Questions tagged [gaussian-noise]

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A smaller modulus-to-noise ratio means more security in LWE

Let $\text{Adv}^{\text{DLWE}}_{n,m,q,\sigma}$ be the advantage of an attacker to distinguish LWE samples from uniform ones, where $m$ is the number of samples, $q$ the modulus and $\sigma$ the ...
2
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1answer
61 views

How is it legal to use a rounded Gaussian for LWE?

As far as I understood, in Regev's initial paper, the error distribution was first constructed as follows: Then rounded in the following way: Using this distribution, the reduction in the theorem ...
0
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0answers
146 views

Is the error distribution in Learning with Errors (LWE), the discrete Gaussian distribution?

In $\mathbb{Z}$, the discrete Gaussian distribution is defined as $D_{Z,s}(x) = \frac{\rho_s(x)}{\rho_s(\mathbb{Z})}, x\in \mathbb{Z}$. In LWE, $(\overrightarrow{a}, b = \langle \overrightarrow{a}, \...
1
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1answer
66 views

Is the discretization of the Guassian distribution on torus still a discrete Gaussian distribution?

Let $\rho_s(x) = e^{-\pi x^2/s^2}$ be the Gaussian measures, then the discrete Gaussian distribution on $\mathbb{Z}$ could be defined as $D_{\mathbb{Z},s}(x) = \rho_s(x)/\sum_{n\in \mathbb{Z}}\rho_s(n)...
3
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1answer
123 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 ...
6
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1answer
308 views

Differential privacy guarantees of Gaussian noise, when each coordinate has different sensitivity

Suppose you have a function $f$ that takes a dataset $D$ as input and returns an output in $\mathbb{R}^d$. If this function has $L^2$-sensitivity $\Delta$, then the analytical Gaussian mechanism (...
2
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1answer
111 views

Standard deviation of gaussian noise in FHEW scheme

I've got two questions regarding the paper FHEW: Bootstrapping Homomorphic Encryption in less than a second. First, the final error of a ciphertext after the refresh procedure is stated as following ...