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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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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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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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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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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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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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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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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How to explain Learning With Errors?
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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1answer
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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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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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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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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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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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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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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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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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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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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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132 views
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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160 views
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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269 views
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, ...
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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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How to provide security against malicious adversary in Ring LWE homomorphic encryption scheme?
I was reading Ring LWE homomorphic encryption scheme referred in http://www.wisdom.weizmann.ac.il/~zvikab/localpapers/IdealHom.pdf
But this encryption scheme provides security mostly in the semi-...
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Why is Ring-LWE more efficient compared to LWE?
Can someone please tell me why is the Ring-LWE more efficient?
By introducing polynomials in place of matrices, what kind of optimizations do we introduce that make Ring-LWE more efficient?
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1answer
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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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LWE status versus modern deployed crypto [closed]
LWE is the one of the most promising post quantum strategy.
How does parameters (key sizes, time for encryption etc) for LWE compare with modern deployed standards such as Factoring or discrete log ...
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153 views
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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Understand LWE(Learning With Error) negligible error probability
According to Regev's paper, p15
Correctness. Note that if not for the error in the LWE samples, $b-⟨a, s⟩$ would be either 0 or ⌊ q ⌋ depending on the encrypted bit, and decryption would always be ...
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Why in BCNS protocol both parties don't follow the exact same key extraction approach
Why in BCNC key-exchange protocol (page 8), Bob instead of:
$K_B \leftarrow \lfloor \bar v \rceil_{2q,2} \in \{ 0, 1\}^n$
do this instead:
$K_B \leftarrow rec(2bs', c) \in \{0, 1\}^n$.
or they are ...
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Security of Lindner-Peikert parameters for lwe
From Regev's paper in 2005, we know that for applying worst-case to average-case reduction to an lwe cryptosystem, one should use error distributions with standard deviation $s$ bigger than $2\sqrt{n}$...
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103 views
What is the purpose of adding secondary error to calculated key in BCNS and NewHope protocols
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 calculated ...
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599 views
Parameters for LWE
In the LWE problem there are two methods in order to choose secure parameters. The Lindner-Peikert and Micciancio-Regev method. The first method is an attack to BDD : find closest vector knowing the ...
