Learning with Errors is easy with quantum samples

Alex B. Grilo; Iordanis Kerenidis; Timo Zijlstra

arXiv:1702.08255·quant-ph·March 27, 2019

Learning with Errors is easy with quantum samples

Alex B. Grilo, Iordanis Kerenidis, Timo Zijlstra

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TL;DR

This paper demonstrates an efficient quantum learning algorithm for the Learning with Errors problem, revealing implications for cryptography and the security of LWE-based schemes against quantum attacks.

Contribution

It introduces a polynomial-time quantum learning algorithm for LWE with cryptographic error distributions, highlighting potential cryptographic vulnerabilities.

Findings

01

Quantum sample complexity for LWE is polynomial.

02

The algorithm does not break existing LWE encryption schemes.

03

Potential for classical samples to approximate quantum states in cryptography.

Abstract

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical…

Equations86

(a, a \cdot s + e (mod q))

(a, a \cdot s + e (mod q))

x \in X \sum \frac{1}{∣ X ∣} ∣ x ⟩ ∣ f (x) ⟩ .

x \in X \sum \frac{1}{∣ X ∣} ∣ x ⟩ ∣ f (x) ⟩ .

\frac{1}{∣ R ∣} r \in R \sum ∣ r ⟩ (\frac{1}{∣ X ∣} x \in X \sum ∣ x ⟩ ∣ g (x) + e (x, r) ⟩),

\frac{1}{∣ R ∣} r \in R \sum ∣ r ⟩ (\frac{1}{∣ X ∣} x \in X \sum ∣ x ⟩ ∣ g (x) + e (x, r) ⟩),

\frac{1}{∣ X ∣} ∣ x ⟩ ∣ g (x) + e (x, r) ⟩,

\frac{1}{∣ X ∣} ∣ x ⟩ ∣ g (x) + e (x, r) ⟩,

\frac{1}{q ^{n}} a \in F_{q}^{n} \sum ∣ a ⟩ ∣ a \cdot s + e_{a} (mod q) ⟩,

\frac{1}{q ^{n}} a \in F_{q}^{n} \sum ∣ a ⟩ ∣ a \cdot s + e_{a} (mod q) ⟩,

(i \in S \sum a_{i} (mod q), b ⌈ \frac{q}{2} ⌉ + i \in S \sum a_{i} \cdot s + e_{i} (mod q)) .

(i \in S \sum a_{i} (mod q), b ⌈ \frac{q}{2} ⌉ + i \in S \sum a_{i} \cdot s + e_{i} (mod q)) .

S \subseteq [m] \sum ∣ S ⟩ i \in S \sum a_{i} (mod q) ⟩ i \in S \sum a_{i} \cdot s + e_{i} (mod q) ⟩ .

S \subseteq [m] \sum ∣ S ⟩ i \in S \sum a_{i} (mod q) ⟩ i \in S \sum a_{i} \cdot s + e_{i} (mod q) ⟩ .

∣ ψ ⟩ = \frac{1}{q ^{n}} a \in F_{q}^{n} \sum ∣ a ⟩ ∣ a \cdot s (mod q) ⟩,

∣ ψ ⟩ = \frac{1}{q ^{n}} a \in F_{q}^{n} \sum ∣ a ⟩ ∣ a \cdot s (mod q) ⟩,

\frac{1}{q ^{n + \frac{1}{2}}} a, j \in F_{q}^{n} \sum j^{*} \in F_{q} \sum ω^{a \cdot (j + j^{*} s)} ∣ j ⟩ ∣ j^{*} ⟩ .

\frac{1}{q ^{n + \frac{1}{2}}} a, j \in F_{q}^{n} \sum j^{*} \in F_{q} \sum ω^{a \cdot (j + j^{*} s)} ∣ j ⟩ ∣ j^{*} ⟩ .

\frac{1}{q ^{n + \frac{1}{2}}} j^{*} \in F_{q}^{*} \sum a \in F_{q}^{n} \sum ω^{0} ∣ - j^{*} s (mod q) ⟩^{2}

\frac{1}{q ^{n + \frac{1}{2}}} j^{*} \in F_{q}^{*} \sum a \in F_{q}^{n} \sum ω^{0} ∣ - j^{*} s (mod q) ⟩^{2}

∣ ψ ⟩ = \frac{1}{v} a \in V \sum ∣ a ⟩ ∣ a \cdot s + e_{a} (m o d q) ⟩,

∣ ψ ⟩ = \frac{1}{v} a \in V \sum ∣ a ⟩ ∣ a \cdot s + e_{a} (m o d q) ⟩,

∣ ψ ⟩ = \frac{1}{v} a \in V \sum ∣ a ⟩ ∣ a \cdot s + e_{a} (m o d q) ⟩,

∣ ψ ⟩ = \frac{1}{v} a \in V \sum ∣ a ⟩ ∣ a \cdot s + e_{a} (m o d q) ⟩,

\frac{1}{q ^{n + 1} v} a \in V \sum j \in F_{q}^{n}, j^{*} \in F_{q} \sum ω^{e_{a} j^{*} + a \cdot (j + j^{*} s)} ∣ j ⟩ ∣ j^{*} ⟩ .

\frac{1}{q ^{n + 1} v} a \in V \sum j \in F_{q}^{n}, j^{*} \in F_{q} \sum ω^{e_{a} j^{*} + a \cdot (j + j^{*} s)} ∣ j ⟩ ∣ j^{*} ⟩ .

\frac{1}{q ^{n + 1} v} a \in V \sum j^{*} \in F_{q}^{*} \sum ω^{e_{a} j^{*}} ∣ - j^{*} s (mod q) ⟩ ∣ j^{*} ⟩^{2}

\frac{1}{q ^{n + 1} v} a \in V \sum j^{*} \in F_{q}^{*} \sum ω^{e_{a} j^{*}} ∣ - j^{*} s (mod q) ⟩ ∣ j^{*} ⟩^{2}

= \frac{1}{q ^{n + 1} v} j^{*} \in F_{q}^{*} \sum (a \in V \sum R (ω^{e_{a} j^{*}}))^{2} + (a \in V \sum I (ω^{e_{a} j^{*}}))^{2}

\geq \frac{1}{q ^{n + 1} v} j^{*} \in F_{q}^{*} j^{*} \leq \frac{γ q}{k} \sum (a \in V \sum R (ω^{e_{a} j^{*}}))^{2}

\geq \frac{γ v cos ( 2 π γ ) ^{2}}{k q ^{n}} .

∣ ψ ⟩ = \frac{1}{q ^{n}} a \in F_{q}^{n} \sum ∣ a ⟩ ∣ a \cdot s + e_{a} ⟩,

∣ ψ ⟩ = \frac{1}{q ^{n}} a \in F_{q}^{n} \sum ∣ a ⟩ ∣ a \cdot s + e_{a} ⟩,

1 - (1 - \frac{v}{20 k q ^{n}})^{L} - (\frac{3 k}{q})^{M} L .

1 - (1 - \frac{v}{20 k q ^{n}})^{L} - (\frac{3 k}{q})^{M} L .

(\frac{2 k + 1}{q})^{M} L \leq (\frac{3 k}{q})^{M} L .

(\frac{2 k + 1}{q})^{M} L \leq (\frac{3 k}{q})^{M} L .

(1 - \frac{v}{20 k q ^{n}})^{L} .

(1 - \frac{v}{20 k q ^{n}})^{L} .

(1 - \frac{v}{20 k q ^{n}})^{L} + (\frac{3 k}{q})^{M} L . \qed

(1 - \frac{v}{20 k q ^{n}})^{L} + (\frac{3 k}{q})^{M} L . \qed

\frac{1}{2 ^{n + \frac{1}{2}}} a \in {0, 1}^{n} \sum j \in {0, 1}^{n + 1} \sum (- 1)^{e_{a} j^{*} + a \cdot (j + j^{*} s)} ∣ j ⟩ ∣ j^{*} ⟩

\frac{1}{2 ^{n + \frac{1}{2}}} a \in {0, 1}^{n} \sum j \in {0, 1}^{n + 1} \sum (- 1)^{e_{a} j^{*} + a \cdot (j + j^{*} s)} ∣ j ⟩ ∣ j^{*} ⟩

\frac{1}{2 ^{n + \frac{1}{2}}} a \in {0, 1}^{n} \sum (- 1)^{e_{a} + a \cdot (s + s)} ∣ s ⟩^{2}

\frac{1}{2 ^{n + \frac{1}{2}}} a \in {0, 1}^{n} \sum (- 1)^{e_{a} + a \cdot (s + s)} ∣ s ⟩^{2}

= \frac{1}{2 ^{2 n} + 1} a \in {0, 1}^{n} \sum (- 1)^{e_{a}}^{2}

P r a \in {0, 1}^{n} \sum (- 1)^{e_{a}} \leq (1 - δ) (1 - 2 η) 2^{n}

P r a \in {0, 1}^{n} \sum (- 1)^{e_{a}} \leq (1 - δ) (1 - 2 η) 2^{n}

< e^{δ^{2} (1 - 2 η)^{2} 2^{2 n} /4}

\frac{1}{2 ^{2 n} + 1} ((1 - δ) (1 - 2 η) 2^{n})^{2} = \frac{1}{2} (1 - δ)^{2} (1 - 2 η)^{2} . \qed

\frac{1}{2 ^{2 n} + 1} ((1 - δ) (1 - 2 η) 2^{n})^{2} = \frac{1}{2} (1 - δ)^{2} (1 - 2 η)^{2} . \qed

∣ ψ ⟩ = \frac{1}{q ^{n}} a \in R_{q} \sum ∣ ϕ (a) ⟩ ∣ ϕ (a s + e_{a}) ⟩,

∣ ψ ⟩ = \frac{1}{q ^{n}} a \in R_{q} \sum ∣ ϕ (a) ⟩ ∣ ϕ (a s + e_{a}) ⟩,

QF T^{\otimes 2 n} ∣ ψ ⟩

QF T^{\otimes 2 n} ∣ ψ ⟩

= \frac{1}{q ^{3 n}} a \in R_{q} \sum x, y \in Z_{q} \sum ω^{ϕ (a) \cdot x} ∣ x ⟩ \otimes ω^{ϕ (a s + e_{a}) \cdot y} ∣ y ⟩

= \frac{1}{q ^{3 n}} a \in R_{q} \sum x, y \in Z_{q} \sum ω^{ϕ (a) \cdot (x + y ϕ (s)) + y \cdot ϕ (e_{a})} ∣ x ⟩ ∣ y ⟩,

ϕ (a) \cdot x + ϕ (a \cdot s + e_{a}) y = ϕ (a) (x + y ϕ (s)) + y ϕ (e_{a}) .

ϕ (a) \cdot x + ϕ (a \cdot s + e_{a}) y = ϕ (a) (x + y ϕ (s)) + y ϕ (e_{a}) .

ϕ (p (x)) = (p (ω_{m}), \dots, p (ω_{m}^{m - 1})),

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Full text

Learning with Errors is easy with quantum samples

Alex B. Grilo1

Iordanis Kerenidis1

Timo Zijlstra2

1 IRIF, CNRS, Université Paris Diderot, Paris, France

2 Lab-STICC, Université Bretagne Sud

Abstract

Learning with Errors is one of the fundamental problems in computational learning theory and has in the last years become the cornerstone of post-quantum cryptography. In this work, we study the quantum sample complexity of Learning with Errors and show that there exists an efficient quantum learning algorithm (with polynomial sample and time complexity) for the Learning with Errors problem where the error distribution is the one used in cryptography. While our quantum learning algorithm does not break the LWE-based encryption schemes proposed in the cryptography literature, it does have some interesting implications for cryptography: first, when building an LWE-based scheme, one needs to be careful about the access to the public-key generation algorithm that is given to the adversary; second, our algorithm shows a possible way for attacking LWE-based encryption by using classical samples to approximate the quantum sample state, since then using our quantum learning algorithm would solve LWE. Finally, we extend our results and show quantum learning algorithms for three related problems: Learning Parity with Noise, Learning with Rounding and Short Integer Solution.

I Introduction

The large amount of data arising in the real world, for example through scientific observations, large-scale experiments, internet traffic, social media, etc, makes it necessary to be able to predict some general properties or behaviors of the data from a limited number of samples of the data. In this context, Computational Learning Theory provides rigorous models for learning and studies the necessary and sufficient resources, for example, the number of samples or the running time of the learning algorithm. In his seminal work, Valiant (Valiant84) introduced the model of PAC learning, and since then this model has been extensively studied and has given rise to numerous extensions.

In another revolutionary direction, Quantum Computing takes advantage of the quantum nature of small-scale systems as a computational resource. In this field, the main question is to understand what problems can be solved more efficiently in a quantum computer than in classical computers. In the intersection of the two fields, we have Quantum Learning Theory, where we ask if quantum learning algorithms can be more efficient than classical ones.

One of course needs to be careful about defining quantum learning and more precisely, what kind of access to the data a quantum learning algorithm has. On one hand, we can just provide classical samples to the quantum learning algorithm that can then use the quantum power in processing these classical data. In the more general scenario, we allow the quantum learning algorithm to receive quantum samples of the data, for a natural notion of a quantum sample as a superposition that corresponds to the classical sample distribution.

More precisely, in classical learning, the learning algorithm is provided with samples of $(x,f(x))$ , where $x$ is drawn from some unknown distribution $D$ and $f$ is the function we wish to learn. The goal of the learner in this case is to output a function $g$ such that with high probability (with respect to the samples received), $f$ and $g$ are close, i.e., $Pr\left[f(x)\neq g(x)\right]$ is small when $x$ is drawn from the same distribution $D$ .

The extension of this model to the quantum setting is that the samples now are given in the form of a quantum state $\sum_{x}\sqrt{D(x)}\left|x\right\rangle\left|f(x)\right\rangle$ . Note that one thing the quantum learner can do with this state is simply measure it in the computational basis and get a classical sample from the distribution $D$ . Hence, a quantum sample is at least as powerful as a classical sample. The main question is whether the quantum learner can make better use of these quantum samples and provide an advantage in the number of samples and/or running time compared to a classical learner.

In this work we focus on one of the fundamental problems in learning theory, the Learning with Errors (LWE). In LWE, one is given samples of the form

[TABLE]

where $s\in\mathbb{F}_{q}^{n}$ is fixed, $a\in\mathbb{F}_{q}^{n}$ is drawn uniformly at random and $e\in\mathbb{F}_{q}$ is an ’error’ term drawn from some distribution $\chi$ . The goal is to output $s$ , while minimizing the number of samples used and the computation time.

First, LWE is the natural generalisation of the well-studied Learning Parity with Noise problem (LPN), which is the case of $q=2$ . Moreover, a lot of attention was drawn to this problem when Regev (Regev05) reduced some (expected to be) hard problems involving lattices to LWE. With this reduction, LWE has become the cornerstone of current post-quantum cryptographic schemes. Several cryptographic primitives proposals such as Fully Homomorphic Encryption (BrakerskiV14), Oblivious Transfer (PeikertVW08), Identity based encryption (GentryPV08; CashHKP12; AgrawalBB10), and others schemes are based in the hardness of LWE (for a more complete list see Ref. (MicciancioRegev) and Ref. (Peikert16)).

Classically, Blum et al. (Blum2003) proposed the first sub-exponential algorithm for this problem, where both sample and time complexities are $2^{O(n/\log{n})}$ . Then, Arora and Ge (AroraG11) improved the time complexity for LWE with a learning algorithm that runs in $2^{\tilde{O}(n^{2\varepsilon})}$ time, for some $\varepsilon<\frac{1}{2}$ , and it uses at least $\Omega(q^{2}\log{q})$ samplesFor LPN, Lyubashevsky (Lyubashevsky05) has proposed an algorithm with sample complexity $n^{1+\varepsilon}$ at the cost of increasing computation time to $O(2^{n/\log{\log{n}}})$ .

II The quantum learning model

In this work, we use the model of learning under the uniform distribution where the learner receives samples according to the uniform distribution and outputs the exact function with high probability. In the quantum setting, the learning algorithm is given quantum samples, namely a uniform superposition of the inputs and function values,

[TABLE]

In this work, we are interested in noisy samples, that can be modeled by setting $f(x)=g(x)+e(x,r)$ , where $g$ and $e$ are deterministic functions, $x\in X$ and $r\in R$ is the randomness necessary to generate the noise. For defining the quantum sample, we start with the superposition

[TABLE]

and then the register corresponding to the randomness is traced out. It means that with probability $\frac{1}{|R|}$ the quantum sample is

[TABLE]

for each possible value $r\in R$ .

We consider the noise model defined in Bshouty and Jackson (Bshouty95), where independent noise is added for each element in the superposition, in other words, $r=(r_{1},...,r_{|X|})$ and $e(x,r)=e^{\prime}(r_{x})$ . This model is a natural generalisation for quantum samples with noise since it can be seen as a superposition of the classical samples.

In contrast, Cross et al. (CrossSS15) proposed a noise function that is independent of $x$ . Although our noise model might require exponentially more resources to implement quantum samples, we show that this does not make the problem intractable. Also, this is the kind of state we would get after solving the index erasure problem.

III Our contributions

In this work we study quantum algorithms for solving LWE with quantum samples. Let us be more explicit on the definition of a quantum sample for the LWE problem. We assume that the quantum learning algorithm receives samples in the form

[TABLE]

where $e_{a}$ are iid random variables from some distribution $\chi$ over $\mathbb{F}_{q}$ .

As expected, the performance of the learning algorithm, both in the classical and quantum case, is sensitive to the noise model adopted, i.e. to the distribution $\chi$ . When LWE is used in cryptographic schemes, the distribution $\chi$ has support on a small interval around 0, either uniform or a discrete gaussian. We prove that for such distributions, there exists an efficient quantum learner for LWE.

Main Result[informal] *For error distributions $\chi$ used in cryptographic schemes, and for any $\eta>0$ , there exists a quantum learning algorithm that solves LWE with probability $1-\eta$ using $O(n\log\frac{1}{\eta})$ samples and running time $poly(n,\log\frac{1}{\eta})$ .

Another interesting feature of our quantum learner is that it is conceptually a very simple algorithm based on one of the basic quantum operations, the Quantum Fourier Transform. Such algorithms have even started to be implemented, of course for very small input sizes and for the binary case (Riste15). Nevertheless, as far as quantum algorithms are concerned, our learner is quite feasible from an implementation point of view.

The approach to solve the problem is a generalisation of Bernstein-Vazirani algorithm (BernsteinV1997): we start with a quantum sample, apply a Quantum Fourier Transform over $\mathbb{F}_{q}$ on each qudit, and then, we measure in the computational basis. Our analysis shows that, when the last qudit is not 0, which happens with high probability, the value of the remaining registers gives $s$ with constant probability. We can then repeat this process so that our algorithm outputs $s$ with high probability.

We also use the same technique in quantum learning algorithms for three related problems. First, we generalise the result proposed by Cross et al. (CrossSS15) and Ristè et al. (Riste15) for the LPN problem. The main difference with their work is that we start with a quantum sample, i.e. a state where the noise is independent for each element in the superposition. Second, we show how to solve the Learning with Rounding problem, which can be seen as a derandomized version of LWE. Finally, we also propose a quantum learning algorithm for another relevant problem in cryptography, the Short Integer Solution problem.

III.1 Related work

We now review some results on quantum algorithms for learning problems. For a more extended introduction, see the survey by Arunachalam and de Wolf (ArunachalamW17).

The first approach on trying to solve learning problems with quantum samples was proposed by Bshouty and Jackson (Bshouty95), where they prove that DNFs can be learned efficiently, even when the samples are noisy. No such efficient learners are known classically.

Despite not presenting it as a learning problem, Bernstein and Vazirani (BernsteinV1997) show how to learn parity using a single quantum sample, while classically we need a linear number of samples.

Some years later, Servedio and Gortler (ServedioG04) showed that classical and quantum sample/query complexity of learning problems are polynomially related, but they showed that for time complexity there exist exponential separations between classical and quantum learning (assuming standard computational hardness assumptions).

Then, Ambainis et al. (Ambainis2004), Atici and Servedio (AticiS05), and Hunziker et al. (Hunziker2010) provided general upper bounds on the query complexity for learning problems that depend on the size of the concept class being learned.

On specific problems, Atici and Servedio (AticiS07) and Belovs (Belovs2015) provided quantum algorithms for learning juntas and Cross et al. (CrossSS15) proposed and implemented quantum algorithms for LPN in a different noise model.

Recently, Arunachalam and de Wolf (ArunachalamW16) proved optimal bounds for the quantum sample complexity of the Quantum PAC model.

III.2 Relation to LWE-based cryptography

As we have mentioned, LWE is used in cryptography for many different tasks. Let us briefly describe how one can build an encryption scheme based on LWE (Regev05). The key generation algorithm produces a secret key $s\in\mathbb{F}_{q}$ , while the public key consists of a sequence of classical LWE samples $(a_{1},a_{1}\cdot s+e_{1}\pmod{q}),...,(a_{m},a_{m}\cdot s+e_{m}\pmod{q})$ , where the error comes from a distribution with support in a small interval around 0. For the encryption of a bit $b$ , the party picks a subset $S$ of $[m]$ uniformly at random and outputs

[TABLE]

For the decryption, knowing $s$ allows one to find $b$ . The security analysis of the encryption scheme postulates that if an adversary can break the encryption efficiently then he is also able to solve the LWE problem efficiently.

The algorithm we present here does not break the above LWE-based encryption scheme. Nevertheless, it has interesting implications for cryptography.

First, our algorithm shows a possible way for attacking LWE-based encryption: use classical samples to approximate the quantum sample state, and then use our algorithm to solve LWE. One potential way for this would be to start with $m$ classical samples and create the following superposition

[TABLE]

This operation is in fact efficient. Then, in order to approximate the quantum sample state, one would need to ’forget’ the first register that contains the index information about which subset of the $m$ classical samples we took. In the most general case, such an operation of forgetting the index of the states in a quantum superposition, known as index-erasure (see Aharonov and Ta-Shma (AharonovTS2003) and Ambainis et al. (Ambainis2004)), is exponentially hard, and a number of problems, such as Graph Non-isomorphism, would have an efficient quantum algorithm, if we could do it efficiently. Nevertheless, one may try to use the extra structure of the LWE problem to find sub-exponential algorithms for this case.

A second concern that our algorithm raises is that when building an LWE-based scheme, one needs to be careful on the access to the public-key generation algorithm that is given to the adversary. It is well-known that for example, even in the classical case, if the adversary can ask classical queries to the LWE oracle, then he can easily break the scheme: by asking the same query many times one can basically average out the noise and find the secret $s$ . However, if we just assume that the public key is given as a box that an agent has passive access to it, in the sense that he can request a random sample and receive one, then the encryption scheme is secure classically as long as LWE is difficult. However, imagine that the random sample from LWE is provided by a device that creates a superposition $\frac{1}{\sqrt{q^{n}}}\sum_{a\in\mathbb{F}_{q}^{n}}\left|a\right\rangle\left|a\cdot s+e_{a}\pmod{q}\right\rangle$ and then measures it. Then a quantum adversary that has access to this quantum state can break the scheme. Again, our claim is, by no means, that our algorithm breaks the proposed LWE-based encryption schemes, but more that LWE-based schemes which are secure classically (assuming the hardness of LWE) may stop being secure against quantum adversaries if the access to the public key generation algorithm becomes also quantum.

A similar situation has also appeared in the symmetric key cryptography with the so called superposition attacks (Zhandry12; Boneh2013; Damgard2014; Kaplan2016). There, the attacker has the ability to query the encryption oracle in superposition, and in this way, she can in fact break many schemes that are assumed to be secure classically. While in the case of symmetric cryptography, the attacker must have quantum access to the encryption oracle in order to break the system, our results show that in the case of LWE-based public-key encryption, the attacker must have quantum access to the public key generation algorithm.

IV Algorithm for LWE

In this section we present the extension of the Bernstein-Vazirani algorithm for higher order fields and analyse its behaviour with LWE samples.

We show now the Field Bernstein-Vazirani algorithm (BernsteinV1997) and its main component is the Quantum Fourier Transform over $\mathbb{F}_{q}$ , $QFT\left|j\right\rangle=\frac{1}{\sqrt{q^{n}}}\sum_{k=0}^{q^{n}-1}\omega^{jk}\left|k\right\rangle$ .

**Field Bernstein-Vazirani algorithm

Input:** $\left|\psi\right\rangle\in(\mathbb{C}^{2})^{\otimes n+1}$

Output: $\tilde{s}\in\mathbb{F}_{q}^{n}\cup\{\perp\}$

Apply $QFT^{\otimes n+1}$ on $\left|\psi\right\rangle$ .

Measure in the computational basis

Let $\left|j\right\rangle\left|j^{*}\right\rangle$ be the output

If $j^{*}\neq 0$ , return $-(j^{*})^{-1}j\pmod{q}$

Else, return $\perp$

For warming-up, we show the behaviour of Field Bernstein-Vazirani for learning linear functions without noise in Section IV.1 and then in Section IV.2 we analyse it for LWE samples.

IV.1 Quantum algorithm for learning a linear function without error

If the input $\left|\psi\right\rangle$ is a noiseless quantum sample of a linear function, namely

[TABLE]

Then the Field-Bernstein Vazirani outputs the correct value with probability $\frac{q-1}{q}$ : after applying the QFT on each qudit of eq. 2, we get the state

[TABLE]

It is not hard to see that the probability that for all $i\in[n]$ , we have $j=-j^{*}s\pmod{q}$ and $j^{*}\neq 0$ is

[TABLE]

Therefore, if $j^{*}\neq 0$ , we can retrieve $s$ by outputting $-(j^{*})^{-1}j_{i}$ (all operations mod q).

IV.2 Analysis of the algorithm for noisy samples

In this section we show that the Field Bernstein-Vazirani algorithm works even if the input is noisy. Instead of the superposition of all elements in $\mathbb{F}_{q}^{n}$ , we prove our result here for a more general case where the quantum sample has the form

[TABLE]

where $v\in[q^{n}]$ is a fixed value, $V$ be a random subset of $\mathbb{F}_{q}^{n}$ of size $v$ and $e_{a}$ is a random noise. In this case, for every quantum sample, a new subset $V$ of size $v$ is picked independently at random.

Theorem IV.1.

Fix $v\in[q^{n}]$ . Let $V\subseteq\mathbb{F}_{q}^{n}$ be a random subset of $\mathbb{F}_{q}^{n}$ such that $|V|=v$ , and let

[TABLE]

where the $e_{a}$ are random variables with absolute value at most $k$ . The Field Bernstein-Vazirani $(\left|\psi\right\rangle)$ outputs $s$ with probability $\frac{v}{20kq^{n}}$ .

Proof.

If we apply QFT on the state $\left|\psi\right\rangle$ , we have

[TABLE]

From the last equation, we have that the probability that $j=-j^{*}s\pmod{q}$ and $j^{*}\neq 0$ is:

[TABLE]

where $\gamma\in(0,\frac{1}{4})$ $\mathfrak{R}(z)$ and $\mathfrak{I}(z)$ are the real and imaginary part of $z$ , respectively. For the first inequality, we have removed some positive quantities, and the last inequality follows from the fact that $\mathfrak{R}(\omega^{e_{a}j^{*}})\leq\cos\left(2\pi\gamma\right)$ for $j^{*}\leq\frac{\gamma q}{k}$ and $\left|e_{a}\right|\leq k$ . The result follows by maximizing the quantity over all $\gamma\in(0,\frac{1}{4})$ . ∎

We now propose an algorithm that tests a candidate solution.

**Test Candidate

Input:** $\tilde{s}\in\mathbb{F}_{q}^{n}$ , $M\in\mathbb{Z}^{+}$

Output: Accept/reject

Repeat $M$ times.

Pick sample $\left|\psi\right\rangle=\frac{1}{\sqrt{v}}\sum_{a\in V}\left|a\right\rangle\left|a\cdot s+e_{a}\mod q\right\rangle$ .

Measure the sample in the computational basis

Let $(a^{\prime},a^{\prime}\cdot s+e_{a^{\prime}})$ be the output

If $|a^{\prime}\cdot s+e_{a^{\prime}}-a^{\prime}\cdot\tilde{s}|>k$ , reject

Accept

Lemma IV.2.

For $\tilde{s}=s$ , Test Candidate $(\tilde{s},M)$ accepts with probability $1$ , while for $\tilde{s}\neq s$ , Test Candidate $(\tilde{s},M)$ accepts with probability probability at most $\left(\frac{2k+1}{q}\right)^{M}$ .

Proof.

Since $|a^{\prime}\cdot s+e_{a^{\prime}}-a^{\prime}\cdot s|=|e_{a^{\prime}}|\leq k$ by the noise distribution, it follows that the test passes with probability $1$ when $\tilde{s}=s$ .

For a value $a^{\prime}$ picked uniformly random from $\mathbb{F}_{q}^{n}$ , it follows that $a^{\prime}\cdot(s-\tilde{s})+e_{a^{\prime}}\mod q$ is uniformly distributed over $\mathbb{F}_{q}$ if $\tilde{s}\neq s$ . Therefore, the probability that it lies in the interval $[-k,k]$ is $\frac{2k+1}{q}$ . Since the probability is independent for every iteration, the probability that $\tilde{s}$ is accepted on $M$ iterations is $\left(\frac{2k+1}{q}\right)^{M}$ . ∎

In Appendix B, we show how to use the previous algorithms to achieve the following theorem.

Theorem IV.3.

For dimension $n$ , let $q$ be a prime in the interval $[2^{n^{\gamma}},2\cdot 2^{n^{\gamma}})$ . Let

[TABLE]

where the $e_{a}$ are random variables drawn from a noise distribution with noise magnitude at most $k=poly(n)$ . There is an algorithm that outputs $s$ with probability $1-\eta$ with sample complexity $O(k\log{\frac{1}{\eta}})$ and running time $poly(n,\log\frac{1}{\eta})$ .

We show then how to extend the result to related problems in Appendix C.

V Open problems

V.1 Generalizing from linear functions

Learning linear functions can be seen as finding a hidden subgroup $H={a|a\cdot s=0}$ of $\mathbb{Z}_{q}^{n}$ . Efficient algorithms for general Abelian Hidden Subgroup Problem are known (CheungM01)(Mosca2014), and we leave as an open question if these algorithms are also tolerant to noise.

V.2 LWE over rings

Due to technical reasons regarding the representation of polynomials in Ring-LWE instances (see Section C.2 for more details), our LWE algorithm cannot be used to solve Ring-LWE with quantum samples and we leave this question as an open problem.

Acknowledgments

AG and IK thank Ronald de Wolf for helpful discussions. AG thanks also Lucas Boczkowski, Brieuc Guinard, François Le Gall and Alexandre Nolin for helpful discussions. Supported by ERC QCC and French Programme d’Investissement d’Avenir RISQ P141580.

References

Appendix A Notation

For $n\in\mathbb{N}$ , we define $[n]:=\{1,...,n\}$ . For a complex number $x=a+ib$ , $a,b\in\mathbb{R}$ , we define its norm $|x|$ by $\sqrt{a^{2}+b^{2}}$ , its real part $\mathfrak{R}(x)=a$ and its imaginary part $\mathfrak{I}(x)=b$ . We denote $\omega$ as the $q$ -th root of unity, where $q$ will be clear by the context. For a field $\mathbb{F}_{q}$ and element $a\in\mathbb{F}_{q}$ , we denote $|a|$ as the unique value $b\in\left[\frac{-(q-1)}{2},\frac{q-1}{2}\right]$ such that $b\equiv a\pmod{q}$ .

We remind now the notation for quantum information and computation. For readers not familiar with these concepts we refer Ref. (NielsenC2011). Let $\{e_{i}\}$ be the standard basis for the $q$ -dimensional Hilbert space $\mathbb{C}^{q}$ . We denote here $\left|i\right\rangle=e_{i}$ and a $q$ -dimensional qudit is a unit vector in this space, i.e. $\left|\psi\right\rangle=\sum_{i\in\mathbb{F}_{q}}\alpha_{i}\left|i\right\rangle$ , for $\alpha_{i}\in\mathbb{C}$ and $\sum_{i\in\mathbb{F}_{q}}\left|\alpha_{i}\right|^{2}=1$ . We call the state a qubit when $q=2$ . A $k$ -qudit quantum state is a unit vector in the complex Hilbert space $\mathbb{C}^{q^{k}}$ and we shorthand the basis states for this space $\left|i_{1}\right\rangle\otimes...\otimes\left|i_{k}\right\rangle$ with $\left|i_{1}\right\rangle...\left|i_{k}\right\rangle$ .

Appendix B An efficient quantum learning algorithm for LWE

In this section we show how to use Theorems IV.1 and IV.2 in order to solve solve LWE with quantum samples using noise distributions proposed in Brakerski and Vaikuntanathan (BrakerskiV14), proving Theorem IV.3. There, the field order $q$ is sub-exponential in the dimension $n$ , generally in $[2^{n^{\gamma}},2\cdot 2^{n^{\gamma}})$ for some constant $\gamma\in(0,1)$ , while the noise distribution $\chi$ produces samples with magnitude at most polynomial in $n$ (for instance linear).

**LWE Algorithm(L, M)

Input:** $L,M\in\mathbb{Z}^{+}$

Output: $\tilde{s}\in\mathbb{F}_{q}^{n}\cup\{\perp\}$

Repeat $L$ times:

Pick a quantum sample $\left|\psi\right\rangle$

Run the Field Bernstein-Vazirani( $\left|\psi\right\rangle$ ) to get output $\tilde{s}$

Run Test Candidate $(\tilde{s},M)$

If $\tilde{s}$ passes the test, return $\tilde{s}$

Return $\perp$ .

Theorem B.1.

LWE Algorithm $(L,M)$ outputs $s$ with probability

[TABLE]

Proof.

LWE Algorithm $(L,M)$ does not output $s$ if either Test Candidate $(\tilde{s},\log{\frac{1}{\eta}})$ accepts some $\tilde{s}\neq s$ before an iteration where Field Bernstein-Vazirani outputs $s$ , or LWE Algorithm outputs $\perp$ . We can upper bound the probability of this event by the probability that at least one of $L$ independent calls to Test Candidate $(\tilde{s},\log{\frac{1}{\eta}})$ accepts some $\tilde{s}\neq s$ or that $L$ independent calls to Field Bernstein-Vazirani do not output $s$ .

From Lemma IV.2 and using the union bound, the probability that at least one of $L$ independent calls to Test Candidate $(\tilde{s},\log{\frac{1}{\eta}})$ accepts some $\tilde{s}\neq s$ is at most

[TABLE]

From Theorem IV.1, the probability that $s$ is not the output of $L$ independent calls to Field Bernstein-Vazirani is at most

[TABLE]

By union bound, LWE Algorithm $(L,M)$ does not output $s$ with probability at most

[TABLE]

Theorem IV.3 follows directly from Theorem B.1 and picking $v=q^{n}$ , $L=20k\ln\frac{1}{\eta}$ and $M=1$ .

Appendix C Quantum learning complexity of related problems

In this section we present learning algorithms for problems that are related to LWE.

C.1 Learning parity with noise

We show here our result for Learning Parity with Noise (LPN) problem, which is the LWE problem for $q=2$ .

Here, the parity bit is flipped independently for each element in the superposition with probability $\eta$ . This is the same noise model proposed by Bshouty and Jackson (Bshouty95). Note that Cross et al. (CrossSS15) studied LPN with different noise models. In the first, all parities in the superposition are flipped at the same time with probability $\eta$ . In the second one, each qubit passed through a depolarising channel. Our algorithm and analysis also works for both of the noise models proposed by Cross et al. (CrossSS15).

The algorithm is the same as in the previous section, where now the QFT is over $\mathbb{F}_{2}$ (also called the Hadamard Transform H).

Lemma C.1.

Let $\frac{1}{\sqrt{2^{n}}}\sum_{a\in\{0,1\}^{n}}\left|a\right\rangle\left|a\cdot s+e_{a}\pmod{2}\right\rangle$ be a quantum sample where $e_{a}$ are iid random variables with value [math] with probability $1-\eta$ and $1$ with probability $\eta$ .

For every constant $0<\delta<1$ , applying a Hadamard transform on all qubits and measuring them in the computational basis, provides an outcome $\left|j\right\rangle\left|j^{*}\right\rangle$ , where $j\in\{0,1\}^{n}$ and $j^{*}\in\{0,1\}$ such that with probability exponentially close to $1$ , $Pr\left[j=s\right]\geq\frac{1}{2}(1-\delta)^{2}(1-2\eta)^{2}$ .

Proof.

If we apply Hadamards on each qubit of the sample state, we have

[TABLE]

We now calculate the probability that $j^{*}=1$ and the first qubits are in the state $\left|s\right\rangle$ :

[TABLE]

From the distribution of each $e_{a}$ , we have that $(-1)^{e_{a}}$ is $1$ w.p. $1-\eta$ and $-1$ w.p. $\eta$ , independently. Therefore $\mathbb{E}\left[(-1)^{e_{a}}\right]=1-2\eta$ and using Hoeffding’s bound we have that

[TABLE]

Therefore, with probability exponentially close to $1$ , the probability that $j=s$ is at least

[TABLE]

We can test a candidate solution $\tilde{s}$ in the lines of Lemma IV.2 and then repeat the process a linear number of times, and in this case the algorithm can find the right $s$ with probability exponentially close to $1$ .

C.2 LWE over rings

The Ring-LWE problem(LPR10), a variant of LWE over the ring of polynomials, has been proposed in order to improve the performance of cryptographic constructions using LWE, at the cost of needing stronger assumptions for proving its hardness.

The Ring-LWE problem uses the structure of the ring $\mathcal{R}_{q}={\raisebox{1.99997pt}{$ \mathcal{R} $}\left/\raisebox{-1.99997pt}{$ q\mathcal{R} $}\right.}$ for a prime $q$ , $\mathcal{R}={\raisebox{1.99997pt}{$ \mathbb{Z}[x] $}\left/\raisebox{-1.99997pt}{$ f(x) $}\right.}$ and a cyclotomic polynomial $f(x)$ . As in LWE, a Ring-LWE sample is the pair $(a,as+e\pmod{q})$ for random $s,a\in\mathcal{R}_{q}$ and $e$ is picked according to some error distribution $\chi$ .

Unfortunately, our algorithm cannot be used to solve Ring-LWE with the noise model proposed by Bshouty et al. (Bshouty95), due to technical issues on representing the polynomials. In order to use the quantum learning algorithm for LWE, we need to find an isomorphism $\phi$ from $\mathcal{R}_{q}$ to $\mathbb{Z}_{q}^{n}$ , where $n=\varphi(m)$ is the number of invertible elements modulo $m$ . With this isomorphism, we can consider a sample $(a,as+e)\in\mathcal{R}_{q}^{2}$ as two vectors in $\mathbb{Z}_{q}^{n}$ , and a superposition of quantum states representing these vectors can be written as:

[TABLE]

and applying the QFT over every register of this state results in

[TABLE]

where the second equality holds because $\phi$ is a homomorphism.

We consider two ways of representing elements in $\mathcal{R}_{q}$ as integer vectors. The first one consists of identifying a polynomial in $\mathcal{R}_{q}$ with the vector containing its coefficients. However, this coefficient embedding is not a homomorphism to $\mathbb{Z}_{q}^{n}$ , and the following identity, used in eq. 3, does not hold

[TABLE]

Therefore, this representation of polynomials cannot be used within our learning algorithm.

The second way of representing a polynomial is through the map

[TABLE]

where $\omega_{m}\in\mathbb{Z}_{q}$ be a primitive $m$ -th root of unity. This map is particularly interesting since multiplication $\mathbb{Z}_{q}^{n}$ is done component-wise(LPR13) and therefore it can be used in implementations of Ring-LWE with efficient multiplication(Mayer16). However, in these constructions, the error is sampled from a distribution over polynomials with small coefficients and when after applying the isomorphism, $\phi(e_{a})$ can be arbitrarily large in $\mathbb{Z}_{q}^{n}$ which cannot be handled by our algorithm if the error is independent for each element in the superposition.

Finally, we show now how to do solve Ring-LWE for the error model presented in Cross et al. (CrossSS15), namely, the noise is the same for all elements in the superposition.

Let $\phi$ be any isomorphism from $\mathcal{R}_{q}$ to $\mathbb{Z}_{q}^{n}$ . We can map the original quantum sample using $\phi$ resulting in

[TABLE]

and using the Field Bernstein-Vazirani algorithm on this state we have

[TABLE]

By measuring the last register, the error becomes a global phase and we are able to retrieve $s$ as shown in Section IV.1.

C.3 Learning with Rounding

LWE has been used in the construction of several cryptographic primitives. However, its usage sometimes is limited. For instance, in the implementation of pseudo-random functions, the output must use little or no randomness, which does not correspond to the inherent randomness in LWE’s input.

For this purpose, Banerjee, Peikert and Rosen(BPR11) proposed a derandomized version of LWE called Learning with Rounding (LWR), which does not compromise hardness. LWR has been used in the construction of pseudo-random functions (BPR11) and deterministic public key encryption (XXZ12).

The main idea of LWR consists in replacing $a\cdot s+e_{a}$ by the ’rounding’ of $a\cdot s$ with respect to some modulus $p\ll q$ , which can be seen as a “deterministic noise”. More precisely, the rounding function is defined as follows:

[TABLE]

An LWR sample is then given by $(a,\left\lfloor a\cdot s\right\rceil_{p})$ for some $a$ sampled from the uniform distribution on $\mathbb{F}_{q}^{n}$ .

Corollary C.2.

Let

[TABLE]

be a quantum LWR sample. Let $\left|\phi\right\rangle$ be the state when we multiply the last register of $\left|\psi\right\rangle$ with $\frac{q}{p}$ . The Field Bernstein-Vazirani $(\left|\phi\right\rangle)$ outputs $s$ with probability at least $\frac{p}{12(q-1)}$ .

Proof.

For a fixed $a$ , we have that

[TABLE]

Since $\frac{-q}{2p}\leq\frac{q}{p}\left\lfloor a\cdot s\right\rceil_{p}-a\cdot s\leq\frac{q}{2p}\pmod{q}$ , the result follows by Theorem IV.1 for $k=\frac{q}{2p}$ . ∎

C.4 Quantum Samples for SIS problem

We present in this section a learning algorithm for another relevant problem in cryptography, the Short Integer Solution problem. As the name indicates, the Short Integer Solution problem (SIS) consists in finding a short integer solution for a system of linear equations, and we present now its formal definition.

Definition C.3 (Short Integer Solution).

Given a random matrix $A\in\mathbb{F}_{q}^{m\times n}$ , a random vector $z\in\mathbb{F}_{q}^{m}$ , the SISn,m,q,β problem is to find a vector $x\in\mathbb{F}_{q}^{n}$ such that $Ax=z\pmod{q}$ with $\left\|x\right\|<\beta$ .

As in the LWE case, the hardness of SIS is also proved through the reduction of (expected to be) hard lattice problems (Ajtai96)(MicciancioR07)(GentryPV08)(MicciancioP13). We remark that if we drop either the constraint of having an integer solution or having a short solution, the problem can be easily solved using Gaussian Elimination.

The SIS problem and its variants have been used to prove security of constructions of signature schemes (Ly12)(DDLL13), and hash functions (LMPR08). In these schemes, samples in the form $(A,Av)$ are public, where $v$ is a small random vector and $A$ is a random matrix.

Inspired in the LWE case, we can define a quantum sample for SIS problem as

[TABLE]

and we are interested in the sample complexity of finding the (fixed) short solution $v$ . Using Field Bernstein-Vazirani brings the same problem of Gaussian Elimination: there is no guarantee of finding a short solution instead of an arbitrary one.

We notice that tracing out $m-1$ rows of $A$ and the corresponding positions of $Av$ , we remain with

[TABLE]

and we show an algorithm that works even for this type of quantum sample.

The algorithm consists by testing all possible values $j\in\{-k,...,k\}$ of $-v_{i}$ . The test on $j=-v_{i}$ passes with probability $1$ , while the test rejects with constant probability for $j\neq-v_{i}$ . By repeating the test $L$ times, the probability of finding the correct value is amplified.

**SIS Algorithm(L)

Input:** $L\in\mathbb{Z}^{+}$

Output: $\tilde{v}\in\mathbb{F}_{q}^{n}$

For $i\in[n]$ do:

For $j\in\{-k,...,k\}$ do:

For $l\in[L]$ :

Pick a quantum sample $\frac{1}{\sqrt{q^{n}}}\sum_{a\in\mathbb{F}_{q}^{n}}\left|a\right\rangle\left|a\cdot v\right\rangle.$

Add $ja_{i}$ to the last register

Apply QFT on the $i$ -th qudit of $a$ and measure it

Test next value of $j$ if outcome is not $\left|0\right\rangle$

Set $\tilde{v}_{i}=-j$ and continue with the next value of $i$ .

Output $\tilde{v}$

Theorem C.4.

Let $v\in\mathbb{F}_{q}^{n}$ whose coefficients are all smaller in absolute value than some bound $k$ . Given the quantum samples in the form

[TABLE]

SIS Algorithm $(L)$ outputs $v$ with probability $1-\frac{2km}{q^{L}}$ .

Proof.

We start by doing the analysis of SIS algorithm for $i=1$ . After adding $ja_{1}$ to the last register of the quantum sample, we have

[TABLE]

If $j=-v_{1}$ , then the previous state is the product state

[TABLE]

and since $QFT\sum_{a_{1}\in\mathbb{F}_{q}}\left|a_{1}\right\rangle=\left|0\right\rangle$ , the test passes for all $l\in[L]$ .

On the other hand, if $j\neq-v_{1}$ , then the state is entangled, and the reduced density matrix of the first register is

[TABLE]

In this case, after applying the QFT on the first register and measuring it, the output is $\left|0\right\rangle$ with probability $\frac{1}{q}$ . Therefore, we have $\tilde{v}_{1}=-j$ if for all $L$ independent samples the measurement outcome after the QFT is $\left|0\right\rangle$ , and this happens with probability $\frac{1}{q^{L}}$ . By the union bound, the probability that the test passes for any value $j\neq-v_{i}$ is at most $\frac{2k}{q^{L}}$ .

Finally, the previous analysis holds for every $i\in[n]$ . Since $v\neq\tilde{v}$ iff there exists an $i\in[n]$ such that $\tilde{v}_{i}\neq v_{i}$ , we can use union bound again to show that this happens with probability at most $\frac{2km}{q^{L}}$ . ∎

By picking $L=\max\{1,\frac{\log\frac{2km}{\eta}}{\log q}\}$ , the algorithm outputs the correct $v$ with probability at least $1-\eta$ .

TL;DR

Contribution

Findings

Abstract

Peer Reviews

Videos

Learning with Errors is easy with quantum samples

Abstract

I Introduction

II The quantum learning model

III Our contributions

III.1 Related work

III.2 Relation to LWE-based cryptography

IV Algorithm for LWE

IV.1 Quantum algorithm for learning a linear function without error

IV.2 Analysis of the algorithm for noisy samples

Theorem IV.1**.**

Proof.

Lemma IV.2**.**

Proof.

Theorem IV.3**.**

V Open problems

V.1 Generalizing from linear functions

V.2 LWE over rings

Acknowledgments

References

Appendix A Notation

Appendix B An efficient quantum learning algorithm for LWE

Theorem B.1**.**

Proof.

Appendix C Quantum learning complexity of related problems

C.1 Learning parity with noise

Lemma C.1**.**

Proof.

C.2 LWE over rings

C.3 Learning with Rounding

Corollary C.2**.**

Proof.

C.4 Quantum Samples for SIS problem

Definition C.3** (Short Integer Solution).**

Theorem C.4**.**

Proof.

References

Theorem IV.1.

Lemma IV.2.

Theorem IV.3.

Theorem B.1.

Lemma C.1.

Corollary C.2.

Definition C.3 (Short Integer Solution).

Theorem C.4.