# Questions tagged [ring-lwe]

Ring learning with errors (RLWE) is a computational problem which serves as the foundation of new cryptographic algorithms, such as NewHope, designed to protect against cryptanalysis by quantum computers and also to provide the basis for homomorphic encryption.

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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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### Forging a new secret key in RLWE

In a RLWE setting where you are given a secret key $s$ and an associated public key $pk = (p_0,p_1) = (-(p_1s+e),p_1)$, is it possible/easy to forge a new secret key $s'$ such that $p_0+p_1s'$ has a ...
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### Ring-LWE instance with errors also in the public polynomial

Consider the ring $R_q = \mathbb{Z}_q[X]/(X^d+1)$, the Ring-Learning-With-Error assumption states that the distribution of $(a, as + e)$ is close to uniformly random, where $s \in R_q$, $a$ is uniform ...
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### Ring LWE: How can the secret be chosen from the “uniform distribution”?

The key generation algorithm for Ring-LWE is as follows. Have a ring $R_p =Z_p[x]/(x^n + 1)$. Then pick a uniformly random $a$ from $R_p$. Pick $s$ from an appropriate distribution. Pick $e$ from the ...
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### Complete Attack on RLWE Key Exchange with reused keys, without signal leakage

I am studying a research paper "Complete Attack on RLWE Key Exchange with reused keys, without signal leakage" . On page number 21 to 28, there is toy example explaining the scheme. I am unable to ...
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### Computational benefits of using exponent as a power of 2 in Ring-LWE

In most of the RLWE based cryptosystems, the parameter $n$, which defines the cyclotomic polynomial $\Phi_{n}(X)$, is chosen to be a power of $2$. Apart from other benefits such as ease of writing ...
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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\...