# 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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### Decision R-LWE parameters for spherical error with worst-case hardness

In Peikert et al.'s most recent work (STOC 2017) a direct reduction of worst-case lattice problems to decision R-LWE is achieved for $\alpha q \ge 2 \cdot \omega(1)$ (Theorem 6.2), where $\alpha q$ is ...
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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,  and , for $K = \mathbb{Q}(\zeta)$ where $\zeta$ is an abstract root of a cyclotomic polynomial, ...
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### Is the ring learning with errors problem still hard if the errors are drawn from some subspace?

Let $R=\mathbb{Z}_p[x]/x^n+1$ be the ring used in normal RLWE, which is linear space over $\mathbb{Z}_p$ with dimension of $n$, let $S$ be a linear subspace of $R$ which described by linear ...
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### Why could the error term be sampled coefficient wise?

In SEAL homomorphic encryption library, it implements the BFV and CKKS. We know the error $e\in R_q$ which is a Guassian distribution. When sampling an error term $e = \sum_{i=0}^{n-1} e_ix^i$, it ...
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### How does the rejection sampling lemma work in the proof of HVZK?

In this protocol, Q1: how does the commitment work? What if the prover sends $\textbf{t}$ directly, and then sends $s_m,s_r,s_{\textbf{e}}$? Q2: How does the rejection sampling lemma work? refer to ...
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### Why should the smudge noise be used?

Consider a threshold FHE scheme based on RLWE like this: Refer to this paper $\textbf{Initialization:}$ Every party generates his own secret key $s_i$, then uses the common polynomial $a$ to generate ...
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### What is optimal error distribution in Ring-LWE?

I am new to Ring-LWE. I had assumed that error distribution in Ring-LWE (or, in any lattice-based cryptography) is always Gaussian. However, while reading a few research papers (e.g. page 5, Section "...
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### Comparison of NTRU-based schemes and LWE-based schemes

What advantages and disadvantages can be distinguished in NTRU-based and LWE-based schemes relative to each other? In what cases which scheme gives advantage? UPD: I'm interesting in two things: 1)...
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### What is the intuition of attacks on ring lwe?

I know that there are several attacks on ring lwe. But I am not sure why they work. Does anyone have intuition of such attacks? What is the common idea used? Reducing the Search space (modulus space)?...
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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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### 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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### CRT decomposition in NTT for BFV

AS explained in section 4.5, decomposition for relinearization (2nd step of ciphertext multiplication in bfv) can be done using RNS components of ciphertext. In other words, they proposed a way to do ...
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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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### 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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### How many ring-LWE samples are required for the (Search) Ring Learning With Errors problem to have a unique solution?

Consider the LWE distribution $\{(\pmb{a}_{i},\left<\pmb{a}_{i} , \pmb{s}\right> + e_{i})\}$ where secret $\pmb{s} \in \mathbb{Z}_{q}^{n}$, randomness is $\pmb{a}_{i} \xleftarrow{\$} \mathbb{Z}_{...
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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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### Canonical embedding vs. plaintext slots in Ring-LWE

I'm working on the canonical embedding mentioned in [LPR10] and [LPR13]. What confuses me is that the difference and the relationship between the canonical embedding and the concept of ''plaintext ...
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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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### Implementation of Post-quantum Authenticated Key Exchange from Ideal Lattices

I am implementing the Authenticated Key Exchange from Ideal Lattices on sage maths. On page number 3 the full key exchange scheme is presented. In this key exchange scheme, they are using the Hash ...