Building lattice reduction (LLL) intuition (opens in new tab)

(kel.bz)

81 pointskkl8y ago7 comments

7 comments

This is probably my favorite crypto blog post of the year. LLL comes up a lot in attacks on asymmetric cryptography.

If you're interested in crypto or linear algebra but don't know what a lattice is, a great starting point is Hoffstein's _Introduction to Mathematical Cryptography_:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.182...

(If you know what a vector space is, you know what a lattice is; it's a vector space where only integers are allowed as coefficients in linear combinations).

Here's Antoine Joux, one of the world's most renowned cryptanalysts, talking about applications of LLL to crypto attacks. Amusingly, he sort of opens by making fun of cryptographers for using LLL without really understanding it:

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.11.7...

Here's Babai's application of LLL to finding close lattice points:

http://www.csie.nuk.edu.tw/~cychen/Lattices/On%20lovasz%20la...

From here you're a [boneh hidden number problem] Google search away from attacking dlog/ecdlog crypto from vulnerabilities like biased nonces.

aruss8y ago

To add to your post, lattice attacks (LLL/BKZ/etc) are becoming even more prevalent in cryptanalysis as they are the best known attacks against next-gen cryptosystems like NTRU, or anything based on the Learning-with-Errors problem, which includes schemes with "fun" properties like full P/poly homomorphism, attribute/predicate encryption, etc.

abetusk8y ago

For context, LLL was used to prove that polynomial factorization can be solved in polynomial time. That is, given r(x) such that:

    r(x) = p(x) * q(x)

where the coefficients of each factor are integral:

    p_i, q_i in Z

LLL can be used to find p(x) and q(x) only given r(x).

There are other problems that LLL solves and there are more modern lattice reduction algorithms (PSLQ, etc.) but LLL was one of the first.

cperciva8y ago

There are other problems that LLL solves

My favourite is integer relationship finding. For example, if you stumble across the value 194.927424491 it's straightforward to figure out that it is pi * 23 + e * 29 + sqrt(2) * 31.

aruss8y ago

So, solving CVP? That's a pretty neat application, though.

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7 comments

tptacek8y ago

This is probably my favorite crypto blog post of the year. LLL comes up a lot in attacks on asymmetric cryptography.

If you're interested in crypto or linear algebra but don't know what a lattice is, a great starting point is Hoffstein's _Introduction to Mathematical Cryptography_:

http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.182...

(If you know what a vector space is, you know what a lattice is; it's a vector space where only integers are allowed as coefficients in linear combinations).

http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.11.7...

Here's Babai's application of LLL to finding close lattice points:

http://www.csie.nuk.edu.tw/~cychen/Lattices/On%20lovasz%20la...

From here you're a [boneh hidden number problem] Google search away from attacking dlog/ecdlog crypto from vulnerabilities like biased nonces.

aruss8y ago

abetusk8y ago

For context, LLL was used to prove that polynomial factorization can be solved in polynomial time. That is, given r(x) such that:

    r(x) = p(x) * q(x)

where the coefficients of each factor are integral:

    p_i, q_i in Z

LLL can be used to find p(x) and q(x) only given r(x).

There are other problems that LLL solves and there are more modern lattice reduction algorithms (PSLQ, etc.) but LLL was one of the first.

cperciva8y ago

There are other problems that LLL solves

My favourite is integer relationship finding. For example, if you stumble across the value 194.927424491 it's straightforward to figure out that it is pi * 23 + e * 29 + sqrt(2) * 31.

aruss8y ago

So, solving CVP? That's a pretty neat application, though.

1 more reply

j / k navigate · click thread line to collapse