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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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Trapdoor for SIS

I am new to lattice cryptography. May I ask why, if one has a trapdoor for SIS, i.e., can compute a short $x$ that satisfies $Ax=0$, then one can have a trapdoor for $Ax_{2}=y$? TIA
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Difference between TFHE and CKKS?

What are the differences in parameters while implementing CKKS vs TFHE? For example modulus size, ring dimensions, bit security. Any pointers to literature would be appreciated
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Is there any bound on the size of ring dimension for Torus FHE?

I see that all implementations of TFHE in opensource supports 2^10 to 2^12 size of ring dimensions. Is there any specific reason (crypto) behind choosing the value or can we choose higher dimensions (...
Green Amber's user avatar
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Reject sampling of lattice ZKP, why need factor of M

I am watching the video https://www.youtube.com/live/N5nKGtugxYY?si=ejLqW8Pk0jD9lVMn&t=2098 Or particularly this slide: May I ask why does the reject sampling output by $f(x)/g(x)*M$ probability? ...
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Decryption of LWE-based Inner-Product Fuctional Encryption of Agrawal-Libert-Stehle Crypto 2016

In below image, we have LWE-based inner-product functional encryption for short integers [ALS'Crypto2016]. Is the encryption correct? Why do they get $\mu\in\{\color{red}{-K+1},\ldots,0,K+1\}$? When ...
user1035648's user avatar
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lattice RLWE encryption and decryption process

I am here trying to solve an issue that I face a lot during solving RLWE. The issue is that I am not able to retrieve the original message after the decryption process. I use the following encryption ...
A. H's user avatar
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ECDSA biased nonce with biased middle bits

The paper "Biased nonce sense" (https://eprint.iacr.org/2019/023.pdf) covers key recovery for MSB and LSB, the latter case being handled by doing modular inverse of 2 to the power of the ...
gquere's user avatar
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Question of section 1.3 Intuition for Aborting in "Fiat-Shamir With Aborts: Applications to Lattice and Factoring-Based Signatures"

When I'm reading this paper "Fiat-Shamir With Aborts: Applications to Lattice and Factoring-Based Signatures" by Lyubashevsky. I have trouble understanding section 1.3 Intuition for Aborting....
Solaris's user avatar
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How to choose the distribution of error and secret vectors in LWE-based KEMs

The general construction of a LWE-based PKE looks like the following: KeyGen : Secret vector, sk :$ s \leftarrow \chi_s^m$ Public Key, pk: ($A \leftarrow \mathbb{Z}_q^{m \times m}, t = As + e)$ where $...
V S.'s user avatar
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How to determine approximate factor (i.e. modulus-to-(bound of)noise ratio) of LWE in advaned primitives is sub-exponential or polynomial?

In advanced primitives like circuit (policy) ABE [BGG+'Eurocrypt2014] or IPFE [ACGU'Asiacrypt2020, appedix A] based on lattices, parameters setting is quite puzzling and vague (for me). (Parameter ...
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Literature on Batching in FHE

From what I understand, the folklore way to batch Ring-LWE style cipher texts is to use the Chinese remainder theorem. I am wondering if there are any different approaches/optimizations to this style ...
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Can this problem reduce to the SIS problem or LWE problem?

The description of the graceless problem is as below. Given a full rank square matrix $A=(R\cdot S+E)\in\mathbb{Z}_q^{n\times n}$,where: $q>2^\lambda$ is a prime; $R\leftarrow\mathbb{Z}_q^{n\times ...
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Gadget Matrix Ring Setting

What would be the analogue of the gadget matrix in the ring setting? Would it be the same matrix? Do the trapdoor algorithms work exactly the same way? Thanks
woah's user avatar
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Questions about SIS hard problem

The definition of $\mathrm{SIS}_{q,n,m,\beta}$ problem is as below. Let $A\in\mathbb{Z}_q^{n\times m}$ be an $n\times m$ matrix with entries in $\mathbb{Z}_q$ that consists of $m$ uniformly random ...
X.H. Yue's user avatar
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Shortest Vector Problem as Dihedral Hidden Subgroup Problem

I’m a mathematician trying to get into cryptography. I have a somewhat silly question, but I can’t seem to find a proper answer anywhere. I am interested in whether or not there is a way to directly ...
Joe's user avatar
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Math of Hint usage in Dilithium

For Dilithium, I understand that $$\text{HighBits}(\mathbf{Ay}) = \text{HighBits}(\mathbf{Ay} - c\mathbf{s_2}) = \text{HighBits}(A_z - c\mathbf{t})$$ The first term of the equation is performed on the ...
Gappu's user avatar
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Is NTRU broken?

Today a new paper appeared on ePrint, "Improved Provable Reduction of NTRU and Hypercubic Lattices". It claims that: this is the first provable result showing that breaking NTRU lattices ...
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Direct quantum reduction from GapSVP to SIS

Looking through Chen's recent paper, if we forget about complex Gaussians and just build uniform superpositions over hypercubes, we could create the state $$\sum_{v\in L}\vert v\rangle \sum_{y\in\...
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Reaching the bound of Boneh and Durfee Attack

According to the paper, theoretically,we can get $\delta=0.292 \lt 1-\frac{1}{\sqrt{2}}$,but how to set the lattice and implement it in sagemath? I generated some data by ...
Cred Mao's user avatar
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Is the cryptography scheme over lattice still secure? [duplicate]

In https://eprint.iacr.org/2024/555, the author proposed a quantum algorithm to solve LWE problem. How serious is its impact on the existing scheme over lattice.
guangyu liao's user avatar
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Polynomial-time Quantum Algorithms for Lattice Problems

A new paper, by Yilei Chen, whose title is Quantum Algorithms for Lattice Problems (https://eprint.iacr.org/2024/555) appeared on eprint and it claims to solve hard lattice problems, such as the ...
Hilder Vitor Lima Pereira's user avatar
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Degree of inverse of f in NTRU?

In NTRU, we know that $f$ is a ternary polynomial in the ring $$R=\frac{\mathbb{Z}_q[x]}{x^n-1}.$$ Here $f$ has $d+1$ coefficients 1 and $d$ coefficients $-1$ and rest are zero. For computing the ...
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Probabilistic proof of multiplying two elements from non-prime finite field

I was reading this paper, and there, they use the ring $\mathbb{Z}_{\large p}[\alpha]/(\alpha^{\large n}+1)$ for all their operations. And that looks like a construction of finite field $\mathbb{F}_{\...
the thinker's user avatar
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How does linearity work with SWIFFT?

I read that SWIFFT is a linear hash function, but I don't understand what this means. The obvious interpretation is that if you have inputs $X1, X2$ each of which is an array of 16 64-dimensional ...
the thinker's user avatar
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Instantiation of norm bound in SIS

Recall Short Integer Solution: $\textbf{SIS}_{n, q, \beta, m}$: Given $\textbf{A} \in \mathbb{Z}^{n\times m}_q$, $\vec{b} \in \mathbb{Z}^{n}$, find $\vec{z} \in \mathbb{Z}^{m}$ of norm $||z|| \le \...
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The necessity for lattice reduction in LWE

I am trying to understand how exactly lattice reduction and LWE are linked. The attacks on LWE I have seen all use lattice reduction in some way or another, dual attacks, uSVP type and so on. Naively, ...
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Finding security constraints for different input domains of Ajtai functions

I know that the normal construction for Ajtai hash functions is as follows: Pick $n, m, q \in \mathbb{Z}^+$ such that $n \log q < m < \frac{q}{2n^4}$ and $q = O(n^c)$ for some $c>0$, and some ...
the thinker's user avatar
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High Hamming Weight Attack on Kyber

I was reading LAC (https://eprint.iacr.org/2018/1009.pdf). They mention about high-hamming weight attacks on the Centered Binomial Distribution (CBD). To counter this, they propose CBD with fixed ...
V S.'s user avatar
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The proof of Claim 5.2 in the "On Lattices, Learning with Errors, Random Linear Codes, and Cryptography"

When I'm reading this paper "On lattices, learning with errors, random linear codes, and cryptography" by O. Regev. I have trouble understanding the proof of claim 5.2. "Hence, it is ...
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How to measure the denseness of Mod-LWR samples in some space?

I tried to understand how dense the Mod-LWR samples are in some space. I tried to see from a view similar to LWE, i.e. using GV-bound(maybe LPN is better example because GV-bound is for codes). But I ...
Sharon's user avatar
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Ring-LWE lattice cryptography and FFT Trick for $X^n+1$

in reference here the FFT trick for $X^n+1$ is discussed with reference to the Number Theoretic transformation. On page 5, the Chinese Remainder Theorem is used to define the mapping. So far so good. ...
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How to reconstruct low order bits of $t$ of CRYSTALS-Dilithium from a small number of signatures?

In FIPS 204 (https://nvlpubs.nist.gov/nistpubs/FIPS/NIST.FIPS.204.ipd.pdf): "The vector $\textbf{t}$ is compressed in the actual public key by dropping the $d$ least significant bits from each ...
Jong Hyeok Lee's user avatar
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1 answer
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Impact of Ryan and Heninger's CRYPTO 2023 paper on post quantum cryptosystems

From Schneier's blog, which seems to have been written in response to a somewhat recent Quanta magazine article: The winner of the Best Paper Award at CRYPTO this year (2023) was a significant ...
kodlu's user avatar
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What does it mean the "Distinguisher" in LWE decision version?

As we know in block ciphers, the distinguisher means that despite thousands ciphertexts (and plaintexts), allows an attacker to distinguish the encrypted data from random data. This attack is ...
R_Jalaei's user avatar
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Initial approximation in CKKS Bootstrapping

In this CKKS bootstrapping paper https://eprint.iacr.org/2018/153 the authors use a Taylor expansion to approximate the complex exponential function within a small range. More precisely, for the input ...
opag's user avatar
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A Smudging Lemma in Lattice

I saw a paper LLW21 in EUROCRYPT 2021 that used this lemma, but there was no proof or references. How should this lemma be proved ?
constantine's user avatar
1 vote
1 answer
240 views

Centred Binomial Distribution and its effects on security in Kyber

I want to concretely understand how exactly choice of error distribution effect the security of KEM in the context of Lattice Based Cryptography. For example, I would like to know the concrete ...
Random Bits's user avatar
3 votes
1 answer
91 views

Definition of Dual Lattice: $\vec{v}\in span_{\mathbb{R}}(\mathcal{L}(\mathbf{B}))$

Consider the definition of the dual lattice for a lattice $\mathcal{L}(\mathbf{B}_{m\times n})\in\mathbb{R}^{m}$ where $\mathbf{B}\in\mathbb{R}^{m\times n}$ and $n\leq m$ [sp2 Seminar, Luxembourg 2019,...
user1035648's user avatar
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Algorithm to solve SVP (shortest vector problem) using LLL reduction

I'm trying to write a C++ program to solve the shortest vector problem. The program is given a basis of a vector space $V$ and needs to find the shortest non-zero vector in the lattice generated by ...
Valéry Giscard d'Estaing's user avatar
3 votes
2 answers
155 views

Connection between (noisy) CVP and LWE

What's actually the difference between a (noisy) CVP and LWE? It seems to me that both are the same. With the definition of LWE: $$A * s + e = b$$ solving for secret vector s is the same than solving ...
lstk44's user avatar
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2 answers
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CRYSTALS-Kyber Compress and Decompress function role

I was reading CRYSTALS-Kyber design. They have used compress_q(x,d) to scale an element of $\mathbb{Z}_q$ to $[ 0,1,...,2^d-1 ]$. The definitions of ...
ABCD's user avatar
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1 answer
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Do you know any library for implementing lattice-based schemes? [closed]

Good afternoon! I'm trying to write a code for a lattice based scheme (based on the SIS problem). I'm looking for a library that may help me in this task without taking care of the implementation of ...
Herbrant's user avatar
1 vote
1 answer
136 views

Decision LWE vs Search LWE: Which one is harder?

Sometimes if we have an attacker who's able to solve decision-LWE problem then we can use them (as a sub-routine) to solve (search) LWE problem, i.e., $\mathsf{sLWE} \leq \mathsf{dLWE}$. Conversely, ...
user1035648's user avatar
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127 views

Approximate SIVP worst-case hardness: proper mathematical formulation used for cryptographic purposes

Is the following a correct formulation for the assumed worst-case hardness of $SIVP_\gamma$? For every PPT algorithm $A$ for every $n\in\mathbb{N}$ there exists a basis $B_{n,A}=\{v_1,\dots,v_n\} \in ...
Alexander Ushakov's user avatar
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What's the lattice dimension of the uSVP for attacking CRYSTALS-Dilithium-128?

I am trying to understand the process of transitioning from a NIST standard to the attacks based on of the Unique Shortest Vector Problem (Unique-SVP). Specifically, I am working with Crystals ...
asdf's user avatar
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1 answer
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Do we know that LWE is harder than Ring LWE?

The plain, normal-form, decisional LWE problem over $\mathbb{Z}/q\mathbb{Z}$ is: given a uniformly random $n\times n$ matrix $A$ and vector $b\in \mathbb{Z}/q\mathbb{Z}^n$, decide if $b=As+e$ for ...
Sam Jaques's user avatar
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Definition of Dual Lattice

1- Can someone explain why we have the definition of dual of a lattice like $\Lambda^*=\{\vec{v}\in span(\textbf{B}): \langle \vec{v},\vec{x} \rangle \in \mathbb{Z}, \forall \vec{x} \in \Lambda\} $. 2-...
user1035648's user avatar
2 votes
1 answer
96 views

Arithmetic in Cyclotomic Number Rings with Shoup's Number Theory Library (NTL)

I wish to do arithmetic on elements in an integer subring of a cyclotomic number field, i.e, in $\mathcal{O}_K = \mathbb{Z}(\zeta) \cong \mathbb{Z}[X] / <\phi_m(x)>$ where $\zeta$ is a root of ...
Rohit Khera's user avatar
1 vote
0 answers
99 views

Understanding Gentry's initial FHE construction based on ideal lattices

I am trying to understand the encryption procedure in Craig Gentry's initial construction for FHE described in Fully Homomorphic Encryption Using Ideal Lattices. Unfortunately after repeated attempts ...
Parham's user avatar
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6 votes
1 answer
309 views

NTRU Cryptosystem: Why "rotated" coefficients of key f work the same as f

In the NTRU cryptosystem, we can use a randomly generated polynomial f that is inversible under modulo p and q to encrypt and decrypt our plaintext. While studying this system, I attempted to ...
Ymi's user avatar
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