Questions tagged [lattice-crypto]

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

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New quantum attack on lattices (or Shor strikes again)?

Lior Eldar and Peter W. Shor published a paper on arXiv.org in which they present a new quantum algorithm against a variant of BDD. They claim that their new algorithm can efficiently solve the ...
29
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1answer
898 views

Uniform vs discrete Gaussian sampling in Ring learning with errors

The Wikipedia article on RLWE mentions two methods of sampling "small" polynomials namely uniform sampling and discrete Gaussian sampling. Uniform sampling is clearly the simplest, involving simply ...
19
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1answer
658 views

Quantum complexity of LWE

As per my understanding, LWE is quantum secure because there is no known quantum algorithm to solve LWE in polynomial time. Due to the reductions given by Regev et al., if there is any algorithm that ...
17
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1answer
4k views

What are the benefits of lattice based cryptography?

Previously we visited the benefits of elliptic curves for cryptography. Lattice based cryptography is starting to become quite popular in academia. The primary benefit of lattice based crypto is the ...
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736 views

Why are only lattice problems used in cryptography?

There are thousands of NP-hard problems out there. Why have only lattice problems been applied to cryptography?
12
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686 views

Given a 'good' basis for a lattice, how can we solve the CVP?

I'm doing a little bit of reading about lattices. I read that if we can find a 'short' basis for our given lattice, we can solve CVP and SVP very efficiently. However, the paper didn't describe an ...
11
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1answer
1k views

Is lattice-based cryptography practical?

How viable is lattice-based cryptography in a "practical" setting? It has been said that lattice-based cryptography would be a "post-quantum" cryptography scheme, but is it feasibly implementable?
11
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1answer
2k views

Is the “New Hope” Lattice Key Exchange vulnerable to a lattice analog of the Bernstein BADA55 Attack?

In the paper, "Post Quantum Key Exhange - A New Hope," the authors present a lattice-based key exchange based on the work of Chris Peikert. In this "New Hope" key exchange the authors try to gain ...
11
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0answers
448 views

Potential Flaws With Lattice Based Cryptography?

From researching post-quantum cryptographic schemes it seems hash-based and lattice-based algorithms are the most promising (MQ-based seem to be covered by patents and have more potential unknowns ...
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2answers
952 views

Concrete evidence for the asymptotics of $\lambda_1(\Lambda^\perp(A))$?

A recent eprint paper claims to bound $\lambda_1(\Lambda^\perp(\mathbf{A}))$ for $\mathbf{A}\in\mathbb{Z}^{n\times m}$, a uniformly random matrix, by $O(1)$, specifically by $4$. This has applications ...
10
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73 views

Decision R-LWE parameters for spherical error with worst-case hardness

In Peikert et al.'s most recent work (STOC 2017) a direct reduction of worst-case lattice problems to decision R-LWE is achieved for $\alpha q \ge 2 \cdot \omega(1)$ (Theorem 6.2), where $\alpha q$ is ...
9
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2answers
450 views

Peikert's framework for attacks on R-LWE: What “reduction modulo q” means?

I am reading Peikert's paper [Pei16] about secure instantiating of R-LWE problem. In section 3.1, The author gives a new attack framework by using "reduction modulo an ideal divisor $\mathfrak{q}$ of ...
9
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1answer
360 views

Converting NewHope/LWE key exchange to a Diffe-Hellman-like algorithm

By a “Diffe-Hellman-like” algorithm, I mean one that has the same API as Curve25519, etc (disregarding trivial differences such as the size of parameters): a function $$F: (P_\text{other}, S_\text{...
9
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1answer
124 views

Collisions in the cyclotomic knapsack function

I've been working my way through the paper “Efficient Collision-Resistant Hashing from Worst-Case Assumptions on Cyclic Lattices” by Peikert and Rosen, and I've come across something that doesn't seem ...
9
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1answer
251 views

Most influential/illuminating papers/books/courses on lattice based cryptography?

I'm interested in some sort of "compendium" on lattice-based crypto. There are a bunch of maths behind FALCON and other stuff. A lot of articles are devoted to lattice crypto, but not of them are of ...
9
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503 views

Why SIVP Is Worst Case Problem?

I just started to study lattice Cryptography. I'm now studying worst-case to average-case reduction for SIS. In previous question, "worst means any and average means random". And I wonder why the ...
8
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2answers
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What does “Worst-case hardness” mean in lattice-based cryptography?

In the wiki page of Lattice-based Cryptography the "Worst-case hardness" is defined as below: Worst-case hardness of lattice problems means that breaking the cryptographic construction (even with ...
8
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1answer
821 views

Lattice Crypto worst case to average case

I am currently reading the ETSI white paper Quantum Safe Cryptography and Security On page 24 one finds the following statement: Lattice problems also benefit from something called worst-case to ...
8
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1answer
1k views

Problem with LLL reduction on truncated LCG schemes

I am struggling to apply Freize et al. paper to break a truncated LCG. A truncated LCG is a pseudo random generator that outputs the $n$ leading bits $y_i$ of $x_i$, where $(x_i)$ is such that $x_{...
8
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2answers
163 views

Cardinality of the group of units in a cyclotomic ring?

In the NTRU key generation, one samples a polynomial from $K = (\mathbb Z/q\mathbb Z)[X]/(X^n+1)$ and tests if it is invertible. What are the chances of this to happen? In other words: Let $q$ be a ...
8
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1answer
348 views

Is HIMMO For Real?

I've come across something out of Philips Research called HIMMO Key Pre-Distribution Scheme (https://eprint.iacr.org/2016/410.pdf and https://www.ietf.org/proceedings/96/slides/slides-96-cfrg-2.pdf) ...
8
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1answer
266 views

Lattice attacks against Multilinear Maps [CLT13]

I am currently studying an article on a construction of Multilinear maps. There are some attacks on the scheme presented by the authors and I got stuck at the one in section 5.1. I will try to ...
8
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1answer
218 views

LWE: error and float operations

Background I'm trying to make sense of the error in implementations of LWE and R-LWE. In LWE and R-LWE error is added to vectors in lattices to make it computationally infeasible to recover any ...
8
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131 views

Differences between “NewHope” and “NewHope-simple”

The well-known paper described a key exchange (KE) scheme named "NewHope" on USENIX 2016. The authors then proposed "NewHope-Simple" - a PKE/KEM scheme. They also submitted "NewHope for NIST" - ...
7
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1answer
425 views

Relation between decisional SIS and leftover hash lemma in lattices

The semantic security of Regev's cryptosystem [Reg05] is based on the LWE assumption and leftover hash lemma. This lemma implies that because $m \approx (n+1)\log q$ is large enough, so for uniform $A\...
7
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1answer
328 views

Are LPN and LWE problems equivalent?

Learning with Error (LWE) problem seems like a generalization of Learning Parity with Noise (LPN) problem, where in the latter one uses bits. But, this also makes LPN seem very related to the problem ...
7
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1answer
1k views

Why is the Lovász condition used in the LLL algorithm?

The LLL algorithm is used to approximate the Shortest Vector Problem, i.e., it outputs a reduced basis. Such a basis will satisfy two conditions: $$ \forall i \gt j. \quad \lvert\mu_{ij}\rvert \le \...
7
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1answer
528 views

Find collision in Ajtai's hash function using short vector

Background What is Ajtai's hash function? Given a matrix $A \hookleftarrow U(\mathbb{Z}_q^{n \times m})$ and a column vector $\vec{m} \in \mathbb{Z}_d^m$, the hash of the message $\vec{m}$ is given ...
7
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1answer
362 views

Trying to implement ring-LWE KE to understand the concept

Today, after reading so much about ring-LWE key exchange, I decided to implement it in java to see if it works. Not a real world implementation, just to see if math works out. My assumption was that ...
7
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3answers
1k views

What is a purpose of reducing lattice basis?

This may be too broad question but it is not. I have been studying lattices for few months now, more specifically I studied: lattice problems ($SVP$, $CVP$ and etc.) lattice cryptography in post ...
7
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1answer
756 views

Gaussian distribution in lattices

In many lattice based cryptosystems, Gaussian distribution is used. Can you explain why only Gaussian distribution is preferred?
7
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1answer
142 views

What's the purpose of the smoothing parameter in lattice-based cryptography?

I see nearly all the lattice-based crypto papers talk about the smoothing parameter $\eta$. And I believe even some parameters are chosen with respect to that. However, I do not quite understand what'...
7
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1answer
937 views

Why is Approximate GCD a hard problem?

There are many Fully Homomorphic Encryption over the Integers schemes whose security is based on the intractability of the Approximate GCD (AGCD) problem. The paper Algorithms for the Approximate ...
6
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1answer
1k views

Help in understanding exactly how lattices used as one way functions for hashing

I am doing a cryptography course via long distance and we have been given an assignment which is based on lattice-based cryptography. I have spent the majority of the past week sifting through papers ...
6
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1answer
481 views

Discrete Gaussian Sampling role in Lattice-Based Crypto?

I'm reading up on how post-quantum cryptography works, and stumbled upon the notion of discrete Gaussian sampling. However, I can't understand where it fits in the greater picture - currently it feels ...
6
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1answer
312 views

Irreducible polynomial in Ring-LWE

In Ring-LWE polynomials are chosen from the ring $R_q=\mathbb{Z}_q[x]/(x^n+1)$, where $n$ is a power of two. As far as I understand, to create a ring the polynomial $x^n+1$ has to be irreducible (see ...
6
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1answer
839 views

Use of q-ary lattices in developing cryptosystems

Why q-ary lattices are used to most cryptosystems rather than lattices. In most of the papers q-ary lattices are used. Is there any advantage? and Given $$B=(v_1,v_2,v_3,.....v_n)$$ is the basis, ...
6
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1answer
461 views

How to generate new LWE samples

Assume we are given a small fixed number of LWE samples with secret $s$ and error $e$, where the error distribution is taken so that the LWE problem is hard. My question: How can one further ...
6
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1answer
722 views

What is the difference between Module-LWE and Ring-LWE?

Recently, the CRYSTALS lattice-based cryptographic suite has been published, which is based on "module lattices". What is Module-LWE? How is it different from Ring-LWE?
6
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1answer
187 views

Provably Secure Password Authenticated Key Exchange Based on RLWE for the Post-QuantumWorld

In this paper (Provably Secure Password Authenticated Key Exchange Based on RLWE for the Post-Quantum World), author describe password authenticated key exchange scheme on page 9 and 10 (see Fig. 1 on ...
6
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3answers
289 views

Is the HNF basis the worst basis for a lattice?

I am researching lattice problems and some methods for solving them. I read some books that mentioned Babai's algorithm for finding the Closest Vector Problem (CVP) cannot be successful with a "bad" ...
6
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1answer
94 views

Is LPN not as important as LWE and SVP?

I've been learning about lattice cryptography and have noticed that most resources such as this survey by Chris Peikart, the Winter School on Lattice Cryptography etc don't include material on LPN, ...
6
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1answer
393 views

When does the SIS (Short Integer Solution) Lattice-problem start becoming easy (According to the parameters size)?

SIS (Short Integer Solution) Problem : Given $m$ uniformly random vectors $a \in Z_q^n$, grouped as the columns of a matrix $A \in Z_q^{n.m}$, find a nonzero integer vector $z \in Z^m$ with $||z|| \...
6
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1answer
119 views

Decision to Search LWE when modulus $q=p^e$

I am reading Applebaum et al.. In Lemma 1. (page 7), Applebaum et al. proved the decision to search reduction when the modulus $q=p^e$ for prime $p$. In the proof, they define the hybrid ...
6
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0answers
104 views

Can LWE be NP-hard?

Regev's reduction shows that LWE is quantumly at least as hard as CVP with an approximation factor of $n/\alpha$ for $0<\alpha<1$. But I just watched this talk which said that if $\sqrt{n/\log n}...
6
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0answers
157 views

Does there exist trapdoor permutation from lattices?

It seems that the lattice functions are either surjective (SIS) or injective (LWE), due to the error that is basically intended to destroy the structure and provide security. I was wondering whether ...
5
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2answers
517 views

Randomness re-use in LWE encryption scheme

Let me describe the scheme first, it is the scheme proposed by O. Regev when he introduced the LWE assumption. $sk = \textbf{s} \in \mathbf{Z}_q^n$ $pk = \textbf{A}\textbf{s}+\textbf{e}$ where $\...
5
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4answers
516 views

R-LWE key exchange why using FFT instead of Karatsuba

In the paper Post-quantum key exchange for the TLS protocol from the ring learning with errors problem one of the authors, Douglas Stebila, uses the FFT algorithm for polynomial multiplication but he ...
5
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3answers
285 views

A Simple Provably Secure Key Exchange Scheme Based on the Learning with Errors Problem

In this paper (A simple provably secure key exchange by Ding et al.) At page number 8, the author gives correctness of the technique as follows then SK A = SKB with overwhelming probability i.e. if ...
5
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1answer
482 views

Why is lattice-based cryptography believed to be hard against quantum computer?

Why is lattice-based cryptography believed to be hard against quantum computer? Learning With Errors(LWE) problem (reduction to SVP) is just one example. Can you provide some intuition of the ...

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