Questions tagged [lwe]

Learning with Errors is a form of lattice problem used in the design of cryptographic primitives. LWE is based on the Closest Vector Problem (CVP).

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Fully Homomorphic Encryption - state of the art

What are the latest advances in fully homomorphic encryption? First of all, I am interested in cryptosystems based on LWE / RLWE and NTRU problems.
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Security of somewhat homomorphic encryption via LSB encoding?

I'm reading this paper https://eprint.iacr.org/2011/344.pdf It says that "The secret-key encryption scheme whose security is based on the LWE assumption is rather straightforward. To encrypt a bit, $...
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Is there an adaptive version of LWE assumption with respect to some potentially non-uniform secret distribution?

There is a version of LWE assumption as follow. Assume that there is a positive number $n$, an integer $q = q(n) \geq 2$, an error distribution $\chi = \chi_{n}$, a vector $\mathrm{\mathbf{s}} \gets \...
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Solving for secret s in LWE problem

in LWE instance $b=A^t s +e$, can we find an orthogonal basis of coefficient matrix A in polynomial time, let it be B.Multiply B to get $Bb=Be$ as term with s will vanish. Then solve for $e$. After ...
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Is LPN not as important as LWE and SVP?

I've been learning about lattice cryptography and have noticed that most resources such as this survey by Chris Peikart, the Winter School on Lattice Cryptography etc don't include material on LPN, ...
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ZKPoK for RLWE secret and error

I came across How to validate the secret of a Ring Learning with Errors (RLWE) key paper by Ding et al., which seems to provide a ZK proof that the given $p$ is of the form $as + e$ with $s, e$ small ...
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Choosing the Bernoulli distribution for LPN encryption scheme

The symmetric-key encryption scheme from [1] is based on the LPN (learning parity with noise) problem. The definition of the problem is, informally, that the adversary cannot recover $\mathbf{s}$ from ...
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LWE secure with one entry without noise

I'd like to know, is Learning With Error (LWE) (with modular noise) "secure" if one entry has no noise? More precisely, I have: a random matrix $A \in \mathbb{Z}_q^{m \times n}$ a random string $s \...
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Why round off vector sampling from continuous Gaussian distribution not directly sample from discrete Gaussian distribution

For any vector $\mathbf{c}$, real $s > 0$, and lattice $\Lambda$, define the probability distribution $D_{\Lambda, s,\mathbf{c}}$ over $\Lambda$ by $$D_{\Lambda, s,\mathbf{c}}(\mathbf{x})=\frac{...
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Why does Learning With Errors require a bunch of samples?

Solving Learning with Errors(LWE) with average case complexity is as hard as solving the SVP with worst case complexity. LWE requires $n$ dimensional lattice and $m$ samples of it, and Decisional-LWE ...
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How to solve a simple case of a RLWE problem

I've been reading up on the Ring Learning with Errors problem and the proposed attacks, in relation to homomorphic encryption. Some of the literature has been quite difficult to understand - what I ...
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How can I geometrically understand LWE ciphertext and decryption step?

In the bottom of the wikipedia article of LWE (https://en.wikipedia.org/wiki/Learning_with_errors), we can see construction of Public-key cryptosystem based on the LWE. But, I cannot understand whole ...
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Number of LWE samples in NewHope

This is regarding the number post-quantum key exchange protocol New-Hope (https://eprint.iacr.org/2015/1092.pdf). In the paper, we can see that the number of samples generated by the protocol is $2n$ ...
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What is the purpose of adding secondary error to calculated key in BCNS and NewHope protocols

The answer to this question might be trivial or very short, but I would like to ask it anyway. In both BCNS and NewHope Ring-LWE key-exchange protocol one party adds a secondary error to their ...
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How does the polynomial module impact the security of ring/lattices-based SIS problem?

Consider the following SIS problem: for a function $f_A(s)$=$As$, where $A$ is a fixed, randomly-chosen matrix in $(R_q)^{r \times n}$=$\left(\mathbb{Z}_q[X]/(X^N+1)\right)^{r \times n}$ and $q$ a ...
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Probability of an RLWE sample

Let $R_q=\mathbb{Z}_q[x]/(x^n+1)$ as usual in the RLWE assumption. Suppoes that I choose a sample of the RLWE distribution, that is, I compute $(a,y=as+e)$ where $a$ is uniform in $R_q$ and $s,e\...
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Hardness of LPN problem with small secret

The Learning Parity with Noise (LPN) assumption states that, for a fixed secret $s$ chosen uniformly from $\{0,1\}^n$, then the distribution that outputs $(a,a\cdot s+e)$, where $a$ is uniform in $\{0,...
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Refreshing Procedure in FHEW: membership test

I am facing an issue regarding the paper FHEW: Bootstrapping Homomorphic Encryption in less than a second. It concerns the MSBextract algorithm during the refresh procedure. Especially, they ...
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Does there exist trapdoor permutation from lattices?

It seems that the lattice functions are either surjective (SIS) or injective (LWE), due to the error that is basically intended to destroy the structure and provide security. I was wondering whether ...
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Adapting LWE Trapdoors for Ring-LWE

In the paper Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller by Micciancio and Peikert, they present the following theorem about the existence of trapdoor for LWE. Theorem 5.1: There is an ...
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Are LPN and LWE problems equivalent?

Learning with Error (LWE) problem seems like a generalization of Learning Parity with Noise (LPN) problem, where in the latter one uses bits. But, this also makes LPN seem very related to the problem ...
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Minimum distance between polynomials in ring-LWE

Let $R_q=\mathbb{Z}_q[x]/\langle f(x)\rangle$ where $f(x)=x^n+1$, as in the ring-LWE problem. Let $a(x)$ be chosen uniformly at random from $R_q$. Question: Is there any theorem that lower bounds ...
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How to explain Learning With Errors? [closed]

I am trying to understand this concept of Learning With Errors. There does not seem to be a layman explanation of it anywhere. Here I describe layman as someone who understands ML concepts a bit (non ...
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Gate count for Ring-LWE

In one of my recent research, I need to compare the number of gates needed in Ring-LWE encryption to AES in the circuit model. For AES we can find some estimation like this site, but for Ring-LWE we ...
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Probability distribution function in Regev Cryptosystem

In Regev - On Lattices, Learning with Errors, Random Linear Codes, and Cryptography, chapter 5, Public Key Crypto System, it is stated that The probability distribution function $\chi$ is taken to ...
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Encoding of the message in Regev encryption

In public key encryption from LWE, we do the following steps $\textbf{PKE.KeyGen($1^n$)}$ takes as input the security parameter n, samples $A \leftarrow \mathbb{Z}_p^{n \times m}$ and $\textbf{e} \...
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A query on Learning with errors(LWE) problem

In generating an LWE sample, we do $s\xleftarrow{$}\mathbb{Z}_q^{n}, A \xleftarrow{$}\mathbb{Z}_q^{n \times m}~$and $e\xleftarrow{$}\mathbb{{\chi}^{m}}$ Then we compute $b^T$ = $s^TA$ + $e^T$ and ...
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Decision-LWE to Search-LWE

Regev requires $q$ to be prime on lemma 4.2 of his paper for LWE. Why does he require that and how this effect the proof of lemma 4.2?
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A RLWE promise problem

Let $(R , \chi$) be a standard RLWE problem instance. I.e. $R$ is a finite degree polynomial ring over a finite field and $\chi$ is some gaussian distribution over R with small variance. I wonder ...
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What are general circuits exactly?

This may be a silly question but while reading this paper Circuit ABE from LWE, I came across this question on 2nd page - Is there an ABE scheme for general circuits with unbounded attribute length ...
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LWE with secret matrix (Reverse LWE?)

I was wondering if there is a version of LWE with secret matrices and public seed vectors? Would it be as hard as the popular definition of LWE?
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Interpreting Figure 1 in the “On Lattices, Learning with Errors, Random Linear Codes, and Cryptography”

I'm reading "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography" by O. Regev. I'm having trouble with understanding graphs in Figure 1. By the definition of $\overline{\psi}_{\...
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Is it secure using LWE-based cryptosystem under RLWE-based parameters?

I'm computer guy having trouble with cryptography. I recently read the BGV Homomorphic encryption paper which was constructed under both LWE and RLWE assumptions. I was implementing Threshold ...
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Does secure LWE implementation leak bit information?

We know RSA leaks one bit about the factors and improper yet secure implementations of Discrete Logarithm leak one bit about the discrete logarithm. Does LWE leak any information?
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Why don't we use an Extendable Output Function to efficiently store the public key of Regev's LWE-based encryption scheme over standard lattices?

In LWE-based schemes the public key is generated by choosing a random matrix (or polynomial) $A$, and outputting the pair $(A, b = A\cdot s + e)$, where $s$ and $e$ are vectors/polynomials with ...
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R-LWE instantiation with non-power of 2 polynomial

In almost all RLWE papers, the polynomials are chosen from a ring $\mathbb{Z}[x]/f(x)$ where $f(x)$ is a polynomial of the form $f(x)=x^{2^n}+1$. That leaves us the choice of polynomials like $x^{256}+...
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Distinguishing advantage in terms of $\Omega(\omega)$

I was going through this paper ("Fully Homomophic Encryption over the Integers Revisited") and the statement written in the first paragraph on page 5 stating $\DeclareMathOperator{\agcd}{AGCD}$ We ...
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ring-LWE: Minkowski Embedding , the Co-Different Ideal, etc

While (trying) to go over the reductions from approx. SVP on ideal lattices to search ring-LWE, [1] and [2], for $K = \mathbb{Q}(\zeta)$ where $\zeta$ is an abstract root of a cyclotomic polynomial, ...
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A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem

In this paper (A simple provably secure key exchange by Ding et al.) At page number 8, the author gives correctness of the technique as follows then SK A = SKB with overwhelming probability i.e. if ...
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Gaussian elimination to equations with errors

I am reading this document and wondering the following part on page 13: "Consider applying Gaussian elimination to the noisy samples to find the first bit" If we take, for example, $n = 3$, $s = (...
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show how LWE errors can have a greater impact on result

Hi weve been given the following question in one of our classes but have not been taught anything about it and is worded strangely. It is to show how the LWE problem works by showing how small errors ...
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Effect of small secret attacks on non homomorphic encryption schemes

The new paper by Albrecht describes a new attack on "unusually" small secrets that are used in homomorphic encryption schemes. In the paper the talk about binary secrets or LWE Normal form i.e $\...
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Batching in LWE based Crypto-System

Is batching possible for a LWE based crypto-system like we do for a RLWE based one?
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Security analysis of LWE with unequal error and secret distribution

Analysis of security of recent LWE based Key-exchange schemes, the error and secret vector is always chosen from the same Gaussian distribution. What will be the impact on the security if $\sigma_s\...
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Pseudorandomness of ring learning with errors

My question is in Ring Learning with Errors, let $a(x)\in \mathbb{Z}_q(x)/(X^n+1)$ where $n$ is a power of $2$, be a random polynomial, $s(x),e(x)\in \mathbb{Z}_q(x)/(X^n+1)$ are the secret and error ...
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CCA2-secure encryption based on LWE/Ring-LWE

As I understand it, cryptosystems based on LWE/Ring-LWE are typically only IND-CPA, and fail to satisfy IND-CCA2. In fact, it appears that chosen ciphertext attacks could allow secret key recovery (!)...
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A Simple Provably Secure Key Exchange Scheme Based on the LWE

In this paper (A simple provably secure key exchange by Ding et al.), I am trying to understand the correctness (which is given on page number 8) of the key exchange technique based on LWE. To ...
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Security estimation of LWE using “On dual lattice attacks against small-secret LWE and parameter choices in HElib and SEAL”

I recently found this paper eprint which estimates security of LWE instantiations using an improved Dual attack. However, I got confused looking at their example in this site. For example, in the case ...
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LWE: error and float operations

Background I'm trying to make sense of the error in implementations of LWE and R-LWE. In LWE and R-LWE error is added to vectors in lattices to make it computationally infeasible to recover any ...
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How the condition $s \geq 8$ is determined in Lindner-Peikert cryptosystem?

In Lindner & Peikert paper, the authors propose that to set the cryptosystem's parameters, one should choose $q$ to be large enough to allow for a Gaussian parameter $s \geq 8$. My question is, ...