An ELI5 Intro to Lattices in Cryptography by@wagslane

January 24th 2020 307 reads

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Bitcoinist, libertarian, atheist, cryptography fan, and founder of http://qvault.io

Lattice-based cryptography has been in the spotlight recently: in January 2019, Many of the semifinalists in theย NIST post-quantum-cryptography competitionย were based on lattices. Letโs explore the basics of lattices and how they apply to cryptosystems.

A Lattice

According toย Wikipedia, a latticeย is the set of all integer linear combinations of basis vectors:

i.e.

More simply put, a lattice is defined by basis vectors, which are only able to be scaled by integersโฆ yay no fractions!

For example, letโs create a lattice of all the integers in a two-dimensional plane:

The definition of our lattice contains only 2 basis vectors,

v1 = (0,1)

v2 = (1,0)

Our lattice is the set ofย **allย **values that can be reached by any combination and scale of our basis vectors. For example, the point (2,0) is in our lattice because it can be reached by 2*v1

Similarly, we could create an entirely new lattice by changing our basis vectors to

v1 = (0,3)

v2 = (3,0)

As you can see, now the intermediary points (0,1) and (0,1)ย **no longer exist**ย in our lattice. There is no way to scale v1 (0,3) and v2 (3,0) to reach those points without using fractional scalars. With lattices, we can only scale by whole integers.

Cryptographic algorithms are typically based on mathematical problems that are easy to verify the answer of, but hard to calculate.

For example, RSA is based on prime factorization. If I told you to find prime factors of 27,919,645,564,169,759, that would be hard. However, if I told you that 48,554,491 and 575,016,749 are prime factors, all you have to do is multiply them together in order to verify my answer.

RSA works great with classical computers. There areย no known solutions to find prime factorsย of a number reliably in less than exponential time.

In the quantum world, things don't look so peachy. Shor's algorithm on quantum computers can crack RSA in less than exponential time. Many believe that lattice math could be the answer.CLICK TO TWEET

In this introductory article, we will take a brief look at one of the more well-known lattice problems that are of use in cryptosystems,ย the shortest vector problem (SVP).

Simply put, the goal of SVP is for the attacker to find the shortest vector from the origin (above in red) when given the basis of a lattice (above in blue). A zero vector doesnโt work as an answer, we consider it trivial.

Like RSA with classical computers, it is hard to find the shortest vector of a large lattice, especially if it exists in many dimensions. One such slow solution for approximating the shortest vector isย Babaiโs algorithm, orย Nearest Plane Algorithm, which you can read about in the links provided.

Lane on Twitter:ย @wagslane

Lane on Dev.to:ย wagslane

Download Qvault:ย https://qvault.io

(**Disclaimer**: *The Author is the Founder of QVault*)

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