Questions tagged [lattice-crypto]
Lattice-cryptography is the study and use of lattice problems applied to cryptography.
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LLL on Knapsack-eque problem
Given integers $s_1, \dots , s_n$ and target integer $t$, I'm trying to find small integer coefficients $x_1, \dots , x_n$ such that:
$$
t \approx x_1 s_1 + \dots +x_ns_n
$$
Taking inspiration from ...
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On the spectral norm in lattice-based cryptography
In the preliminaries section of a paper$^\color{magenta}{\star}$ on lattice-based cryptography, the matrix norm $\| \cdot \|_{2}$ is used. Why do we define such norm? What's the purpose of defining ...
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Proof regarding a property of "$q$-ary" lattices
In this question we are dealing with "$q$-ary" lattices. I will give the definition available to me and I'm interested in proving the lemma. As a reference see the PDF on page 2 from Peikert'...
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Gaussian distribution propoprties
Good day,
I've a question regarding Gaussian distribution properties over lattices :
Let $\mathcal{L}$ := $ \mathcal{L}(\,b_{1}$,..., $b_{m})$ be a lattice over $\mathbb{R}^{n}$, and $W$ = span($b_{1}$...
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Average- and worst-case complexity
The terms "average-case", "worst-case" hardness are quite confusing.
What do they mean when they say certain problems (like lattices)
have an average-case to worst-case ...
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Two problem about noise management of BFV
I have stuck in two problems when understanding the noise management of BFV scheme, and I don't have any idea about the two problem, help me please.
Problem 1:
In the ...
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Ring LWE distribution definitions
This may be a stupid question but I've been stuck on parsing these definitions for a while.
I am reading the paper "On Ideal Lattices and Learning with Errors Over Rings" by Lyubashevsky, ...
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Hiding sum of vectors. Hardness based on CVP
This is the problem
Let $\mathcal{L}$ be a lattice and $v_1,v_2,\ldots,v_n\notin\mathcal{L}$. Given the values $a_1,\ldots,a_n$ such that
$$a_1=\lfloor v_1\rceil+v_2+\ldots+v_n$$
$$a_2=v_1+\lfloor v_2\...
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What are the most important parameters when it comes to lattice based cryptography security?
When utilizing the closest vector problem for decrypting data, does lattice size matter. For example, is a 1000x1000 grid necessarily more safe than a 100x100 grid? And if so, why would these affect ...
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Tensor and power bases for SIS?
What is there to say about using a power basis or a tensor basis or some combination of them for the RSIS problem in lattice cryptography?
Restricting to dimension 3 for illustration, usually the ...
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How long time per operation to crack Kyber1024 compared to AES256 for quantum computers?
How long time does quantum computers take per operation when search the key of Kyber? Grover's algorithm weakens 256-bit AES to 128-bit security, quantum computers at most take 2^128 operations to ...
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LWE KEMs and message coding
In many proposed lattice PKE schemes, the plaintext is encoded or modulated in a simple fashion, e.g. using Kyber-ish notation:
key gen: $pk=(A, t=As+e)$, $\quad sk=s\quad$ ($A$ random, $s$, $e$ ...
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Attacks on Ring-LWE exploiting structure of ideal lattice?
Currently every LWE-based cryptographic schemes analyze their security using lattice estimators and lattice estimators analyze the security of standard LWE even though the actual scheme is based on ...
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Do we need the quantum random oracle model (QROM)?
I am currently studying the proof of the Dilithium signature in the quantum random oracle model (QROM). I am curious to hear if anyone have any thoughts on the importance of having proofs in the QROM ...
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Reducing exact SVP to exact SIVP
In "Efficient reductions among lattice problems" by Micciancio (2007) it is said, that
SVP reduces to SIVP in their exact versions.
I did not found anything about this fact, is a reduction ...
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Why was A doubled in size
Why was the dimension of A doubled in kyber?
LWE encryption uses a public matrix A of dimension K but kyber uses a double matrix A resulting in $A ^{ k * k * n }$
When deriving the results of the ...
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CRYSTALS-Dilithium - How do the supporting algorithms work?
I am studying the Dilithium signature from Ducas et al's CRYSTALS-Dilithium: A Lattice-Based Digital Signature Scheme.
Wanting to understand how the supporting algorithms work together, I am trying ...
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Hardness of LWE with Uniform Secrets and Error Distributions
I have seen various papers discussing the security of the Learning with Errors problem with very small uniform secrets and errors but I have not found any papers on the general LWE problem with ...
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Difference between Decryption-failure and Plaintext-checking oracles
I am reading this paper, which in the introduction, tells about two main types of key recovery SCAs :
Reaction_type SCAs, which uses a decryption failure oracle
Message-recovery-type SCAs, which uses ...
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LWE Decryption: Generating errors for (c1, c2) that match binary message m
In the encryption process, the ciphertexts c1 and c2 are added to errors e1 and ...
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Learning with rounding: uniformity
Naively, when one applies rounding to a uniform random value one anticipates that the change is uniformly distributed. In lattice-based cryptography, is there a formal notion or proof of equivalence ...
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Unable to retrieve the binary string using LWE and Lattice-based decryption
I am new to this encryption scheme, so I may not be exactly sure of its implementation.
I have a list of (u, v) ciphertext pairs to decrypt, each of them are 1-bit.
...
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LWE and Lattice-based cryptography: How to recover binary message $M$ from $(u, v)$ values?
I am given a set of $(u, v)$ values, matrix $A$, primary key vector, private key vector, error vector and prime $q$. I wanted to recover the binary value of each $(u, v)$ pairs using LWE decryption.
...
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Equivalence of lattice definitions
I have come across two supposedly identical definitions of lattices in the lattice crypto literature. There are mainly these two definitions of lattices, the first considers lattices as discrete ...
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How & where is concepts of Good basis and bad basis used in Crystal kyber?
I've read the documentation of Crystal Kyber, but nowhere it is mentioned about good basis and bad basis.
Please explain how and where is the good basis and bad basis is used in crystal kyber.
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Are lattice-based cryptography and error-correcting codes mathematically unsound?
From Ronald de Wolf's The potential impact of quantum computers on society:
The first is so-called post-quantum cryptography. This is classical cryptography, based on computational problems that are ...
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What is this parameter? in Lyubashevsky's ID-scheme
I am studying Lybashevsky's ID-scheme from the article Fiat-Shamir With Aborts: Applications to Lattice and Factoring-Based Signatures(https://www.iacr.org/archive/asiacrypt2009/59120596/59120596.pdf) ...
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Public seed expansion for uniform reference strings
Many cryptographic protocols are parameterized by a uniformly random reference string (e.g. the commitment key for Pedersen commitments).
Our goal is to publicly generate the random values of this ...
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How "unorthogonal" can a LLL-reduced basis be?
I have been recently studying LLL-reduction. I get from the size condition and Lovasz condition that the basis are guaranteed to be somewhat orthogonal. But I couldn't figure out how orthogonal the ...
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The successive minima of a lattice
I am new to lattice theory. I hope(will be grateful) that one could explain to me this claim 7 in REGEV course(this claim appears in this file page 6 : https://cims.nyu.edu/~regev/teaching/...
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Why do we need "selective security" for ABE?
The general question is: Why are ABE schemes usually/sometimes proven in the selective-set of attributes model of security? Or even co-selective (both attributes and policy function)? Is it just ...
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Hardness of a modified version of NTRU
Let the modified NTRU be $h=f/g$ such that $f$ is not necessarily a short polynomial, is the NTRU problem still hard in this case?
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NTRUEncrypt proof that there are plenty of keys
In NTRU algorithm one is supposed generate a vector which is invertible as a polynomial in both $(\mathbb{Z}/p\mathbb{Z})[x]/(x^n-1)$ and $(\mathbb{Z}/q\mathbb{Z})[x]/(x^n-1)$. But is there a ...
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How to show additive subgroup of $R^n$ is not discrete? [closed]
Suppose we have the additive subgroup of reals generated by $\sqrt{3}$ and $\sqrt{5}$. How would you show you that this subgroup does not form a lattice?
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Hardness of LWE
I was reading "TFHE Deep Dive" from Ilaria Chillotti, and I am a bit confused over the sample given in 31:08
In the above toy sample, isn't it possible to directly eliminate noise by ...
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Which lattice-based encryption scheme/signatures is fundamental?
If I would like to focus on only one signature scheme, and only one encryption based on lattices in a pedagogical context (to introduce the concept of lattice-based crypto to people familiar with ...
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Avoid CKKS Bootstraping
CKKS is a levelled scheme, because the rescale $\lfloor\frac{x}{\Delta}\rceil$ operation requires truncating a modulus to be efficiently evaluated, and rescale is (usually) needed after every ...
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CRYSTALS-KYBER versus FrodoKEM, what makes each of them different than the other?
NIST's main recommendation for encryption/decryption mechanism is CRYSTALS-KYBER. Whereas, the BSI (German equivalent) chooses FrodoKEM.
As far as my knowledge goes both these mechanisms use LWE ...
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Kyber and Dilithium explained to primary school students?
Kyber and Dilithium are post-quantum cryptographic designs, but the resources are hard to understand. Is it possible to explain those ciphers to children?
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ISIS problem in the case of $m=n$
The Inhomogeneous Short Integer Solution (ISIS) problem is as follows: given an integer $q$, a matrix $A\in \mathbb{Z}^{n\times m}_q$, a vector $b\in \mathbb{Z}^{n}_q$, and a real $\beta$, find an ...
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Base of $(n+1)$ elements in a lattice
Does there exist a lattice in $\mathbb{R}^n$, with an independent generative family $(b_1, \dots, b_{n+1})$ of $(n+1)$ vectors (without any loss of generality I suppose $(b_1, \dots, b_{n})$ is a $\...
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High density SIS and Low density SIS
I am searching for the exact definition of High density SIS and Low density SIS, but there is something unclear about it.
SIS problem is to find $x\in \mathbb{Z}^m$ such that for random $A\in\mathbb{Z}...
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NTRUEncrypt fails on sedonion algebra
This question is a direct follow-up (hopefully - the last) of my previous one; please see it for full information. I would like to further generalise NTRU cryptosystem on higher-order algebras. ...
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NTRUEncrypt fails on quaternion algebra
This is a follow-up of my previous two questions (1 and 2), might be relevant to check them out first for a full context. I am trying to re-create results from this paper. The basic algorithm is ...
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"Shifting" a dual-Regev keypair away from a trapdoored instance
This question pertains to identity-based key encapsulation mechanisms (IB-KEMs). To recap the functionality:
$\mathsf{KeyGen}(1^\lambda) \to (\mathsf{msk}, \mathsf{mpk})$ Generates the master keypair
...
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Why is the best way to solve LWE (and Cryptographic related Systems) with SVP (approx)?
Community,
I'm new into lattice based cryptography, and I'm interested about the security of cryptography schemata like Kyber and why the focus of solving this problem lead into solving approx. SVP.
...
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Choice of Polynomial Quotient Ring
In (lattice-based) cryptography, the quotient ring $\mathbb{Z}[X]/(X^n+1)$ where $n = 2^e$ is a power of 2 is used in various cryptographic schemes (e.g., CRYSTALS-Kyber). It is my understanding that ...
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Good libraries for lattice-crypto [closed]
I'm searching good libraries to manipulate lattice tools to do cryptography.
I'm mainly interested by C/C++. But I'm also interested if it is in python.
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Break Lattice-Based Cryptography with Variational Quantum Algorithm (only 25 k. Qbits for Kyber1024)?
I am currently writing a seminar paper on Kyber and other lattice-based methods. I was so excited about the lattice-based methods that I also currently searched quantum algorithms to solve the methods....
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Question about Theorem 2 in CRYSTALS-Kyber Paper
I have some questions about the Kyber paper, especially about Theorem 2 on page 6, which I would like to ask here. First of all I quote the following theorem from the paper and ask my questions ...
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