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11 votes
Accepted

Number of LWE samples in NewHope

They're actually sampling $5n$ elements from $\Psi_{16}$. Perhaps Protocol 2 on page 5 shows this most clearly, where $\textbf{s}, \textbf{e} \stackrel{\$}{\leftarrow} \Psi_{16}^n$ and $\textbf{s}', \...
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11 votes

NewHope and NIST's Post-quantum standardization

From Status Report on the Second Round of the NIST Post-Quantum Cryptography Standardization Process 3.12 NewHope NewHope is a KEM based on the presumed hardness of the RLWE problem. At its core is ...
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8 votes
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Are LPN and LWE problems equivalent?

Yes, LPN is (essentially by definition) equivalent to the hardness of decoding a random linear code over $\mathbb{F}_2$. No, there is no known reductions between LPN and LWE. It is usually believed ...
7 votes
Accepted

Pseudorandomness of ring learning with errors

Let $R$ be the ring $\mathbb{Z}_p[X]/(X^n+1)$, where $n$ is a power of 2. The Ring-LWE assumption says that for any randomly chosen, fixed $s\in R$, the distribution of $$((a_1,a_1s+e_1),\ldots,(a_k,...
  • 1,126
7 votes

Ring-LWE in other fields

We don’t always use power-of-two cyclotomics for RLWE. Many cryptosystems use other cyclotomics, or subfields thereof, or even other fields altogether. For example, many FHE schemes use non-two-power ...
7 votes

Distribution of the Difference of Uniformly Random Elements

Firstly, why is this true? This is easy to see, if we consider the finite field as a finite group with the addition operation (and ignore the multiplication operation) If we consider the value $X - Y$...
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6 votes
Accepted

Why is Ring-LWE more efficient compared to LWE?

The previous (galvatron's) answer gave two good reasons why Ring-LWE is more efficient. They explain why the running time of LWE schemes is worse than of Ring-LWE ones. (The n^2 vs n equations is ...
  • 1,126
6 votes
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MLWE (and RLWE) to LWE reductions proof

There is no known reduction from LWE to MLWE (or to RLWE). That is, it could be that both MLWE and RLWE are broken, yet LWE is secure. However, this seems highly unlikely. To support the security of ...
5 votes
Accepted

Minimum distance between polynomials in ring-LWE

I'm assuming $n$ is a power of $2$ and that $q$ is an odd prime larger than $n$. I'm discarding the trivial case $s_1 = s_2$. If you consider everything $\mod q$, then it is most likely over the ...
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5 votes
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Parameters for LWE

As mentioned in the comments, there is still a lot of active research being done in the area of algorithms for solving Learning With Errors, as clearly it's an important topic for estimating the ...
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5 votes
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Understand LWE(Learning With Error) negligible error probability

Denote by $X$ the random variable which is the sum over all $S$. As mentioned, this is a Gaussian of standard deviation at most $\sqrt{m}r$ with $r = \alpha q$. Hence, by properties of the (sub-)...
  • 3,812
5 votes

Understand LWE(Learning With Error) negligible error probability

The probability of error is negligible "as a function of $n$", meaning that the probability of error will decrease (quickly) as $n$ grows. Increasing $n$ should solve your issue.
  • 1,119
5 votes
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Hardness of LPN problem with small secret

There is a simple trick (known in the LWE literature as the Hermite normal form of the problem) that takes an existing LPN problem and transforms it into a problem in which the secret has the same ...
  • 11.9k
5 votes
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Distribution of the Difference of Uniformly Random Elements

It's worth mentioning that the conditions needed for $f(X_0, X_1)$ to be uniformly random based off the distributions of $X_0, X_1$ are quite mild usually. In particular what you need is: $X_0$ and $...
  • 8,777
5 votes

What is the difference between Poly-LWE and Ring-LWE?

One main difference is that in Ring-LWE, the ring $R$ is the full ring of integers $\mathcal{O}_K$ of a number field $K$, whereas in Poly-LWE it is of the form $R=\mathbb{Z}[x]/f(x)$ for some ...
4 votes
Accepted

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?

I believe it is also used in other lattice based schemes that use standard LWE. For example, the Frodo paper. They used a $seed_A$ and a Gen($\cdot$) function to compute $A$. Then Alice sends $seed_A$ ...
4 votes
Accepted

Why does Learning With Errors require a bunch of samples?

The number of rows in matrix $\textbf{A}$ shows the number of LWE samples and $m=1$ means the adversary has access only to one sample. This would be a naive adversary model and would not be suitable ...
  • 990
4 votes

A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem

This lemma is used to conclude that a sample from $\mathcal{D}_{\mathbb{Z}^n,\alpha q}$ is (with overwhelming probability) less than or equal to $\alpha q \sqrt{n}$. Now, because all values $\textbf{...
  • 990
4 votes
Accepted

Security estimation of LWE using "On dual lattice attacks against small-secret LWE and parameter choices in HElib and SEAL"

I asked the question to the author directly. To answer the first question, authors of NewHope estimate their security very conservatively, whereas the estimator takes many other things into account. ...
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4 votes
Accepted

LWE with secret matrix (Reverse LWE?)

The LWE assumption tells you that $(\mathbf{a},\mathbf{a}\mathbf{s}+e)$ looks random for a hidden random vector $\mathbf{s}$, a random vector $\mathbf{a}$ and small error $e$. Invoking the LWE ...
4 votes

what does output parameters of lwe estimator stands for?

δ_0: the root Hermite factor required β: the BKZ block size d: the dimension of the lattice being reduced m: the number of LWE samples used
4 votes
Accepted

Is Type I lattice trapdoor hard to find even given oracle access to compute inverse of trapdoor function?

If the adversary is a classical algorithm, then the answer to your question is not known. But if the adversary is a quantum algorithm that can query the oracle in superposition, then the answer is ...
4 votes

LWE: Round a continuous Gaussian to a true Discrete Gaussian

Out of curiosity, what is the current state of the art on the sampling over $D_{\mathbb{Z},\alpha q}$ This is a fairly involved question to answer. There are a number of competing ways to sample it, ...
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4 votes
Accepted

Why does bootstrapping (R)LWE homomorphic encryption produce small noise?

The output of bootstrapping has relatively small noise because it starts from an encryption (of the secret key) that has very small noise, and performs some homomorphic operations on it. These ...
4 votes
Accepted

*-LWE equivalent of Diffie-Hellman $g^{x^2}$ vulnerability

Given $(A, Ax + e)$ and $(A, x^tA+e')$, you can do (at least) one potentially interesting thing to solve LWE. Namely, compute the sample $$(A+ A^t, Ax + e + (x^tA+e')^t) = (A+A^t, (A + A^t)x + e + {e'}...
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4 votes
Accepted

LWE and pseudorandom functions

You can. There is a certain caveat that should be mentioned here --- the LWE problems hardness is controlled (in part) by the size of the modulus $q$. Two important parameter regimes are $q$ being ...
  • 8,777
4 votes
Accepted

RLWE Explanation

The cyclotomic polynomials are used in the proofs that worst-case lattice problems reduce to the RLWE. If you try to instantiate RLWE with other polynomials, then you don't have such formal guarantees ...

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