Section 1.3, as well as the impact of the reductions made.

Leonardi and Ruiz-Lopez [20] dismissed the abelian instantiation due to an argument that there should exist a polynomial-time quantum algorithm for the LHN-PKE problem; the reduction is more complex than is suggested in [20] but their statement is true as we show in Section 3.5. Nevertheless, the unbounded homomorphic property of the proposed cryptosystem is sufficiently powerful that a classically secure construction would also be of great interest to the cryptographic community.

3.1 A reduction of LHN-PKE with abelian noise to abelian groups

In this section we take $G$ to be finite and $H$ to be finite and abelian. We show that, when the natural projection $G\rightarrow\overline{G}=G/[G,G]$ from $G$ to its abelianization is efficiently computable, then, for cryptanalysis, one can consider $\overline{G}$ instead of $G$, with the distribution $\overline{\xi}$ that is induced on $\overline{G}$ by the projection.

Lemma 1

Assume $H$ is abelian. Then the LHN-PKE problem for $G,H,K,\xi,$ and $\chi$ is at most as hard as the LHN-PKE problem for $\overline{G}=G/[G,G],H,K,\overline{\xi}$, and $\chi$.

Proof

Since $H$ is abelian, the commutator subgroup $[G,G]$ of $G$ is contained in the kernel of $\varphi$. Define $\overline{G}=G/[G,G]$. Then $\varphi\colon G\to H$ induces a well-defined homomorphism $\overline{\varphi}\colon\overline{G}\to H$ and any ciphertext $(g,h)$ can be interpreted as the ciphertext $(\bar{g},h)$ in the cryptosystem given by $\overline{\varphi}\colon\overline{G}\to H$, $\psi\colon H\to K$ as before and public key given by $(\overline{g_{i}},\bar{\varphi}(\overline{g_{i}})h_{i})$ for $i=1,\dots,m$ together with the same $\tau$ as before. Indeed, if Bob used the word $w_{1}\cdots w_{\ell}$ to encrypt $(g,h)$, then the same word gives rise to the encryption $(\bar{g},h)$ in the new cryptosystem of the same message. ∎

3.2 Reductions of LHN-PKE in the finite abelian case

In Section 3.2 we discuss the LHN-PKE problem for $G$ and $H$ finite and abelian.

3.2.1 From abelian groups to $2$-groups

In this section, we take $G$ and $H$ to be finite groups for which $(\mathrm{A}1)$ and $(\mathrm{A}2)$ hold, i.e. we assume that both $G$ and $H$ are finite abelian groups and that the odd parts of $|G|$ and $|H|$ are known. We let $G_{2}$ and $H_{2}$ denote the unique Sylow $2$-subgroups of $G$ and $H$, respectively, and $\xi_{2}$ and $\chi_{2}$ denote the distributions $\xi$ and $\chi$ restricted to $G_{2}$ and $H_{2}$ respectively. In the new language, the elements $\tau$ and $\tau^{\beta}$ belong to $H_{2}$. In the following, we show that in the case of abelian groups, the LHN-PKE problem for the original data reduces to the LHN-PKE problem for the 2-parts $G_{2}$ and $H_{2}$ (with the induced data).

Lemma 2

Suppose that $G$ and $H$ are finite groups satisfying $(\mathrm{A}1)$ and $(\mathrm{A}2)$. Then the LHN-PKE problem for $G,H,K,\xi$, and $\chi$ is at most as hard as the LHN-PKE problem for $G_{2},H_{2},K,\xi_{2}$, and $\chi_{2}$.

Proof

Write the orders of $G$ and $H$ as

[TABLE]

where $q_{G}$ and $q_{H}$ are odd numbers. Let $q$ be a common multiple of $q_{G}$ and $q_{H}$ that can be computed efficiently. Then for any elements $g\in G$ and $h\in H$, the orders of $g^{q}$ and $h^{q}$ are both powers of $2$, i.e. $g^{q}\in G_{2}$ and $h^{q}\in H_{2}$. Moreover, $G$ and $H$ being abelian, the assignment $x\rightarrow x^{q}$ defines homomorphisms $G\to G_{2}$ and $H\to H_{2}$.

Equip $G_{2}$ and $H_{2}$ with the induced distributions $\xi_{2}$ and $\chi_{2}$ and write $\varphi_{2}\colon G_{2}\to H_{2}$ and $\psi_{2}\colon H_{2}\to K$ for the restrictions of $\varphi$ and $\psi$ to $G_{2}$ and $H_{2}$, respectively. Assume that $\beta$ can be recovered from $(G_{2},H_{2},K,\xi_{2},\chi_{2})$. We show that $\beta$ can be determined from $(G,H,K,\xi,\chi)$.

Recall that the pair $(g,h)=(g_{1}^{r_{1}}\cdots g_{m}^{r_{m}},\varphi(g)h_{1}^{r_{1}}\cdots h_{m}^{r_{m}}\cdot\tau^{\beta})=(g,h^{\prime}\tau^{\beta})\in G\times H$ is public. Raising both entries to their $q$-th power, one obtains

[TABLE]

From the last equation it is clear that the elements $g_{1},\dots,g_{m}\in G$ and $h_{1},\dots,h_{m}\in H$ are replaced with their $q$-th powers $g_{1}^{q},\dots,g_{m}^{q}\in G_{2}$ and $h_{1}^{q},\dots,h_{m}^{q}\in H_{2}$. Moreover, as $\tau=\tau^{q}$, it holds that $\tau\in H_{2}\setminus\ker(\psi_{2})$ and so the pair $(\tau,\beta)$ is preserved. By assumption $\beta$ is determined from the data associated to the $2$-parts and so the proof is complete. ∎

3.2.2 From LHN-PKE to eDLP and membership

In this section, we work under Assumption $(\mathrm{A}1)$ of Section 1.3 for $\Gamma\in\{G,H\}$ and Assumptions $(\mathrm{A}3)$ and $(\mathrm{A}4)$ of Section 1.3 for $\Gamma=G$ and $\{\gamma_{1},\ldots,\gamma_{m}\}=\{g_{1},\ldots,g_{m}\}$. Under Assumptions $(\mathrm{A}1)$, $(\mathrm{A}3)$, and $(\mathrm{A}4)$ the following problem is well-posed.

Definition 2

The extended discrete logarithm problem (eDLP) for the finite abelian group $\Gamma=\langle\gamma_{1}\rangle\oplus\ldots\oplus\langle\gamma_{m}\rangle$ is the problem of determining, for each $x\in\Gamma$, the unique vector

[TABLE]

such that

[TABLE]

Note that eDLP is just called discrete logarithm problem (DLP) in [30]. For a discussion of existing algorithms to solve it, we refer to Section 3.4.1. In the following, we additionally define a natural variation of the classical membership problem for groups.

Definition 3

Let $\Gamma$ be a group. The generator membership problem for $\Gamma$ is the problem of determining whether, given a finite subset $Y$ of $\Gamma$ and an element $x\in\Gamma$, one has $x\in\langle Y\rangle$.

Lemma 3

Assume that $G$ and $H$ are both finite and abelian and that $G=\langle g_{1}\rangle\oplus\cdots\oplus\langle g_{m}\rangle$ additionally satisfies $(\mathrm{A}3)$ and $(\mathrm{A}4)$. Then, the LHN-PKE problem for $G,H,K,\xi,$ and $\chi$ is at most as hard as the hardest of the eDLP problem for $G$ and the generator membership problem for $H$.

Proof

Let us write $G$ as $G=\langle g_{1}\rangle\oplus\ldots\oplus\langle g_{m}\rangle$. Given the public pair $(g,h)=(g,h^{\prime}\tau^{\beta})\in G\times H$ as in Section 1, let $r_{1}\ldots,r_{m}\in\mathbb{Z}$ be the non-negative integers chosen to write

[TABLE]

For each $i\in\{1,\ldots,m\}$, set $\ell_{i}=\varphi(g_{i})h_{i}$ and, given that the order of $\varphi(g_{i})$ divides $|\langle g_{i}\rangle|$, compute

[TABLE]

As a consequence, the subgroup $M$ of $H$ that is generated by $Y=\{\ell_{i}^{|\langle g_{i}\rangle|}:i=1,\ldots,m\}$ is contained in $\ker(\psi)$, and thus $M$ does not contain $\tau$.

Assume that eDLP is solvable in $G$. With $h$ being as above, let $s_{1},\ldots,s_{m}$ be such that, for any choice of $i$, one has $s_{i}\equiv r_{i}\bmod|\langle g_{i}\rangle|$. Denote by $t_{i}$ the integers satisfying $r_{i}-s_{i}=|\langle g_{i}\rangle|t_{i}$ and

[TABLE]

where we easily observe that $h_{M}\in M$. We now see that $\beta=0$ if and only if $x\in M$ and thus, if we are able to solve the generator membership problem for $H$, we are also able to determine whether $\beta=0$ or $\beta=1$.

∎

3.3 Proof of Theorem 1.1

We are given the following public information:

$\{(g_{1},\varphi(g_{1})h_{1}),\ldots,(g_{m},\varphi(g_{m})h_{m}),\tau\}$ and 2. 2.

$(g,h)=(g,h^{\prime}\tau^{\beta})\in G\times H$

One can proceed as follows:

• •

By Lemma 2, as $(\mathrm{A}1)$ and $(\mathrm{A}2)$ are both satisfied for both $G$ and $H$, the LHN-PKE problem for $G,H,K,\xi,$ and $\chi$ is at most as hard as the LHN-PKE problem for $G_{2},H_{2},K,{\xi}_{2}$, and ${\chi}_{2}$, where $G_{2}$ and $H_{2}$ are the Sylow 2-subgroups of $G$ and $H$, respectively.

• •

As $G=\langle g_{1}\rangle\oplus\cdots\oplus\langle g_{m}\rangle$ satisfies $(\mathrm{A}3)$, and by $(\mathrm{A}2)$ the largest odd factor $q_{G}$ of $|G|$ is known or easily computable, we also have that $G_{2}=\langle g_{1}^{q_{G}}\rangle\oplus\cdots\oplus\langle g_{m}^{q_{G}}\rangle$ satisfies $(\mathrm{A}3)$; note that some components may now be trivial, but they are necessarily still distinct as no power of $g_{i}$ was contained in $\langle g_{j}\rangle$ for $i\neq j$. Also, as $G_{2}$ is a 2-group, checking the order of any element is polynomial-time, just by repeated squaring and checking for the identity, so $(\mathrm{A}4)$ is also satisfied. Therefore, by Lemma 3, the LHN-PKE problem for $G_{2},H_{2},K,\xi_{2},$ and $\chi_{2}$ is reduced to the eDLP problem in $G_{2}$ and the generator membership problem in $H_{2}$.

The complexity of the eDLP problem in a 2-group $G_{2}$ with $m$ cyclic components is given by

[TABLE]

by [30, Cor. 1] (see Section 3.4.1 for more details). The membership problem for $H_{2}$ has polylogarithmic complexity by [4, Lemma 3(a)]; note that the required decomposition of $H_{2}$ discussed in [4, Section 3.1] exists by assumption.

∎

3.4 Assumptions and reductions

Assume in this section that $G$ and $H$ are finite and that $H$ is abelian. One can then, as before under some reasonable assumptions, reduce to the abelian case of Theorem 1.1 as follows.

Thanks to $(\mathrm{R}3)$, for cryptanalysis purposes, we can replace $G$ with $\tilde{G}=\langle g_{1},\ldots,g_{m}\rangle$ and we do so. Moreover, in view of Lemma 1, the LHN-PKE problem on $(\tilde{G},H,K,\xi,\chi)$ is reduced to the LHN-PKE problem on $(\overline{G},H,K,\overline{\xi},\chi)$, where $\overline{G}$ denotes the abelianization of $\tilde{G}$ and $\overline{\xi}$ is as in Section 3.1. Assumption $(\mathrm{A}1)$ holds for $\overline{G}$, the elements $g_{1},\ldots,g_{m}\in G$ are replaced with their images $\overline{g_{1}},\ldots,\overline{g_{m}}$ in $\overline{G}$, and the homomorphism $\varphi:\tilde{G}\rightarrow H$ is replaced with the induced homomorphism $\overline{G}\rightarrow H$, which we identify with $\varphi$, for simplicity.

Assuming now that $(\mathrm{A}2)$ holds for $H$ and $\overline{G}$, Lemma 2 allows to reduce the LHN-PKE problem on $(\overline{G},H)$ to the LHN-PKE problem on $(\overline{G}_{2},H_{2})$, where $\overline{G}_{2}$ and $H_{2}$ denote the Sylow $2$-subgroups of $\overline{G}$ and $H$. Here the elements $\overline{g_{1}},\ldots,\overline{g_{m}}$ and $h_{1},\ldots,h_{m}$ are replaced by

[TABLE]

where $q$ denotes the least common multiple of the odd parts of $|\overline{G}|$ and $|H|$.

Note that $\overline{G}_{2}=\langle\overline{g_{1}}^{q},\ldots,\overline{g_{m}}^{q}\rangle$ and suppose, at last, that $(\mathrm{A}3)$ holds for $\overline{G}_{2}$ together with $\overline{g_{1}}^{q},\ldots,\overline{g_{m}}^{q}$; that is

[TABLE]

Then, assuming that we can efficiently compute in $\overline{G}_{2}$, $(\mathrm{A}4)$ also holds: The orders of each $\overline{g_{i}}^{q}$ can be found in polynomial-time by repeated squaring as all of the orders are powers of two. Then, as a consequence of Lemma 3, the LHN-PKE problem for $(\overline{G}_{2},H_{2})$ is reduced to the eDLP problem in $\overline{G}_{2}$ and the generator membership problem in $H_{2}$.

Both this more general attack and the attack on the finite abelian case presented in Theorem 1.1 rely in some capacity on the eDLP, the generator membership problem, and our assumptions $(\mathrm{A}1)$–$(\mathrm{A}4)$. While the eDLP and the generator membership problem are relatively well-understood in the cases we need for our attacks, their complexity depends heavily on the instantiation, and we discuss these problems in Section 3.4.1 and Section 3.4.2 respectively. Also, while assumptions $(\mathrm{A}1)$–$(\mathrm{A}4)$ are natural assumptions to make when constructing groups that both admit efficient computation and satisfy the security requirements $(\mathrm{S}1)$–$(\mathrm{S}4)$ of Section 2.3.5, there exist examples of groups where these assumptions are not satisfied or may be at odds with our security requirements. In Section 3.4.3 and Section 3.4.4 we discuss assumptions $(\mathrm{A}1)$–$(\mathrm{A}4)$ with a view towards constructing instantiations of Leonardi–Ruiz-Lopez encryption to which our classical attack does not apply, or at least is not polynomial-time.

3.4.1 The eDLP in finite abelian $p$-groups

Let $p$ be a prime number and let $G$ be a finite abelian $p$-group given as

[TABLE]

where the orders of the summands are known. Let $e\in\mathbb{Z}$ be such that $p^{e}$ is the exponent of $G$. Then, according to [30, Cor. 1], the eDLP in $G$ can be solved using

[TABLE]

group operations on a classical computer. In particular, when $m$ is polynomial in $\log\log|G|$ and $p$ a fixed (small) prime, the eDLP has complexity polynomial in $\log|G|$. When $p=2$, setting $n=\log_{p}|G|$, we deduce the following from the performance result in [30, Table 1]:

• •

When $m=1,2,4,8,$ the counts are dominated by the first term of (1). This explains the initial cost decrease when $m$ increases for a fixed $n.$

• •

Using Shank’s algorithm, there is a possibility of improving the factor $n/m$ by $\sqrt{n/m}$ though this is not relevant for applications: these normally require that $n/m$ is close to $1.$

To the best of our knowledge, there is no existing work on the eDLP that beats [30].

3.4.2 The generator membership problem

Here we briefly discuss some known results on the generator membership problem in more generality than finite abelian 2-groups satisfying our assumptions: Let $H$ be a finite group and let $Y$ be a (finite) subset of $H$. Below we collect some known results regarding membership testing for $\langle Y\rangle$, depending on how the group $H$ is given.

If $H$ is given as a subgroup of some finite $\operatorname{GL}(n,q)$ and $\mu$ denotes the largest prime dividing $|\langle Y\rangle|$ and not dividing $q$, then testing whether $x\in H$ belongs to $\langle Y\rangle$ can be achieved in time that is polynomial in $|Y|+n+q+\mu$; cf. [21, Theorem 3.2(2)].

If $H$ is given as a permutation group and satisfies $(\mathrm{A}3)$, then membership testing in $Y$ is NC (polylogarithmic on a polynomial number of parallel processors); cf. [1, Theorem 1.2(a)] and [23, Theorem 1].

3.4.3 On the security assumptions $(\mathrm{A}2)$ and $(\mathrm{A}4)$

In Section 3.2.2 above we described how and under which assumptions the LHN-PKE problem for $G$, $H$, and $K$ can be reduced to solving the eDLP in $G$ and the membership problem in $H$. The necessity for setting assumptions $(\mathrm{A}2)$ and $(\mathrm{A}4)$ in particular, came from the fact that there are known examples of abelian groups, some of them already being used successfully in existing cryptographic protocols, that do not have to obey them.

One type of such groups are the known RSA groups. These are given in the form $(\mathbb{Z}/N\mathbb{Z})^{\times}$ where $N=pq$ is hard to factor. In this case, Alice would know the factorisation and hence the order of the group, but an adversary should not be able to compute it.

Another type of such groups, where even the creators of the cryptosystem might not know the group’s order, is that of ideal class groups of imaginary quadratic number fields. This is an interesting category of groups for cryptography since it allows one to work in a trustless setup. In other words, we do not need a trusted third party to generate groups of secure order, in contrast to cryptosystems that employ RSA groups for example, where a trusted third party needs to generate a secure, i.e. hard to factor, modulus $N\in\mathbb{N}$ for the groups $(\mathbb{Z}/N\mathbb{Z})^{\times}$. Despite the fact that neither the structure nor the order of the ideal class group is known, the group operation is efficient and the elements of the group have a compact representation, via reduced binary quadratic forms. An excellent reference for trusted unknown-order groups is the paper by Dobson, Galbraith and Smith [8]; they also take into account Sutherland’s algorithm [29] and they propose new security parameters for cryptosystems that employ ideal class groups. In the same paper the authors also discuss other groups of unknown order that can be used, namely genus-$3$ Jacobians of hyperelliptic curves, initially introduced by Brent [5]. Even though these groups appear to have some computational advantages when compared to ideal class groups, they offer less security than the genus-$1$ and genus-$2$ curves, as one can see for example in [12].

3.4.4 Regarding Assumption $(\mathrm{A}3)$

In this section, we discuss Assumption $(\mathrm{A}3)$. For the sake of simplicity and in view of Section 3.2.1, we restrict to $2$-groups but everything can be said similarly for arbitrary finite abelian groups using Lemma 2.

The case in which Assumption $(\mathrm{A}3)$ holds for the subgroup $\langle g_{1},\ldots,g_{m}\rangle$ of $G$, i.e., $g_{1},\dots,g_{m}$ satisfy $\langle g_{1},\dots,g_{m}\rangle=\langle g_{1}\rangle\oplus\ldots\oplus\langle g_{m}\rangle$ (equivalently $g_{1},\ldots,g_{m}$ are independent), seems to be the key case for the following reasons. First of all, when sampling $g_{1},\dots,g_{m}$ from $G$ uniformly, it is very likely that these are independent, at least if $m$ is small in comparison with the number of cyclic factors in $G$. For example, if $G=C_{2}^{\lambda}$ is a direct product of cyclic groups of order $2$, then the probability of sampling $g_{1},\dots,g_{m}$ that are independent is

[TABLE]

which is very close to $1$ if $m\ll\lambda$. In addition, the case in which $g_{1},\dots,g_{m}$ are independent can be seen as the generic case; we expect that an attacker could use similar strategies as developed in Section 1.3 and design an attack for the dependent case. Indeed, if $g_{1},\dots,g_{m}$ are dependent, the attacker will obtain more information from the public key than in the independent case.

There is also another strategy for an attack if $(\mathrm{A}3)$ does not hold for $G$ and the number of cyclic factors in $H$ is small. More precisely, assume that $|\operatorname{Hom}(H,\mathbb{F}_{2})|$ is sub-exponential, or in other words, the number of maximal subgroups of $H$ is sub-exponential. In addition, it would be necessary to list these maximal subgroups $M$ and to be able to work in the quotients $H/M$ efficiently. If Eve can write any element $g\in G$ as a product in $g_{1},\dots,g_{m}$ then she can recover the secret message $\beta$ from the ciphertext $(g,h)$ as follows. For every maximal subgroup $M$ of $H$ not containing $\tau$ (by assumption, there are only sub-exponentially many of these) convert $h$ into an element $\tilde{h}$ in $H/M$. Decrypt $(g,\tilde{h})\in G\times\tilde{H}$ as if the noise was erased completely in $H/M$ (as explained in Section 2.2) and check on a number of self-encrypted messages if this provides a correct decryption function. Since $\ker(\psi)\neq H$, there will always be a maximal subgroup $M$ of $H$ that contains $h_{1}\dots,h_{m}$ but does not contain $\tau$, so eventually this procedure will indeed provide a decrpytion function.

3.5 Comparison with the quantum attack

Throughout this section, assume that $(\mathrm{A}3)$ holds for the abelian group $G$ together with $g_{1},\ldots,g_{m}$. Even though we only did cryptanalysis for finite groups until now, we show in the following two subsections how one can perform quantum attacks when $G$ is either torsion-free or finite. The torsion-free case was handled already in [20] but we include it here for completeness; the finite case is new. The mixed case can be considered as a combination of the two cases: first dealing with the free part of the group and then recovering $\beta$ as explained in the finite case.

3.5.1 The torsion-free case

Assume as in [20, Sec. 8.2], that $G=\langle g_{1}\rangle\oplus\ldots\oplus\langle g_{m}\rangle$ is isomorphic to $\mathbb{Z}^{m}$, i.e. the orders $|\langle g_{i}\rangle|$ are all infinite. We briefly recall the discussion from [20, Sec. 8.2]. To this end, let $f:\mathbb{Z}^{m+1}\rightarrow G$ be defined by

[TABLE]

Then the kernel of $f$ is equal to $\langle(r_{1},\ldots,r_{m},1)\rangle$ and it can be determined, using Shor’s algorithm [27], in quantum polynomial time in $m$; cf. [16]. Once $(r_{1},\ldots,r_{m},1)$ is known, it is easy to recover $\beta$ from the encrypted message $(g,h)$ and the public information.

3.5.2 The torsion case

Assume in this section that $G$ is finite and that $(\mathrm{A}4)$ holds in addition to $(\mathrm{A}1)$ and $(\mathrm{A}3)$. Let, moreover, $f:\mathbb{Z}^{m+1}\rightarrow G$ be defined by

[TABLE]

Then a set of generators of $\ker(f)$ can be determined in time polynomial in $\log|G|$ on a quantum computer [17, 27, 28]; see also [11]. Note that $\ker(f)$ contains $|\langle g_{1}\rangle|\mathbb{Z}\times\ldots\times|\langle g_{m}\rangle|\mathbb{Z}\times|\langle g\rangle|\mathbb{Z}$.

Let now $(s_{1},\ldots,s_{m+1})$ be one of the generators found. Then $g=g_{1}^{-s_{1}s_{m+1}}\cdots g_{m}^{-s_{m}s_{m+1}}$ and so it follows that

[TABLE]

This is the same as saying that, for each $1\leq i\leq m$, one has

[TABLE]

Modding out $H$ by the subgroup generated by all $h_{i}^{|\langle g_{i}\rangle|}$ one can recover $\beta$ and thus solve the LHN-PKE problem; cf. Lemma 3.

4 An attack strategy that reduces to the abelian case

In this section, we assume $G$ and $H$ are finite. A general strategy for an attacker to solve the LHN-PKE problem could be to convert the $H$-part of the ciphertext as follows. Assume that Eve has access to the ciphertext

[TABLE]

encrypted by Bob using Alice’s public key

[TABLE]

Eve can then choose a homomorphism $\vartheta\colon H\to\overline{H}$ for some group $\overline{H}$ such that $\vartheta(\tau)\neq 1$ and compute $\vartheta(h)$. Her new pair $(g,\overline{h})=(g,\vartheta(h))$ is then of the form

[TABLE]

where, for all $i$, we set $\overline{h_{i}}=\vartheta(h_{i})$ and write $\overline{\varphi}=\vartheta\circ\varphi$ and $\overline{\tau}=\vartheta(\tau)$. Since $\overline{\tau}\neq 1$, this pair still contains the information on $\beta$ and if $\vartheta$ is chosen cleverly, it might be much simpler to deduce $\beta$ from $(g,h^{\prime})$. Note that in general, we cannot define a suitable counterpart $\overline{\psi}$ of $\psi$ here, so the information that $\tau$ is not contained in the kernel of $\psi$ cannot be directly converted into a statement on $\overline{\tau}$ and has to be considered individually (if necessary).

A special case of this strategy is to reduce, if possible, to a finite abelian group $\overline{H}$, i.e., the goal is to eventually apply the attack that we describe in Section 3.4 even when $G$ and $H$ are nonabelian.

Suppose that, given $\tau$ and $H$, Eve is able to find a finite abelian group $\overline{H}$ and an efficiently computable homomorphism $\vartheta:H\rightarrow\overline{H}$ such that $\vartheta(\tau)\neq 1$ and such that it can be checked that $\vartheta(\tau)$ is not contained in the subgroup generated by certain powers of $\vartheta(\varphi(g_{i})h_{i})$. This condition can be checked using the public key and we will specify below which powers are sufficient.

Eve follows the following three steps based on Lemma 1, Lemma 2 and Lemma 3.

Step 1.

Let $\overline{G}=G/[G,G]$ be the abelianization of $G$ and $\bar{\varphi}\colon\overline{G}\to\overline{H}$ be the homomorphism obtained from $\vartheta\circ\varphi\colon G\to\overline{H}$ by reducing modulo $[G,G]$. For all $i$, define $\bar{h}_{i}=\vartheta(h_{i})$, set $\bar{\tau}=\vartheta(\tau)$ and, for all $g$ in $G$, let $\bar{g}\in\overline{G}$ be the image of $g$ in $\overline{G}$.

As in Lemma 1, Eve replaces the encrypted message by

[TABLE]

in order to work inside abelian groups $\overline{G}$ and $\overline{H}$.

Step 2.

Provided $(\mathrm{A}2)$ holds for the finite abelian groups $\overline{G}$ and $\overline{H}$, Eve proceeds in a similar way as in the proof of Lemma 2: First, she computes the largest positive odd factor of the group orders of $\overline{G}$ and $\overline{H}$ and finds a common multiple $q$ of their odd parts. Then she converts the tuple $(\bar{g},\vartheta(h))$ above into a tuple with entries inside the $2$-Sylow subgroups $\overline{G}_{2}$, $\overline{H}_{2}$ of $\overline{G}$ and $\overline{H}$ by taking both entries to their $q$-th powers:

[TABLE]

Step 3.

Finally, assuming $(\mathrm{A}3)$ and $(\mathrm{A}4)$ hold for $\overline{G}_{2}$, Eve proceeds in a similar way as in the proof of Lemma 3. For all $i$ define $\bar{\ell}_{i}=(\bar{\varphi}(\bar{g}_{i})\bar{h}_{i})^{q}=\vartheta(\varphi(g_{i})h_{i})^{q}$ and let $\beta_{i}$ be the order of $\bar{g}_{i}^{q}$. Eve computes the elements $\bar{\ell}_{i}$ and the numbers $\beta_{i}$ using the public key. Then $(\bar{\ell}_{i})^{\beta_{i}}=(\bar{h}_{i}^{q})^{\beta_{i}}$ can also be computed from the public key. Assuming the eDLP is solvable in $\overline{G}_{2}$ and the generator membership problem is solvable in $\overline{H}_{2}$, and if $\bar{\tau}$ is not contained in the subgroup $M$ generated by $(\bar{\ell}_{1})^{\beta_{1}},\dots,(\bar{\ell}_{m})^{\beta_{m}}$, the value of $\beta$ can be recovered as in the proof of Lemma 3.

Remark 2

Observe that the above attack applies to any finite group $H$ if $\tau\not\in[H,H]$ and the image of $\tau$ in $H/[H,H]$ is not contained in the subgroup generated by the images of certain powers of $\varphi(g_{i})h_{i}$. In particular, it is advisable (but likely not sufficient) to sample $\tau$ from $[H,H]$.

5 Normal forms and the solvable case

In this section we consider the case in which $G$ and $H$ are finite and solvable and give evidence of why, given the efficiency constraints attached to the system, the groups should not be expected to provide secure postquantum cryptosystems. This is done in analogy to Section 3.5, but with more iterations. Roughly speaking, since solvable groups have natural filtrations yielding abelian quotients, as we explain below, one could perform a quantum attack on solvable groups by iterating quantum attacks on abelian groups.

For the background on finite solvable groups we refer to the very friendly [15, Ch. 3].

Definition 4

For a finite group $\Gamma$, the derived series of $\Gamma$ is the series $(\Gamma^{(i)})_{i\geq 1}$ defined inductively by

[TABLE]

If for some index $m$ the group $\Gamma^{(m)}$ is trivial, then $\Gamma$ is said to be solvable.

In a finite group $\Gamma$ each quotient $\Gamma^{(i)}/\Gamma^{(i+1)}$ is abelian and, if $\Gamma$ is solvable and $\Gamma^{(i)}/\Gamma^{(i+1)}$ is trivial, then $\Gamma^{(i)}=\Gamma^{(i+1)}=\Gamma^{(i+2)}=\ldots=\{1\}$.

Until the end of this section, assume that $H$ is solvable. Then, for cryptanalysis purposes and in analogy with Lemma 1, we assume without loss of generality that $G$ is also solvable. We let $s$ be the derived length of $H$, i.e. $s$ is such that $H^{(s+1)}=1$ but $H^{(s)}\neq 1$. Since $\varphi(G^{(s+1)})$ is contained in $H^{(s+1)}=\{1\}$, in analogy to Section 3.1, we assume without loss of generality that $G^{(s+1)}=1$. We remark that, if $s=1$, then $G$ and $H$ are abelian.

Remark 3** (Efficient communication and computation)**

Alice and Bob, as part of their message exchange, need to be able to communicate elements and perform operations in the groups efficiently. An often favourable approach (also proposed in [2, § 4.2]) is that of using normal forms of elements with respect to a polycyclic presentation, cf. [10, Ch. 2]. For instance, when working with $G$ and $H$ abelian, $(\mathrm{A}3)$ and $(\mathrm{A}4)$ holding for $G$ is almost the same as saying that $G$ is given by a polycyclic presentation and the expression $g=g_{1}^{r_{1}}\cdots g_{m}^{r_{m}}$ is the normal form of $g$ with respect to this presentation. The word “almost” in the previous sentence is there to stress that the $r_{i}$’s are not uniquely identified by their class modulo $|\langle g_{i}\rangle|$, which in turn is what happens for normal forms (see below).

We briefly explain here what it means for an element $g$ of the solvable group $G$ to be communicated in a normal form with respect to a polycylic presentation respecting the derived filtration. To do so, for each $i\in\{1,\ldots,s\}$:

1. $(a)$

let $m_{i}$ denote the minimum number of generators of $G^{(i)}/G^{(i+1)}$, 2. $(b)$

let $g_{i1},\ldots,g_{im_{i}}$ be elements of $G^{(i)}$ such that

[TABLE] 3. $(c)$

for each $j\in\{1,\ldots,m_{i}\}$ let $o_{ij}$ denote the order of $g_{ij}$ modulo $G^{(i+1)}$.

Then any element $g\in G$ can be uniquely represented by a vector

[TABLE]

of integers $0\leq\delta_{ij}<o_{ij}$ of length $n=m_{1}+\ldots+m_{s}$ such that

[TABLE]

which is precisely the normal form of $g$ with respect to the chosen generators $\{g_{ij}\}$.

Note that the data $(a)$-$(b)$-$(c)$ mentioned above should be public for Alice and Bob to be able to share the elements with each other. If the chosen generators fit into a polycyclic sequence, then the group operation is performed through the collection process [10, § 2.2]. It should be mentioned that, though polycyclic presentations generally yield a good practical performance, it is, to the best of our knowledge, not clear whether multiplication in these presentations can always be performed in polynomial time [19]. Note that the expression of $h^{\prime}$ also depends on the vector $\delta$:

[TABLE]

Shor’s algorithm, applied on each level $G^{(i)}/G^{(i+1)}$, is polynomial in the log of the size of this quotient. Moreover, Shor’s algorithm really does determine the vector $\delta$ of $g$ because of the condition $0\leq\delta_{ij}<o_{ij}$. In particular Eve can recover the vector $\delta$ of $g$ and use it to compute

[TABLE]

The complexity of this algorithm on each quotient $G^{(i)}/G^{(i+1)}$ is polynomial in $\log|G^{(i)}/G^{(i+1)}|$, which makes the overall complexity to be polynomial in $\log|G|$. There is therefore no separation in complexity between the algorithm being run by the adversary, and the one being run by the user. This implies that for the system to be secure, we would need to have $|G|>2^{2^{\lambda}}$, making it hard to say if it would even be possible to represent elements of $G$ in a computer.

Remark 4

Given that the quotients of consecutive elements of the derived series are abelian, it is natural to ask whether the classical attack we designed for abelian groups could be generalised to an attack in the solvable context. It seems, however, that the nonabelian solvable case is substantially different from the abelian one. Among the limitations are:

• •

$\tau$* could for instance belong to $H^{(s)}$ while $g_{1},\ldots,g_{m}$ will typically live in $H^{(1)}\setminus H^{(2)}$ (in the general setting we are indeed not necessarily working with normal forms);*

• •

without knowing $\varphi$, it is not at all clear at which depths in the derived filtration $\varphi(g_{1}),\ldots\varphi(g_{m})$ are to be found in $H$ (as publicly given only with noise);

• •

dealing with quotients is more delicate as one always has to consider normal closures of subgroups.

6 Future work

In future work, we plan to consider several types of finite groups $G$, $H$, and $K$ for instantiating Leonardi–Ruiz-Lopez encryption and explore whether we can either construct attacks for the corresponding cryptosystems or prove security results.

A first candidate would be the group $C_{2}^{\lambda}$ for all groups $G,H,K$. As none of the classical attacks presented in this paper apply in this case, Leonardi–Ruiz-Lopez encryption might prove to be classically secure for this choice. Other abelian candidates are the RSA groups and ideal class groups mentioned in Section 3.4.3.

As a first nonabelian example, we plan to work with certain $p$-groups and use strategies that allow us to circumnavigate the attacks presented in Section 5. In particular, it would be beneficial to work with presentations of groups that are not based on normal forms and yet allow efficient computation. The advantage of working with nonabelian groups is that it may be possible to construct a post-quantum additive homomorphic cryptosystem.