# Questions tagged [lattice-crypto]

Lattice-cryptography is the study and use of lattice problems applied to cryptography.

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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) ...
1 vote
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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 ...
1 vote
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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 ...
1 vote
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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 ...
102 views

### 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?
177 views

### 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 ...
111 views

### 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 ...
134 views

### 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 ...
327 views

### 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 ...
6k views

### 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?
162 views

### 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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### 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. ...
114 views

### 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 ...
99 views

### "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 ...
208 views

### 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. ...
1 vote
110 views

### 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 ...
1 vote
62 views

### 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....
176 views

### 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 ...
1 vote
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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 ...
1 vote
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### NTRUEncrypt fails on complex algebra

I am following the NTRUEncrypt cryptosystem as described on the wikipedia. I have implemented it in Sage Math engine (with small problems along the way, but in the end - succesfully resolved) and the ...
130 views

### Why does NTRUEncrypt fail on different values for large modulus?

I am trying to closely follow the algorithm here (keeping the same variable names) and reconstruct the cryptosystem in Sage Math engine. It seems to work on parameters ...
1 vote
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### Decryption analysis for Regev's Public Key Cryptosystem

Regev's Public Key Cryptosystem is defined as follows: I want to proof the correctness. For this it must be shown that a 0 is decoded correctly and equally that a 1 is decoded correctly. I would ...
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### Do ideal non-cyclotomic lattices provide better compression in lattice-based cryptography?

Let $f \in \mathbb{Z}[x]$ be an irreducible polynomial of degree $N$ and $q \in \mathbb{N}$. Consider the rings $R := \mathbb{Z}[x]/f$ and $R_q := R/q$. Obviously, an element of $R_q$ can be ...
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### Question on the proof of correctness in CRYSTALS-Kyber

I am currently trying to follow the proof of correctness in the CRYSTALS-Kyber paper. The following is an excerpt of the proof: On the one hand, I am interested in how exactly one justifies/argues ...
322 views

### Compression and Decompression in CRYSTALS-Kyber

I am currently studying the Kyber Paper. I have a question about section 2.2 Compression and Decompression, but first I would like to quote the statement: Compression and Decompression. We now define ...
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### How does the SWIFFT algorithm relate finding hash collisions to a lattice problem?

I've been messing around with lattice based cryptography and came across the SWIFFT algorithm, a provably secure cryptographic hash function which has a security proof stating that finding collisions ...
1 vote
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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 ...
1 vote
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### Basis matrix of NTRU lattice

In NTRUEncrypt, we choose polynomials $\mathbf f,\mathbf g$ (with suitably small coefficients) such that $\mathbf f$ admits inverses $\mathbf f_p, \mathbf f_q$ with respect to the moduli $p,q$. The ...
59 views

### How do lattice-based proofs with Reed-Solomon codes simultaneously avoid aborts and multiple repeats?

I'm trying to understand how lattice-based schemes with the Reed-Solomon proximity testing work and why the scheme in Bootle's et. al. Fig 3 has no aborts at all (nor big number of repetitions). TLDR:...
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### Dense sphere packings and lattice-based cryptography

It is known that there are two popular applications of lattices: dense sphere packings and lattice-based cryptography. I didn't find any information on the Internet about possible interaction of these ...
76 views

### Lattice-based Signatures and Hashes

Although many different lattice-based signature schemes exist, Hash and Sign signatures schemes, like [GPV08], are prevalent. On the other hand, it is well known that collision-resistant hash ...
60 views

### Question about the definition of the CVP as an approximation variant

I have a question about the definition of the (Closest Vector Problem) CVP. In the literature you can find for example this definition of the approximation variant of CVP: $CVP_\gamma$, Search: Given ...
83 views

### Closest Vector Problem in RLWE

I am interested in a polynomial form of the lattice problem Closest Vector Problem (C.V.P), or in other words if C.V.P. can be ''transferred'' to Ring-LWE. My idea about this question is that a ...
50 views

### MITM against NTRU

In MITM attacks against the NTRU cryptosystem, we exploit the fact that in the ring of truncated polynomials of degree $n-1$ it holds that $$fg=h\mod q$$ for our secret and public keys $f,h$. The ...
1 vote
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### value bound of r⋅e for LWE Decryption correctness

For LWE decryption, Someone told me that If we can bound r⋅e by q/4 then we can retrieve M by checking if this is closer to <...
154 views

### The sum of independent discrete Gaussians is a discrete Gaussian

I am currently learning about lattice-based cryptography and, reading from A Decade of Lattice Cryptography by Peikert, specifically section 2.3, it emerges that [...] if the parameter s is greater or ...
1 vote
185 views

### Encryption and decryption for LWE

For https://asecuritysite.com/public/lwe_ring.pdf#page=9 , could anyone explain how the encryption and decryption for LWE work ? When I do more reading on https://summerschool-croatia.cs.ru.nl/2015/...
1 vote
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### Coefficient Growth

In this survey, I don't understand the necessary of coefficient (paragraph 4.1.2) growth and the choice of $X^d\pm 1$ or $X^d \pm X^{d/2} +1$, since later introduces $q$ which doesn't mention the ...
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### Comparison of Complete NTT and Incomplete NTT Multiplication

is the complete NTT is the fastest algorithm to multiply polynomials or there are hybrid versions that are faster than complete NTT multiplication?
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### Statistical Distance and Learning with Rounding

Given an integer $b$ modulo a prime $q$, one can define a `rounding’ function $\lfloor b\rceil_p$ for a prime $p$, $p<q$, as follows: $$\lfloor b\rceil_p = \lfloor \frac{p}{q}\cdot b\rceil\bmod p.$$...
1 vote
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### Breaking RSA with P,Q LSB bits using LLL Lattice reduction

This question is someway correlated to Breaking RSA with P,Q LSB bits but more specific. I would like to use LLL to fully reconstruct P,Q given some LSB bits of P and Q in an arbitrary base B. Let's ...
1 vote
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### Type 1 Trapdoor Sampling in LWE

In the BGN-like LWE cryptosystem, section $2.2$, we sample a $m \times m$ trapdoor matrix $T$ that is full rank such that $TA = 0 \pmod q$. Suppose that $q$ is prime so we are in a finite field: if $T$...
I already asked a question about the feasibility of LWE when the matrix A is sparse or small here. Let $q$ be a prime, let $\chi$ be a distribution of $\textit{small}$ elements over $\mathbb{Z}/q$, ...