Questions tagged [lattice-crypto]
Lattice-cryptography is the study and use of lattice problems applied to cryptography.
643 questions
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Effect of multiplying message by $q/2$ LWE
I have recently started learning about the LWE problem, and I am wondering about how multiplying the message bit $\mu \in \mathbb{Z}_2$, affects the encoding of this message.
The LWE-based ...
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CRYSTALS-Kyber Central Binomial Distribution reference inplementation in c
In lines 8 to 10, why it does not get input bytes as it is (i.e., line 8)? It first does some changes (line 9 and 10).
Do authors do this changes as we cannot work with binary form of a variable ...
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Security of Ring-LWE and Module-LWE encryption scheme
Regev-05 encryption under plain LWE consists in using a public key $\mathsf{pk} = (\mathbf{A}, \mathbf{b} = \mathbf{A}^\top\mathbf{s}+\mathbf{e})$, where $\mathbf{A}\in \mathbb{Z}_q^{n\times m}$ is ...
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Modulus in Lattice-Based Cryptography？
The common modulus methods are standard modulus (the result is confined to the range $[0, q-1]$) and centered modulus (the result is confined to the range $[- \lfloor q/2 \rfloor, \lceil q/2 \rceil)$)....
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Understanding Canonical-embedding vs Coefficient-embedding in Ideal Lattices: Relation to NTT?
I'm trying to understand the relationship between different representations of ideal lattices, particularly the canonical embedding and coefficient embedding. While studying these concepts, I noticed ...
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What is a Lattice trapdoor?
In "A New Approach for Non-Interactive Zero-Knowledge from
Learning with Errors" , Brent Waters uses lattice trapdoors to build hidden bit generators. Can someone explain to me what a ...
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Can the message space for Ajtai Hash be extened?
I have a question regarding the Ajtai hash function. Typically, the message space for this function is the binary space $\{0, 1\}^m$. However, I am considering extending the message space to $\{-1,0, ...
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Is there an efficient algorithm to compute the inverse of a small-norm element in a special polynomial ring？
The paper "Short, Invertible Elements in Partially Splitting Cyclotomic Rings and Applications to Lattice-Based Zero-Knowledge Proofs" presents a corollary stating that in a polynomial ring $...
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Is $A \in \mathbb{Z}_q^{m \times m}$ is a one-way function?
In lattice-based Cryptography, Ajtai [1996] demonstrated the family of one-way functions. For a uniformly chosen matrix $A \in \mathbb{Z}_q^{n \times m}$ and $x \in \{0,1\}^m$, $A \cdot x \mod q$ is a ...
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Proof of random-self-reucibility of LWE (worst- to average-case) in Regev paper, Lemma 4.1
I'm reading proof of [Regev'arXive2401.03703, lemma 4.1, page 30].
I can't figure out how Oded Regev estimates acceptance probability of the average-case distinguisher by Chernoff bound
Thorem (...
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Do the specific powers of two $2^{23}$ and $2^{13}$ in the modulus $q = 2^{23} - 2^{13} + 1 = 8380417$ have any special purpose in its design?
I recently started exploring Post-Quantum Cryptography, particularly Lattice-based Cryptography, and came across the modulus $q = 2^{23} - 2^{13} + 1 = 8380417$, which is used in schemes like ...
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Why are matrix-secrets used on lattice based signatures?
I was wondering why in the Fiat-Shamir with aborts paradigm on lattices such Lyuba12, matrices S and E are used to generate the public key ($AS + E = B$ or $AS = B$).
And not just vectors like on ...
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How does lattice reduction produce $s$ and $e$ from public parameters of an LWE instance?
It has been shown that by embedding $A$ and $t$ from an LWE instance $A$, $t=As+e$ inside a lattice, we can find a vector from the reduced lattice that contains $s$ and $e$.
For example, in the ...
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How does the size of the parameter $n$ in $Kyber$ affect the success of lattice reduction of LWE instances?
I'm learning about using lattice reduction to find the secret $s$ of an LWE instance. When testing it on code, I have noticed that when $n$ is 32 or lower, I am able to retrieve $s$ successfully, but ...
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Why consider/formulate Shortest Vector Problem as a Promise Problem and not as a Decision Problem?
We know (search) approximate Shortest Vector Problem ($\mathsf{SVP}_{\gamma}$): Given an arbitrary basis $\mathbf{B}$ of some lattice $\mathcal{L}=\mathcal{L}(\mathbf{B})$, find a shortest non-zero ...
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Is the variant of LWE-based encryption scheme secure?
Suppose that there exists a SIS instance generator $Gen_{SIS}(n,m-1,q)\to(A\in\mathbb{Z}^{n\times m}_q,s\in\{0,1\}^{m-1})$, where $(s,1)$ is the short integer solution for the SIS instance $A$, s.t. $...
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How is it that a matrix of polynomials can be represented as a circulant matrix for purposes like lattice reduction?
When using lattice reduction on polynomial matrices, I have seen examples where a matrix can be represented as a circulant matrix before lattice reduction. Is there an explanation on why this works?
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Understanding LWE's Correctness Proof
In this paper, on page 33, we are presented with Claim 5.2 which is supposed to show that for the selected parameters in LWE, namely:
$\alpha(n)=o(1/(\sqrt{n}\log ...
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Why RLWE is typically implemented using unsigned integers?
Every RLWE implementation I know uses unsigned integers even when it needs to represent signed values. Why?
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Choosing $x^n+1$ as an irreducible polynomial in $\mathbb{Z}[x]$ instead of $x^n-1$ for ring $\mathbb{Z}[x]/\langle f(x)\rangle$ of Ring-LWE
In the note of ["Ring-SIS and Ideal Lattices by Noah Stephens-Davidowitz (for Vinod Vaikuntanathan’s class", footnote 3], it has written:
3 The ring $\mathbb{Z}[x]/(x^n + 1)$, ideal ...
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I have the best basis possible for a lattice. How well can I solve CVP?
Let's suppose I've given a lattice $L$. I'm allowed to spend as much pre-computation as I want to produce a poly-sized advice string, and I use it to find the best basis possible for the lattice.
Then ...
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Why are only primal and dual attacks dealt with among the various attacks on LWE?
As we see in https://estimate-all-the-lwe-ntru-schemes.github.io/docs/, only dual and primal attacks are the most important attacks against LWE and NTRU schemes. On the other hand, many attacks such ...
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When can radix-3 NTT faster than radix-2 NTT?
I am a beginner in Lattice based cryptography and now I am encountering some cryptography algorithms using NTT.
And I am trying to change the NTT in algorithms and I am going to make an appoche to use ...
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How to batch-prove the knowledge of Module-SIS secrets
If the prover wishes to prove knowledge of Module-SIS secrets $\mathbf{s}_i$ satisfies $\mathbf{A}_i \cdot \mathbf{s}_i = \mathbf{u}_i\ \text{mos}\ q, i \in \{1,2,\cdots,N\}$ where $R_q$ is a ...
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If $\Vert cs\Vert$ is short in a lattice signature, how large can $s$ be?
I would like a result of the following form:
"Let $s\in \mathcal{R}_q^n$ where $\mathcal{R}_q = \mathbb{Z}_q[x]/(x^d+1)$. If $\Vert s\Vert > q$, then (with all but negligible probability) for ...
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SIVP to SIS: Why we must use uniformly random input for an hypothetical oracle in reduction?
I'm reading a handout about worst-case to average-case reduction by Peikert and Banerjee: https://github.com/cpeikert/LatticesInCryptography/blob/main/lec11.pdf
In the reduction from SIVP to SIS, we ...
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Is it possible to prove with high certainty part of a preimage must exist to produce an LtHash output?
Given a H calculated using the LtHash algorithm, is it possible to prove a section of the preimage must have been part of the specific original preimage?
i.e. given H = lt(A | X | B) give proof that X ...
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Conditional distribution of an integer error vector, taken from an appropriate discrete Gaussian, given its syndrome [GPV'STOC2008]
I'm reading lemma 5.2 of [GPV'STOC2008, page 18] about conditional distribution of an integer error vector $\mathbf{e}\in\mathbb{Z}^{m}$, taken from an appropriate discrete Gaussian $\mathbf{e}\sim D_{...
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Estimate security parameter $n$ for Ajtai commitment
Let $x$ be a binary string of length $m \approx 2^{20}$, $A_{n \times m}$ be a random matrix with entries independently and uniformly drawn from $\mathbb{Z}^*_{q}$, and $q$ be a $128$-bit prime number....
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A few questions about smooth paramter in lattice signature
I was recently studing lattice signature and reading the GPV08 paper.
For my comprehension, given a continuous guassian distribution, the smooth parameter decides a bound that, by placing the same ...
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How to get a sample from $D_{q\mathbb{Z}+c,r}$？
In the paper Compact Lattice Gadget and Its Applications to Hash-and-Sign Signatures, the preimage sampling boils down to only n times sampling of $D_{q\mathbb{Z}+c,r}$. How can we obtain a sample ...
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Relationship between LWE, SIS, and ISIS
Suppose I have a short-secret LWE instance $As+e=b\mod q$. If I treat this as a single matrix, it becomes an ISIS problem:
$$ \begin{pmatrix} I &A\end{pmatrix}\begin{pmatrix} e \\ s\end{pmatrix}=b\...
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A problem about fundamental region of a lattice
This lemma is from Peikert and Carter's handout of lecture 2. $\mathbf{\tilde{B}}$ is the Gram-Schmidt orthogonalized matrix of $\mathbf{B}$. I want to prove this lemma.
My idea is to prove it by ...
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Gaussian width in lattice setting
In the lattice setting (like LWE, RLWE) , the Gaussian function is often defined as
$$
\rho_{\Sigma}(x) = e^{-\pi x^T\Sigma^{-1}x}
$$
The discrete Gaussian distribution $\mathcal{D}_{\Lambda, \Sigma}$ ...
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Smaller module ranks of MLWE imply more algebraic structure?
In CRYSTALS-Kyber, they mentioned that the module rank $k$ should not be too small, they argued "having a smaller $k$ implies more algebraic structure, making the scheme potentially susceptible ...
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Why is the public key included in the private key in Kyber KEM?
We know that CRYSTALS-KYBER's KEM (key encapsulation mechanism) key generation algorithm is:
Output:
Public key pk $\in B^{12\cdot k\cdot n/8+32}$,
Secret key sk $\in B^{24\cdot k\cdot n/8+96}$
$z \...
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In GPV08 lattice signature, why the fact that the bijective from quotient group to the set of syndromes holds?
I was recently reading the lattice-based signature scheme GPV08 (full paper), and the following statement confuses me alot (which is in the section 5.1, page 17&18):
Throughout the paper, we use ...
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Is there split-radix (mixed-radix) number theoretic transform?
Can I use multiple radices (for example, radix-2 and radix-4) for one NTT? If so, what would the twiddle factors be?
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How to compute the vector $\mathbf{v}$ in verifiable encryption scheme?
In the subsection 6.3 of Lattice-Based Zero-Knowledge Proofs and Applications: Shorter, Simpler, and More General, if the module p of encryption scheme is co-prime to the module q of commitment scheme,...
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Questions about LWE in NIST standards
LWE instances have the form $\vec{a}_i,b_i = \langle\vec{a}_i,\vec{s}\rangle+e_i\bmod q$ for some integer $q$ and for $i=1,\dots,m$.
My questions are about the NIST proposed standards. In the ...
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A question about smoothing lemma and proof
I'm a beginner on lattice-based cryptography and currently reading a handout of Prof. Vinod Vaikuntanathan.
The first question
In section 2.4 Smoothing Lemma and Proof, I cannot figure out why the ...
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Compare Saber and Kyber, about their techniques of message bit layout in encryption
I'd like to discuss message bit layout in the Saber and Kyber's IND-CPA encryptions.(Details of these two schemes follows behind these question paragraphs). From my understanding, both Saber and Kyber ...
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Centered Binomial Distribution vs Fixed Weight Ternary Sampler
Fixed weight ternary polynomial vector sampler randomly selects a polynomial from ${R_q[x]}/{(x^n+1)}$ with coefficients $-1,1,0$. Notable KEMs that uses fixed weight ternary samplers include NTRU, ...
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Noise flooding in Lattices
I noticed that in the paper [HLL24], the authors used the noise flooding technique to choose parameters and complete the proof.
But I am confused that why set $\sigma \ge 2^{\kappa+6}y$ to guarantee ...
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In lattice, does converting a "bad" basis to a "good" basis constitute a hard problem?
In lattice, a "good" basis are the set of "almost" orthogonal set of vectors which have small norms and a "bad" basis are those basis which are in certain sense hardest ...
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Doubts regarding the ternary vector sampler in SMAUG-T KEM
SMAUG-T is an efficient post-quantum key encapsulation mechanism (KEM). It is the winner of Korean PQC Competition.
SMAUG-T uses a Hamming Weight Sampler $HWT_h$ to sample secret polynomial vectors $s,...
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What is the difference between PLWE (Polynomial Learning with Errors) and RLWE (Ring Learning with Errors)？ [closed]
Recently, I have been studying lattice-related concepts, and I want to understand the differences between PLWE and RLWE, such as how their security compares, as well as their structure and value ...
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Issue building RLWE based program
I've successfully built a LWE based program now moving onto building a RLWE based python program using: https://blog.openmined.org/build-an-homomorphic-encryption-scheme-from-scratch-with-python/ as a ...
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Tutorial of lattice estimator to compute a specific function with BGV
I recently try to go into lattice estimator https://github.com/malb/lattice-estimator
I did find the document of it, but unfortunately, can not follow.
May I ask how to run the lattice estimator to ...
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About the notion of "full rank" in lattices
I'm a newer to the lattice theory, so there is a basic notion about "full rank" confusing me.
In some papers focusing on the lattice theory, they always use the full rank lattice (the number ...
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