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BrombergShpilrainVdovina SL₂ Homomorphic Hashing
This is an implementation of the TillichZémorstyle hash function presented in the paper "Navigating in the Cayley Graph of SL₂(𝔽ₚ)" by Bromberg, Shpilrain, and Vdovina.
Warning
This module is not produced by cryptography experts, but by some random guy. Furthermore, the algorithm was published in 2017, and is itself not at all battletested. Only use this library if you either (a) know what you're doing and have read and understood our code, and/or (b) are building something that does not rely heavily on the cryptographic properties of the hash function.
If you are a cryptography expert, we welcome any bug reports or pull requests! We also welcome them if you're not a cryptography expert; this library is quite simple, and should be easy to grok over a coffee with a copy of the paper linked above in hand.
What is this library for?
This library implements a putativelystrong hash function H with the
useful property that it gives a monoid homomorphism. This means there
is a cheap operation *
such that given strings s1
and s2
,
H(s1 ++ s2) = H(s1) * H(s2)
.
This property is especially useful for applications where some very long string may be constructed via many different routes, but you'd nonetheless like to be able to quickly rule out unequal strings.
It also allows you to hash parts of your data as you acquire them, and then merge them later in whatever order is convenient. This allows for very flexible hashing schemes.
H has some other cool properties, and is in some limited but potentiallyuseful sense "provably secure". See Bromberg et al. for details.
How to use this library
This library provides the means to construct HashMatrix
es, using
hash()
, which takes a slice of bytes. These hashes can be compared,
or serialized to hex strings using to_hex
.
use bromberg_sl2::*;
assert_eq!(
hash("hello, world! It's fun to hash stuff!".as_ref()).to_hex(),
"01c5cf590d32654c87228c0d66441b200aec1439e54e724f05cd3c6c260634e565594b61988933e826e9705de22884ce007df0f733a371516ddd4ac9237f7a46");
Hashes may also be composed, using the *
operator:
use bromberg_sl2::*;
assert_eq!(
hash("hello, ".as_ref()) * hash("world!".as_ref()),
hash("hello, world!".as_ref())
);
Technical Details
We use the A(2) and B(2) matrices as generators, and p = 2^127  1 as our prime order, for fast modular arithmetic.
We have not yet attempted to seriously optimize this library, and performance is a secondary goal. As of right now our procedure is about 1/3 as fast as SHA3512.
We needed an architectureagnostic cryptographic hash procedure with a monoid homomorphism respecting string concatenation, written in a lowlevel language. While there are a few implementations of related algorithms, e.g. the venerable but broken TillichZémor hash, from "Hashing with SL₂" , none of them fulfill these desiderata.
Dependencies
~240–560KB
~13K SLoC