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Questions tagged [lattice-crypto]

Lattice-cryptography is the study and use of lattice problems applied to cryptography.

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Comparasion of NTRU-based schemes and RLWE-based schemes

What advantages and disadvantages can be distinguished in NTRU-based and RLWE-based schemes relative to each other? In what cases which scheme gives advantage?
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Concrete evidence for the asymptotics of $\lambda_1(\Lambda^\perp(A))$?

A recent eprint paper claims to bound $\lambda_1(\Lambda^\perp(\mathbf{A}))$ for $\mathbf{A}\in\mathbb{Z}^{n\times m}$, a uniformly random matrix, by $O(1)$, specifically by $4$. This has applications ...
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1answer
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Refreshing Procedure in FHEW: membership test

I am facing an issue regarding the paper FHEW: Bootstrapping Homomorphic Encryption in less than a second. It concerns the MSBextract algorithm during the refresh procedure. Especially, they ...
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Does there exist trapdoor permutation from lattices?

It seems that the lattice functions are either surjective (SIS) or injective (LWE), due to the error that is basically intended to destroy the structure and provide security. I was wondering whether ...
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1answer
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Help understanding lattice-based aggregate signature scheme

I came across this paper about aggregate lattice-based signatures, however, I'm not able to fully understand it. Specifically, I'm wondering if someone could help answer the following questions: In ...
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Hardness or negligibility of finding small non-trivial addition coefficients for random values to sum to zero

In my cryptographic scheme, I would like to rely on the hardness or negligibility of the following problem or situation, respectively. Note the original motivation: it shall be impossible to find two ...
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1answer
65 views

Are LPN and LWE problems equivalent?

Learning with Error (LWE) problem seems like a generalization of Learning Parity with Noise (LPN) problem, where in the latter one uses bits. But, this also makes LPN seem very related to the problem ...
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1answer
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Solutions to $\gamma \equiv \sum_{i=1}^m \xi_i\cdot x_i\bmod p$ with $|x_i| < \ell$

Are there any clear conditions on $p,\ell$ and $m$ under which the equation $\gamma \equiv \sum_{i=1}^m \xi_i\cdot x_i\bmod p$ has at most one solution with $|x_i|<\ell$ with high probability over ...
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Size of reduced bases of orthogonal lattice

I consider the following setting. Let $L$ be a lattice of rank $d$ in $\mathbb{Z}^m$ ($d\leq m$). The orthogonal lattice of $L$, denoted by $L^{\perp}$, is defined as the intersection of the ...
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1answer
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Rejection Sampling reasoning for Lattice Based Signatures

I'm new to lattices. According to Lattice Signatures and Bimodal Gaussians in the Rejection Sampling section. In Schnorr, GQ you can simply commit to $y$, use it to hide a secret key $s$. But this ...
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2answers
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How is the matrix A related to the lattice space L in SIS?

Is the matrix $A= (b_1|,...,|b_m)$ where B=$(b_1,...,b_m)$ is the basis of the lattice space, $L$(B)? Not sure if the answer is trivial however I'm having trouble seeing how SIS is a lattice hard ...
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Minimum distance between polynomials in ring-LWE

Let $R_q=\mathbb{Z}_q[x]/\langle f(x)\rangle$ where $f(x)=x^n+1$, as in the ring-LWE problem. Let $a(x)$ be chosen uniformly at random from $R_q$. Question: Is there any theorem that lower bounds ...
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In Lattice Cryptography, why is it hard to find short vectors if given long vectors?

In lattice cryptography it seems like giving out long vectors for a lattice that can be drawn from much shorter vectors (generating an identical lattice) is somehow useful for public-private key ...
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1answer
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Hash and sign via trapdoors for lattices

Both the papers GPV'08 and MP'11 present trapdoors for lattices that allow to recover $s\in\mathbb{Z}_q^n$ and the error vector $e\in\mathbb{Z}_q^m$ when given $y=As+e$, for $A\in\mathbb{Z}_q^{m\times ...
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1answer
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Adapting LWE Trapdoors for Ring-LWE

In the paper Trapdoors for Lattices: Simpler, Tighter, Faster, Smaller by Micciancio and Peikert, they present the following theorem about the existence of trapdoor for LWE. Theorem 5.1: There is an ...
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1answer
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Regarding Lattice atttacks on ECDSA with a portion of known bits of the nonce k

I am new in the field of cryptography, and I am having some troubles understanding a concept regarding the lattice dimension needed in the attack on ECDSA using several messages with L known bits of ...
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Why is Approximate GCD a hard problem?

There are many Fully Homomorphic Encryption over the Integers schemes whose security is based on the intractability of the Approximate GCD (AGCD) problem. The paper Algorithms for the Approximate ...
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IND-CCA2 post-quantum key exchange

QUIC requires that servers reuse keys so that session resumption works. That breaks many post-quantum key exchange systems. I am looking for a post-quantum key exchange algorithm with the following ...
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1answer
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How does forking lemma work in regard to Digital signatures produced by GPV hash and sign algorithm based on lattices?

I am working on understanding the concepts and approach behind digital signatures that are based on lattices specifically the GPV algorithm. During the security reduction of this method, the forking ...
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1answer
62 views

Original NTRU : How to calculate the size of private key?

In the original NTRU paper:NTRU: A Ring-Based Public Key Cryptosystem,1996, the author proposes 3 choices of implementation parameters: moderate, high and highest. Let's take moderate security level ...
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1answer
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Counting the number of binary solutions of quadratic system

I have a quadratic system of equations related to a balanced RSA modulus $n=pq$ (i.e. $\log p\approx\log q$), and I want to give an upper bound on the number of solutions. Indeed, let $p_i,q_i$ be ...
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Extending the basis

Suppose I have $A \in \mathbb{Z}_q^{n \times m},A_1 \in \mathbb{Z}_q^{n \times m},A_2 \in \mathbb{Z}_q^{n \times m}$. I am following the $\textbf{ExtBasis}$ algorithm of this (Page No. 13). I ...
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1answer
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Encoding of the message in Regev encryption

In public key encryption from LWE, we do the following steps $\textbf{PKE.KeyGen($1^n$)}$ takes as input the security parameter n, samples $A \leftarrow \mathbb{Z}_p^{n \times m}$ and $\textbf{e} \...
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1answer
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LLL - Lattice Reduced Basis Algorithm question?

I have two related questions: Version 1: Let $B=\{b_1,b_2,\dots,b_n\}$ be an orthogonal basis for $R^n$. What is the associated reduced basis obtained by applying LLL algorithm to $B$? I know how ...
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1answer
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Cryptanalysis on affine like matrix based strange cryptographic scheme [closed]

This is a garage made encryption scheme provided as cryptanalysis practice during 34C3 CTF. The challenge is done under the following assumptions All Mersenne twister instances are MT19937 64bit ...
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1answer
653 views

Lattice-based cryptography

How viable is lattice-based cryptography in a "practical" setting? It has been said that lattice-based cryptography would be a "post-quantum" cryptography scheme, but is it feasibly implementable?
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SampleLeft function in lattice trapdoors

We have SampleLeft function in lattice trapdoors as Algorithm $\textbf{SampleLeft}(A,M_1,T_A,u,\sigma)$: $\textbf{Input}$: a rank $n$ matrix $A$ in $\mathbb{Z}^{n×m}_q$ and a matrix $M_1$ in $\...
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1answer
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A query on Learning with errors(LWE) problem

In generating an LWE sample, we do $s\xleftarrow{$}\mathbb{Z}_q^{n}, A \xleftarrow{$}\mathbb{Z}_q^{n \times m}~$and $e\xleftarrow{$}\mathbb{{\chi}^{m}}$ Then we compute $b^T$ = $s^TA$ + $e^T$ and ...
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Multi-party Key Exchange protocol from lattice

There are many two-party or three-party key exchange protocols from lattice. But, it seems that there is no famous multi-party key exchange protocol. Does anyone know the relevant knowledge? Or ...
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2answers
87 views

Lattice generation from basis?

This might be a very short very obvious answer, because I've yet to come across a question similar to mine in my searches. Given a lattice L, with a good base B1 and a bad base B2, what stops an ...
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1answer
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Can i use Babai algorithm in q-ary lattice

Let's assume we have the q-ary lattice $$ \mathcal{L}_q({\bf A})=\{ {\bf z}\in \mathbb{Z}^{n} : \exists {\bf s}\in \mathbb{Z}^{n}_{q} \ , \ {\bf z}={\bf A s}^{T} \mod q \},$$ where ${\bf A}\in \...
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About practical implementation of Ring-SIS/LWE Based Signature and IBE

In paper: Practical Implementation of Ring-SIS/LWE Based Signature and IBE, authors provided the source code for IBE. In the Extract algorithm, they said: $a \cdot x = u \bmod p$ but I can not ...
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Effitiently sampling the error (noise) distribution in ring-LWE

In LPR12, page 4 is described a ring-LWE encryption in which we are working in a ring $R = \mathbb{Z}[x]/(x^n + 1)$ for a $n$ a power of 2. The public key is of the form $(a, b= a\cdot s + e)$ where $...
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Discrete Gaussian Sampling in Authenticated key exchange from ideal lattices

I am implementing the key exchange scheme proposed by zhang et al. on Sage. In the implementation of the scheme, they have used the two distributions $\chi_{\alpha}, \chi_{\beta}$. How to choose $\...
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1answer
98 views

Decision-LWE to Search-LWE

Regev requires $q$ to be prime on lemma 4.2 of his paper for LWE. Why does he require that and how this effect the proof of lemma 4.2?
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1answer
156 views

Discrete Gaussian Sampling role in Lattice-Based Crypto?

I'm reading up on how post-quantum cryptography works, and stumbled upon the notion of discrete Gaussian sampling. However, I can't understand where it fits in the greater picture - currently it feels ...
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determinant of intersection of two lattices

Say $L_1,L_2$ are contained in $\mathbb Z^r$ with \begin{gather*} \operatorname{rank}(L_1) = \operatorname{rank}(L_2) = r, \\ \gcd(\det(L_1), \det(L_2)) = 1. \end{gather*} How do I prove $\...
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Size of $q$ in reductions from lattice problems to R-SIS

The Short integer solution problem is parameterized by four values: $n$, the dimension of the vectors that must be added $m$, the number of samples (dimension of the solution) $\beta$, upper-bound ...
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1answer
118 views

Hardness of Short Interger Solution in lattices

Short Integer Solution ($SIS_{n,m,q,\beta}$) is defined as Given a matrix $A \in \mathbb{Z}_{q}^{n \times m}$, find a non-zero vector $x \in \mathbb{Z}^{m}$ such that $A \cdot x = 0\mod q$ and $||x|| ...
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1answer
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Dividing elements in $R_q$ by $z$ in Grag-Gentry-Halevi (GGH) Graded Encoding Scheme

I'm trying to understand the GGH graded encoding scheme, but something there leaves me very confused and I can not figure out how to explain it: Let $R := \mathbb{Z}[X]/(X^n+1)$, where $n$ is a power ...
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1answer
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Breaking Truncated Linear Congruential Generator with known parameters

There is an elaborate discussion on the breaking of TLCG on the link below, where they show how to break the generator with known parameters given the most significant bits. Problem with LLL reduction ...
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1answer
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Function families from lattices

On this course, Micciancio talks about function families (functions parametrized by some value) that can be used in cryptography. On page 2, he presents the following function family parametrized by ...
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1answer
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Effect of tail cutting and precision of discrete Gaussian sampling on LWE / Ring-LWE security

How does tail cutting and precision of discrete Gaussian sampling implementations affect LWE / Ring-LWE security? Is there a rule of thumb or guideline for choosing the tail cut and the precision for ...
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1answer
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Is there any course video for lattice cryptography? [closed]

Recently, I started doing research about Lattice Based Cryptography. and searched on YouTube a lot of public talks or seminars about it. But is there any course video (graduated course) related to ...
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1answer
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Is it secure using LWE-based cryptosystem under RLWE-based parameters?

I'm computer guy having trouble with cryptography. I recently read the BGV Homomorphic encryption paper which was constructed under both LWE and RLWE assumptions. I was implementing Threshold ...
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Lattice Reduction Method to solve multivariate equation

I have seen very small work in multivariate RSA polynomial modular equation solutions using Coppersmith's based lattice reduction algorithm (LLL). Is there any mechanism to solve the following type ...
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1answer
55 views

Practical Key exchange for Internet

In section 3.2 (page 10) of Vikram Singh's paper A practical Key Exchange for the internet using Lattice Cryptography, he gives the number of elements in each set for odd $q$. However, the results do ...
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Lattice cryptography, key size of public and private key?

An answer to this question what are the NTRU keysize and application in industry? mentions that lattice cryptography has public keys and private keys of the same size. That seems like a property that ...
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1answer
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Why do the game-hops in Kyber and related papers contain 2 steps at a time?

In the Kyber paper in section 3 about the Kyber IND-CPA Encryption there is a proof by sequence of games containing three games. I understand that in the first game hop the M-LWE advantage is used to ...
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Relation between k-th shortest vector of a lattice and (n-k+1)-th shortest of its dual

Let $\Lambda$ be an $n$-dimensional lattice and $\Lambda^*$ be its dual lattice. For any $k \in \{1, 2, ..., n\}$, let $\lambda_k(\Lambda)$ be the $k$-th successive minima of $\Lambda$ (analogously ...