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How can a lattice attack be applied to ECDSA signatures?

The aim is to check if it is possible to break the ECDSA cryptosystem under the following criteria. Suppose that each ECDSA signature is generated by using the GLV method for point multiplication ...
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1answer
74 views

Ring-LWE elliptic gaussian distribution

I am currently trying to understand this Ring-LWE article: http://www.cims.nyu.edu/~regev/papers/ideal-lwe.pdf and I have a question. Firstly, it is mentioned in the paper that we can view the ...
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1answer
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What is the meaning for a vector mod a matrix in a lattice?

I'm reading about the lattice recently.In the paper, it gives a method of a vector mod a matrix: ⃗c mod B as ⃗c−⌊⃗c×B^(−1)⌉×B = [⃗c×B^(−1)]×B. I know that a integer A mod the other integer B is ...
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1answer
82 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 ...
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2answers
88 views

What is the difference between the standard representants of $\mathbb Z/q\mathbb Z$?

The symbol $\mathbb Z/q\mathbb Z$ (given that $q$ is prime) represents the prime field $\mathbb Z_q$. Basically, the elements of this field are represented by $\{0, 1, \dots, q-1\}$, let's call this ...
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1answer
166 views

What is a “lattice” in cryptography?

There are some questions here concerning lattice-based cryptography and this kind of cryptography seems to be especially useful if quantum computers are assumed to exist. When reading such questions ...
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69 views

How to recover $e$ from $f_A(e) = Ae \mod q$ when knowing trapdoor

I have a silly question, but I don't know a solution, so I need to question. Assume, with a algorithm $TrapGen(1^\lambda)$, it generates $A\in\mathbb{Z}_q^{n\times m}$ with a basis $B ...
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113 views

The Inhomogeneous Short Integer Solution (ISIS) problem with a clue

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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1answer
134 views

Gap problem for Learning With Errors

Informally, a "Gap problem" arises when solving the computational (or search) version using an oracle for the decisional version. This definition of Gap Problem was introduced by Okamoto and ...
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2answers
135 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 ...
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1answer
80 views

Arithmetic modulo 1

In the context of lattice-based cryptography, in particular the Learning With Errors (LWE) problem, I see some definitions given in terms of equations modulo 1 (see for example, Appendix A of this ...
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161 views

The Short Integer Solution (SIS) problem for square matrices

The short (or small) integer solution (SIS) problem is as follows: given an integer $q$, a matrix $A \in \mathbb Z^{n\times m}_q$, and a real $\beta$, find a nonzero integer vector $e\in \mathbb Z^m$ ...
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1answer
102 views

Trapdoors for lattices

I refer to an article https://eprint.iacr.org/2011/501. I focus on (a bit modified) Algorithm 1 which runs as follows (in my understanding): For given $n, m\in \mathbb N$, $q=2^k$ and a distribution ...
4
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1answer
213 views

Using Lattice-based cryptography for TLS\SSL

Given the general benefits of Lattice-based cryptography, such as: Post quantum Security Security from worst case scenario Efficiency What could the outlook of shifting from RSA \ ECC-based ...
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127 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 ...
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0answers
126 views

Understanding Lattice based cryptosystem [closed]

I have heard that there is one paradigm of Public key cryptosystem, called Lattice based crytography. Further, its security claims are such that it will not be affected even if the quantum computers ...
2
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1answer
243 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 ...
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1answer
210 views

Illustrate NTRU using lattices

I studied some papers related to NTRU. All these papers describe NTRU as a lattice based cryptosystem but I could not find any paper which illustrates NTRU algorithm from lattice point of view. It ...
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1answer
128 views

What are alternatives to number theory based crypto? [closed]

Quantum crypto,lattice based crypto, Neurocryptography and cellular automata based cryptography are alternatives to number theory based crypto. I need to know what are the other hard problems like ...
3
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2answers
432 views

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 ...
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1answer
178 views

Resources for basics of lattice crypto [closed]

I'm looking to fill in the gaps of my knowledge of lattices. Can anyone point me towards papers or books that introduce lattice crypto assuming a fairly solid math background? Mods: Feel free to ...
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1answer
1k 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 ...