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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What if the bitlength of the value evaluated in Barrett reduction is greater than 2k the modulus?
For $c\equiv a \pmod n$, in Barrett Reduction, $\mu = \lfloor{\frac{2^{2k}}{n} \rfloor}$ is precomputed, where $k = \lceil{\log_2{n}} \rceil$ and the bitlength of $a$ is assumed to be less than $2k$. ...
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Why does bootstrapping (R)LWE homomorphic encryption produce small noise?
Why does homomorphic evaluation of the decryption circuit produce a ciphertext with "fresh" or small noise?
Rough description of bootstrapping homomorphic encryption:
Suppose we have a ...
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What does the “scale invariant” mean in some FHE schemes?
In some paper about FHE, the term "scale invariant" often appears. What does it means?
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LWE: Round a continuous Gaussian to a true Discrete Gaussian
Short version: how is it possible to round a continuous Gaussian into a true discrete Gaussian (usually denoted $\mathcal{D}_{\mathbb{Z},\alpha q}$)? The goal is to obtain a reduction from continuous ...
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LWE with identity sub-matrix and reused sampled from [MP12]: why is it secure?
I studied this paper a while ago, but now I'm confused by the paper Trapdoors for Lattices:Simpler, Tighter, Faster, Smaller by Micciancio and Peikert. Page 24 and 25, they present an algorithm that ...
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59 views
Computational LWE-Trapdoor without tag
In the paper Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller, Micciancio and Peikert mention that it is possible to save an additive $n$ term in the dimension $\bar{m}$ in paragraph $\...
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47 views
How small can the error be in LWE?
For modulus $Q$ and stddev $\sigma$, [GHS12] suggests that, to achieve 128-bit security, just choose the dimension $N$:
$$
N\geq(Q/\sigma)\cdot 33.1
$$
This seems to suggest flexibility to choose ...
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34 views
Module LWE with an even modulus
Does module-LWE remains hard for an even modulus $q$, or a power of two?
This is true for Ring-LWE (pseudorandomness) and Module-LWR (SABER).
I can't find any reference to it!
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Is there a LPN estimator like LWE?
I recently studied the LPN problem and BKW algorithm for solving LPN problem. Several papers deal with the concrete parameters of LPN to be secure against BKW algorithm, but I cannot measure the ...
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Advantage of distinguishing LWE using short vector in dual lattice
In analyzing the advantage of distinguishing LWE via dual attack, many papers uses the result of [LP11] as follows :
Given an LWE instance parameterized by $n$, $\alpha$, $q$ and a vector
$\mathbf{v}$...
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1answer
91 views
Decrypting message as MSB in regev scheme
I was watching this FHE video and it define Regev encryption scheme as fallow :
kyegen:
sk : choose $t = (1,s)^t \in \mathbb{Z}_q^{n+1}$
pk = $A \in \mathbb{Z}_q^{m*(n+1)}$ random except $[A * t]_q$ ...
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Attacks on LWE when q is a power of 2
I am working on an LWE instance where q is a power of 2 and I'm wondering if there is any literature about attacks in this context, especially if there are any attacks which work significantly better. ...
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57 views
Solving RLWE modulo a prime ideal
Suppose you have the following set up for RLWE: $K$ is a cyclotomic field of degree $n$ over $\mathbb{Q}$, and $p\in\mathbb{Z}$ is a prime integer that splits as follows in $R = \mathcal{O}_K$: $p\...
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1answer
119 views
What is the difference between Poly-LWE and Ring-LWE?
I am often confused by Poly-LWE and Ring-LWE, always thinking that they are different names for the same thing. In some literature, Poly-LWE is a simplified version of Ring-LWE? What is the difference?...
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CKKS security estimation for Palisade
My question is rather practical and specific. I am trying to setup an efficient CKKS scheme in Palisade. To this end, the automatic choice for secure parameters has to be turned off and I rely on the ...
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66 views
Rounding function used in Saber Key Exchange
In Saber: Module-LWR based key exchange, the authors use a rounding function called $\textit{bits}$, defined (in page 3) as follows:
$bits(x, i, j)$, with $j \leq i$, gives $j$ consecutive bits of a ...
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37 views
Can we possible to solve the LPN problem using Arora-Ge algorithm?
I recently studied how to solve the LWE (learning with errors) using algebraic relations proposed by Arora and Ge (paper : New Algorithms for Learning in Presence of Errors).
As I understood, for LWE ...
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Security of LWE when a small leakage is allowed on the noise
Short version:
Are there some results that are known on the security of the Learning With Error problem when there are some leakages, notably on the noise? (In my case these leakages come from a bit ...
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NewHope and NIST's Post-quantum standardization
Where can I find NIST's reasoning to eliminate NewHope from the 3rd round of the post-quantum competition? I see all the lattice KEMs finalists are based on modules.
Is being a ring-based KEM ...
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1answer
61 views
Replay attacks and LWE
Just a small question. Since in LWE the error is rather small, is there a problem with replay attacks? What I mean is that if we use the typical scheme of Regev [1] to encrypt a vector m, but this ...
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MLWE (and RLWE) to LWE reductions proof
In crypto papers, cryptanalysis of MLWE/RLWE/etc. is often reduced to LWE. Why can we do this? Is there strict proof of such reductions?
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Guessing the Secret in RLWE Search-to-Decision
In On Ideal Lattices and Learning with Errors over Rings, the authors prove a search-to-decision reduction by guessing the RLWE secret $s$, and using the guess to transform a sample from $\mathfrak{q}...
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Distribution of the Difference of Uniformly Random Elements
In the search to decision reduction of 'On Ideal Lattices and Learning with Errors over Rings', the authors implicitly use the fact that the difference of distinct, uniformly random elements of a (...
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What are the prerequisites for understanding Lattice based Cryptography, LWE or RLWE based on SVP?
I'm new to Quantum Resistant Cryptography, so, I thought of diving into Lattice based crypto, LWE and ring LWE. I realise that the hard problem involving them is the "shortest vector problem"...
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Ring-LWE in other fields
Can someone please tell me why in R-LWE we always make use of Cyclotomic fields, and specially those with degree equals to a power of $2$?
Can we use another fields without losing in hardness of the ...
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Binomial distribution sampling - concrete example
Can anyone give me an explicit example of how one can samples from the binomial distribution defined in NewHope's paper? What is the difference of sampling from rounded Gaussian in practice?
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Difference between polynomial embedding and canonical embedding
Can anyone tell me the difference between working in the polynomial embedding for $R$-LWE, and working in the canonical embedding?
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Proof for, Vectors sampled from $D_{(L,r)}$ have Euclidean norm at most $r\sqrt{n}$ with a high probability
For any $n$-dimensional lattice $L$ and $r > 0$, a point sampled from $D_{L,r}$ has Euclidean norm at most $r\sqrt{n}$ except with probability at most $2^{-2n}$ (where $r$ refers to the standard ...
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LWE versus neural nets
It seems like that the construction of the LWE problem: $As + e = b$ resembles how neural nets work: $Ax + b = y$.
In LWE, we are given the problem instance $A$, and the product with errors $b$ and ...
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63 views
Hash chain based secret revealing using homorphic princples?
I have recently been looking into Homomorphic encryption and I am looking for a specific hash-based encryption/decryption scheme.
I don't need a full implementation but I am not sure if what I want ...
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1answer
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Hardware Gaussian random numbers for lattice-based cryptography
I have been recently reading about lattice-based cryptography.
I read that a key aspect of such protocols rely on added Gaussian noise on lattices, and which therefore require highly efficient and ...
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Is Type I lattice trapdoor hard to find even given oracle access to compute inverse of trapdoor function?
Consider the Type I lattice trapdoor in [GPV08]: https://eprint.iacr.org/2007/432.pdf
Suppose a PPT adversary is given the LWE trapdoor function in the picture:
$g_{A^\top} (s,e) = A^\top s + e = b (...
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Multibit LWE Encryption
What's the simplest way to encrypt multiple bits with LWE public key cryptosystem? Some paper say use a different secret key for each bit of the message. Does that mean the length of the message ...
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learning with errors is hard even when given some information about the secret vector
In this case we are given the distribution D from which secret s is sampled. If I can get some hints to proceed with the questions, it would be of help.
Thanks,
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Trying to Understand Ring Learning With Error Encryption
I'm trying to understand the following RLWE encryption scheme from this Chris Peikert's paper
Say i choose $q = 97$, $n=8$ and the polynomial
$a = 96 x^8+30 x^7+76 x^6+12 x^5+57 x^4+77 x^3+70 x^2+49 ...
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1answer
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Decision to Search LWE when modulus $q=p^e$
I am reading Applebaum et al..
In Lemma 1. (page 7), Applebaum et al. proved the decision to search reduction when the modulus $q=p^e$ for prime $p$.
In the proof, they define the hybrid ...
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1answer
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Randomness of Decision Learning With Error Problem
I read the statement of the Decision Learning with error problem is:
distinguish between $(\vec a, \langle \vec a, \vec s \rangle + e)$ from uniformly random samples.
Can anyone explain what does ...
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1answer
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understanding LWE public key algorithm
I'm trying to understand this LWE public key system
say I use matrix A = [[44, 73, 20, 54],[92, 19, 78, 22],[31, 34, 94, 29],[82, 32, 70, 68]]
q = 97
bit = 1
and secret key s: [56, 90, 0, 46]
and ...
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2answers
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what does output parameters of lwe estimator stands for?
I want to use lwe estimator to find classical and quantum security of my proposed key exchange protocol. On this website, I want to understand the output of sage code on lwe estimator given bellow.
...
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1answer
250 views
Threshold decryption in multi-key homomorphic encryption
I have a problem understanding the security of threshold decryption in multi-key homomorphic encryption (MKHE) with so called "noise flooding". In particular I think that it is not secure, so probably ...
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199 views
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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1answer
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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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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 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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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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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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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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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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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 ...
