Questions tagged [lattice-crypto]
Lattice-cryptography is the study and use of lattice problems applied to cryptography.
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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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Why the refresh (modulus and key switching) is required in BGV after addition?
I am reading the BGV paper. On page 18, after addition, the protocol will also refresh (modulus and key switching), may I ask why is this required? It seems to me that I can still use the same secret ...
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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-...
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Sigma parameter from Trapdoors for Lattices
In the document Trapdoors for Lattices, section 5.4 Gaussian Sampling, they introduce the parameter $\sqrt{\Sigma_{\bf G}}$, which is related to the lattice $\Lambda^\perp(\bf G)$. They use it as a ...
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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 ...
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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 ...
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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 ...
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GSW Homomorphic Encryption
In GSW homomorphic encryption scheme proposed here. The integers are over $Z_q$ where $q$ is a modulus parameter of the scheme. It is not clearly mentioned in paper if the ordinary representation of $...
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Is BGV encryption using different secret keys indistinguishable?
Assume that the same message is encrypted using two different keys within the BGV encryption scheme. Can we assume that the resulting ciphertext are indistinguishable?
I.e., given $c_1 = \text{Enc}(...
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Where do we put known bits of nonce when performing lattice attack on ECDSA?
I have read so many papers and posts about lattice attacks on ECDSA but none of them used an example of different MSB values for k but instead they all used fixed MSB.
So here i am trying to ...
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Is there a many-to-one reduction from GapSVP to GapCVP?
I was wondering if by now any poly-time Karp reduction between GapSVP and GapCVP (exact or approximate) exist. I know of the Cook reduction between these problems, but I couldn't find anything about ...
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Is there an efficient way to check if a lattice has a point with all non-zero components?
Given a basis $\{v_1,\dots,v_k\}$ for a $q$-ary lattice $L$ in ${\mathbb Z}_q^n$, is there an efficient (deterministic/randomized) way to find a point in $L$ with all non-zero components, or decide ...
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[error reducing techinique in lattice based commitment]
I am aware there are many techniques to reduce the error of lattice-based homomorphic encryption. But is there any technique to deal with lattice-based homomorphic commitment, e.g., More Efficient ...
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True Lovàsz condition and definition of a LLL-reduced basis
I am studying the Shortest Vector Problem and I have some troubles understanding the actual Lovàsz condition used in the LLL algorithm.
On the one hand, the original LLL article, the Springer book &...
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Differences between the theory and implementation of a lattice attack against ECDSA
I know the theory of lattice attacks against ECDSA from Minerva. So, as far as I can understand, the lattice that they build is
$$
L_M = \begin{bmatrix}
2^ln & 0 & 0 & \cdots & 0 & ...
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Non-lattice NIST candidates affected by SVP problems
I would like to know if there are non-lattice based NIST submissions that are affected by a polynomial time algorithm to Shortest Vector Problem. Are there known reduction from (e.g.) code based ...
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Understanding Unique-SVP and Kannan's Embedding
I am trying to understand the Kannan embedding technique. But I am confused about the formation of the B' and the finding of the short vector inside that basis. How does this basis matrix in the ...
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LWE with a binary matrix A
In LWE, we know that given reasonable public parameter $A\in \mathbb{Z}_q^{n\times \lambda}$, secret $s\in \mathbb{Z}_q^{\lambda}$ and noise $e\in \mathcal{X}^{n}$, random $r\in \mathbb{Z}_q^{n}$, $(A,...
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[About choosing params in BGV like ciphertexts]
I am new to lattice-based cryptography, so sorry that this question might seems stupid
May I ask that how can I choose the BGV parameter of ciphertext with plain text in mod 128, and error in ...
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About multiply by constant of LWE
I am new to lattice-based cryptography
May I ask that for a lattice-based encryption
$$enc(m) = A^{T}R+m \bmod q$$
If I set the $q$ to be able to decrypt to $m$ (and suppose the bound of $q$ is tight ...
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Estimating BKZ block size in Kyber
In Section 5.2.1 of the Kyber documentation, it states that the BKZ block size of 413 was chosen using the tool from this paper, i.e., this tool. How was the block size derived from this? Currently, ...
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Approximate-GCD problem in polynomials
I am trying to understand the two main hard problems that have been explored in the context of homomorphic encryption: Learning with Errors Problem (LWE) and the Approximate-GCD (AGCD) problem. I have ...
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How to set the variance of LWE when using the lwe estimator
based crypto
And I would like to use the lwe estimator to calculate bound for ring LWE
Found in this issue It seems to me I can set up parameters like params = LWE.Parameters(n=2^14, q=2^438, Xs = ...
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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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How are public and private keys generated and used for encryption and decryption in a lattice based cryptosystem?
I've recently become quite interested in lattice based cryptosystems, and I wish to understand them more deeply. I have only a rudimentary understanding of the shortest vector problem (SVP), and its ...
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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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in NTRU, can g be recovered given f and h?
The NTRU key generation involves polynomials and their arithmetic in polynomial rings, which is a bit different from arithmetic in modular integers.
In the NTRU cryptosystem, the public key $h$ is ...
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question for lemma 4 of the BGV paper
I would like to ask a question that arose when reading the proof of lemma 4 on page 10 of this BGV paper:
The assumption is:
And the inequality:
So it seems that
$$ \sum_{j=1}^{n} \parallel c'[j]-(p/...
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Can lattice attack work MSB or LSB are unkown but 16 bytes of private key are known?
I have been reading about lattice attack on ECDSA when partial bits of nonce are known for amount of signatures, So i went through some source code trying to understand how it works.
First of all, ...
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Is lattice encryption susceptible to Grover's algorithm?
So Grover's algorithm, also known as the quantum search algorithm, can find an entry, with a high probability, in an unstructured database.
Well can't we consider the basis of a lattice problem an ...
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What are the implications for the proof when we substitute matrix multiplication with a bitwise XOR operation in Definition 5.1 (LWE degree-k PRF)?
In the paper located at https://eprint.iacr.org/2011/401.pdf, suppose we replace matrix multiplication with bitwise XOR operations in Definition 5.1 to create an LWE degree-k PRF. I'm seeking ...
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Finding two inputs [i, j] of a custom Hash function where their Hashes are [H(i), H(j)] = [H(i), H(i)^2] [closed]
I came upon the following hash function (pseudo-code):
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$\epsilon$ parameter choice in lattice-based schemes
I am trying to implement Pei10 and BB13, but I am confused about what concrete parameters to use.
In Pei10, Algorithm 1 takes a rounding parameter $r = \omega(\sqrt{\log n})$ as parameter, but it does ...
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Discrete Gaussian distribution on a lattice vs. the periodic Gaussian function on a lattice
Gaussian distribution on lattices generally seems esoteric (at least for me, for now). My question is:
Does Gaussian distribution on a lattice mean to add a Gaussian noise on a single point of a ...
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Why the output of G-lattice sampling is spherical in the paper GM18?
In the paper GM18, they say that the sampling algorithm, SampleG, is shown in Figure 2. It takes as input a modulus $q$, an integer variance $s$, a coset $u$ of $\Lambda^{\perp}(g^T )$, and outputs a ...
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Computing the intersection of two lattices
Given two lattices $L_1$ and $L_2$ represented by bases $B_1$ and $B_2$, is there an efficient algorithm to compute $L_1\cap L_2$?
I can show, I think, that if $\gcd(\det(B_1),\det(B_2))=1$, then $...
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What is the difference between discrete-then-gaussian and gaussian-then-discrete?
In lattice cryptography, we always face the probem of discrete gaussian sampling. To the beginners, it is a bit complex. However, gaussian sampling from a continous space is much easier to understand, ...
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Sagemath help: Introduction to Lattices
Hi im doing a problem from the Chapter Lightweight Introduction to Lattices in "Learning and Experiencing Cryptography with CrypTool and SageMath"
I'm curious if my implementation is wrong ...
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[Questions about a proof in the prelim of paper "Lattice-Based Zero-Knowledge Proofs and Applications"]
May I ask that in section 2.7 challenge space in the paper Lattice-Based Zero-Knowledge Proofs and Applications:
Shorter, Simpler, and More General
What is rot(c), why does rot(c) $\in Z^{d*d}$, and ...
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Question about the description from ring SIS to SIS in the survey paper: A Decade of Lattice Cryptography
I am currently reading "A Decade of Lattice Cryptography"
At page 30, section 4.3.2, it descrip left multiplication by any fixed ring element a
It mention something about curcilant matrix ...
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[About parameters effect LWE and SIS to be computation or perfect secure]
Hello I am new to lattice cryptography
I am reading the paper More Efficient Commitments from
Structured Lattice Assumptions
They define bound B in page 3
Then In figure 1 in page 9
Can ...
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learning with errors
If I talk about efficiency of system of learning with error, is it it fine for q to be composite in Z_q, the ring of integers. As when q would not be prime, Z_q will not be field anymore, won't it ...
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Kyber-CCA-KEM - Deterministic implicit rejection
In Kyber-CCA-KEM, there's a step in the Fujisaki-Okamoto transformation, where decryption failure results in a random shared secret returned from the decapsulation call.
I have a C language project ...
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Matrix multiplication circuit
I am trying to understand which operations are computable by an $\texttt{NC}^1$ circuit. However, I am struggling to understand whether there is such a circuit for multiplying a matrix with a vector ...
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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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The asymptotic form of Hermite's constant in lattice
The are some linearly upper bounds on Hermite's constant $\gamma_d$, such as $\gamma_d \leq 2d/3$, $\gamma_d \leq d/4+1$. So we can claim that $\gamma_d=O(d)$. There is also a rather tight ...
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Literature on (concrete) hardness of Short Integer Solution (SIS)
I am interested in what the state of the art results on the hardness of the Short Integer Solution (SIS) instances are. The one I am the most familiar with (and the most discussed) is to use lattice ...
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Why use small error vectors in LWE instead of big ones?
In LWE systems, why is it recommended to add only small error vectors to the system of equations and not big error vectors? Can someone come up with an example?
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Is LWE easy when the matrix $A$ is sparse?
Recall that the LWE problem is the following.
LWE problem. Let $q$ be a prime, let $\chi$ be a distribution of small elements over $\mathbb{Z}/q$, and let $n,m$ be two integers (dimensions of vectors ...
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Can you instantiate Ring-LWE with coefficients from a prime-power field?
Generally, we instantiate Ring-LWE with the polynomial ring $R = \mathbb{F}_q\ /\ (X^N+1)$ for prime $q$ and some power-of-two $N$.
Can we instead do Ring-LWE over the ring $R = \mathbb{F}_q\ /\ (X^N+...
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