Questions tagged [lattice-crypto]
Lattice-cryptography is the study and use of lattice problems applied to cryptography.
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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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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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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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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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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 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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[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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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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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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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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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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[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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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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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$$
If I set the $q$ to be able to decrypt to $m$ (and suppose the bound of $q$ is tight ...
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in NTRU, can g be recovered given f and h?
The NTRU key generation involves polynomials and their arithmetic in polynomial rings, which is a bit different from arithmetic in modular integers.
In the NTRU cryptosystem, the public key $h$ is ...
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question for lemma 4 of the BGV paper
I would like to ask a question that arose when reading the proof of lemma 4 on page 10 of this BGV paper:
The assumption is:
And the inequality:
So it seems that
$$ \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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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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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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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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$\epsilon$ 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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[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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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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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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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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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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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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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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Misuse Attacks on Lattice Crypto
I've been reading "Misuse Attacks on Post-Quantum Cryptosystems" (https://eprint.iacr.org/2019/525). In what scenarios are the attacks described in the paper applicable? Is it specifically ...
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LWE and distributions
In LWE, the error term $e$ is "classically" obtained from the discrete normal distribution. Why is it so often found that this distribution is used? Are there other possibilities for ...
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Does randomization make a big difference in the output of the BKZ algorithm?
We all know that block Korkine-Zolotarev (BKZ) algorithm is essentially a deterministic lattice reduction algorithm. However, in the actual implementation, the BKZ algorithm contains some ...
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Trapdoor Quality for Lattice Crypto
In these two papers the authors mention the "quality" of a trapdoor
[GPV] https://eprint.iacr.org/2007/432
[MP] https://eprint.iacr.org/2011/501
But the best detail on this matter I could ...
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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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Why Module-LWE and not Ring-LWE?
I am trying to understand the NIST-submissions for post-quantum cryptography a bit better, and I noticed that the submissions from the CRYSTALS-family in particular is based on Module-LWE.
I ...
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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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Finite index quotient group, lattice crypto
A $m$-dimentional full-rank integer lattice $\Lambda\in\mathbb{Z}^{m}$ can be defined as the set of all integer linear combinations of $m$ linearly independent over $\mathbb{R}$ basis vectors $\textbf{...
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Post-quantum secure trapdoor function
I am looking for examples post-quantum secure trapdoor functions. Ideally, the inversion knowing the trapdoor should be "simple" in the sense that it can be computed by a circuit in NC^1.
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KYBER.CPAPKE: IND-CCA Security of Lyubashevsky, Peikert, Regev (LPR) Encryption
The NIST Kyber KEM spec. defines an encryption scheme, KYBER.CPAPKE, that's a variant of the so called Lyubashevsky, Peikert, Regev ("LPR") encryption scheme [1]. While LPR encryption is ...
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About learning with error rings with only constant coefficient
I am new to RLWE, would like to ask whether what I am thinking make sense
Suppose I have a message e.g.: x=5
And I have a lattice based encryption scheme, e.g.: BGV
could I encrypt x with BGV by ...
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How does big Galois groups yield better security in NTRU Prime?
I'm still kinda new to Galois theory so I apologize if this question is very obvious to some people.
Basically I'm reading this paper by the NTRU Prime team and in section 2.5 it's explaining how ...
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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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