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https://github.com/pornin/paradox-compress

Paper and Demo Implementation of Paradoxical Compression with VDF
https://github.com/pornin/paradox-compress

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Paper and Demo Implementation of Paradoxical Compression with VDF

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# Paradoxical Compression



A lossless compression algorithm that can actually reduce the size of

at least one input file must also necessarily expand the size of at

least one other possible input file. This is a direct application of the

[pigeonhole principle](https://en.wikipedia.org/wiki/Pigeonhole_principle).

A compression algorithm that claims to never increase the size of any

input, while at the same time reducing the size of at least *some*

inputs (it's not the trivial do-nothing compression that, by definition,

preserves the length of the input), is called *paradoxical compression*.



Paradoxical compression is mathematically impossible. This repository

contains a [description](doc/paradox-compress.pdf) of how to nonetheless

achieve it, as well as [a demo implementation](src/).



Of course I am slightly cheating here, but the cheat is conceptually

interesting (and fun). The idea is that paradoxical compression might

have only a small density of problematic cases (inputs where the

algorithm misbehaves, e.g. by expanding the length or not decompressing

to the original input) and tools from the field of cryptography can be

used to "push away" these cases so that they mathematically exist but

nobody can find one with nonnegligible probability. In essence, this is

transforming the question from "is it mathematically possible?" to "can

I claim to do it and not being detected as a fraud?". This change of

model makes paradoxical compression possible.



**Warning:** "Paradoxical compression" is only about *not increasing*

the length of any input. This is NOT "infinite compression", in which it

is claimed that every possible input can be *reduced* in size. There

is no shortage of scammers who peddle bogus infinite compression

schemes; what I describe here will not help them. Infinite compression

is mathematically impossible, but a lot more impossible than paradoxical

compression (so to speak). Paradoxical compression can be achieved in

practice because problematic inputs (inputs where two pigeons must be

crammed into the same hole, in the pigeonhole metaphor) are rare and can

be hidden away with cryptographic tools: you won't find such an input

randomly, out of pure luck, the probability of hitting one is way too

low, and the crypto tools can even make it infeasible to find a bad

input on purpose. But with infinite compression, this is not the case:

at least half of all possible inputs are "problematic" (i.e. they reveal

that the claim is bogus, by not being preserved across a

compression/decompression cycle), and thus easy to find (just try random

values!). This is inescapable: one cannot achieve infinite compression

without being quickly detected as a fraud.



**Warning 2:** Paradoxical compression is, practically speaking,

useless. Data compression is useful if used in a context where some

advantage can be leveraged out of shortened data, i.e. in the presence

of some metadata that can identify the length of each individual

message. In almost all cases, an extra "compressed / not compressed"

flag can be smuggled somewhere in such metadata, which is enough to

avoid paylooad expansion through compression without using any of the

stuff described here. Thus, there should be little point in reusing the

methods and code from this repository in any tangible software project.

As a mathematical recreation, though, paradoxical compression is sort of

fun.



## Mathematical Description



The [paper](doc/paradox-compress.pdf) includes a detailed description

of the concept and its realization. Two instantiations are described:



  - A simple instantiation requires the compressor and decompressor to

    share a secret key; this is a restrictive usage context, but it

    allows for an efficient, low-overhead implementation using only

    a MAC (message authentication code).



  - A more complex instantiation is keyless: no secret is needed, so that

    the compressor and decompressor may, for instance, be code that runs

    on anybody's machine. The cryptographic tool used here is a

    *verifiable delay function* (VDF), specifically the one [described

    by Wesolowski at Eurocrypt 2019](https://eprint.iacr.org/2018/623).



## Implementation



The [src/](src/) subdirectory contains a demo implementation of the

keyless method, with Wesolowski's VDF. It is written in pure C# and is

not especially efficient (it's just a demonstration). It can be built

and executed on Windows, macOS or Linux. On macOS and Linux, you need to

install some C#/.NET support, e.g. [Mono](https://www.mono-project.com/)

(on Ubuntu systems, just install the `mono-devel` package).



To build the code, use the `build.cmd` (on Windows) or `build.sh` (on

Linux and macOS) script. This produces three command-line executables,

`Compress.exe` (to compress a file), `Decompress.exe` (to decompress a

file), and `TestParadoxCompress.exe` (to run self-tests).



In the source code, the [BigInt/](src/BigInt/) directory contains a

general-purpose big integer implementation; it is somewhat faster than

the one that comes with .NET 4.0 in the specific case of big integer

values that happen to be small (i.e. they could fit on 32 bits). It also

offers integer primality tests, which the .NET implementation does not

have; this is used in the paradoxical compression implementation.



The [Crypto/](src/Crypto/) directory contains a straightforward pure C#

implementation of the SHA3 hash function, and the SHAKE extensible

output function (XOF), as specified in [FIPS

202](https://csrc.nist.gov/publications/detail/fips/202/final).



The real core of the implementation is in the

[ParadoxCompress.cs](src/ParadoxCompress.cs) file. I included comments.

Refer to the paper for details.



## License



If you think about reusing this code anywhere, read again the warnings

above.



The SHA3/SHAKE implementations, and the big integer code (`ZInt`

structure), can be useful. Feel free to reuse; license is, formally

speaking, MIT (see the [LICENSE](LICENSE) file), which I understand to

be sort of the default "no worry" license that allows anybody to reuse

the code without making me responsible for anything.