plzin / mba

Mixed Boolean-Arithmetic

Mixed Boolean-Arithmetic Obfuscation

This algorithm transforms expressions like x+y into monstrosities like this:

$ cargo run --example quadratic_mba 8 x+y
212 * -1 * (x ^ y) + 215 * -1 * (x & y) + 5 * -1 * ~(x | y) + 39 * -1 * ~x + 87 * -1 * ~y + 98 * -1 * z + 107 * -1 * x + 237 * -1 * y + 152 * -1 * (y & z) + 36 * -1 * (x | z) + 76 * -1 * (~x & z) + 162 * -1 * (y | ~z) + 68 * (x & y) * (x ^ y) + 85 * (x ^ y) * ~(x | y) + 184 * (x ^ y) * ~x + 140 * (x ^ y) * ~y + 96 * z * (x ^ y) + 55 * x * (x ^ y) + 223 * y * (x ^ y) + 63 * (y & z) * (x ^ y) + 101 * (x | z) * (x ^ y) + 159 * (~x & z) * (x ^ y) + 10 * (y | ~z) * (x ^ y) + 50 * (x & y) * ~(x | y) + 252 * (x & y) * ~x + 72 * (x & y) * ~y + 6 * z * (x & y) + 192 * x * (x & y) + 240 * y * (x & y) + 190 * (y & z) * (x & y) + 58 * (x & y) * (x | z) + 184 * (x & y) * (~x & z) + 59 * (x & y) * (y | ~z) + 194 * ~x * ~(x | y) + 193 * ~y * ~(x | y) + 191 * z * ~(x | y) + 54 * x * ~(x | y) + 174 * y * ~(x | y) + 187 * (y & z) * ~(x | y) + 69 * (x | z) * ~(x | y) + 209 * (~x & z) * ~(x | y) + 222 * (y | ~z) * ~(x | y) + 58 * ~y * ~x + 138 * z * ~x + 78 * x * ~x + 227 * y * ~x + 183 * (y & z) * ~x + 251 * (x | z) * ~x + 187 * (~x & z) * ~x + 131 * (y | ~z) * ~x + 21 * z * ~y + 179 * x * ~y + 17 * y * ~y + 54 * (y & z) * ~y + 147 * (x | z) * ~y + 152 * (~x & z) * ~y + 189 * (y | ~z) * ~y + 115 * z * x + 81 * z * y + 26 * z * (x | z) + 127 * z * (~x & z) + 216 * y * x + 203 * x * (y & z) + 115 * x * (x | z) + 85 * x * (~x & z) + 251 * x * (y | ~z) + 213 * y * (y & z) + 48 * y * (x | z) + 13 * y * (~x & z) + 110 * y * (y | ~z) + 196 * (y & z) * (x | z) + 163 * (y & z) * (~x & z) + 153 * (~x & z) * (y | ~z)

These kinds of expressions involving both normal arithmetic as well as boolean operations are known as mixed boolean-arithmetic expressions. This particular transformation is only valid when x and y (and z) are 8-bit integers and the usual rules of computer arithmetic apply (e.g. when adding/multiplying numbers and there is an overflow then the most significant bits that can not be represented are cut off). In particular, this will not work when the numbers are floating point numbers. Rust itself will panic (at least in debug builds) when addition/multiplication overflows, so in order to use this with Rust you will have to use the Wrapping types.

How it works

See my blog post about it.

Usage

If you want to try this for yourself, check out my implementation that compiles to WASM here, and is hosted as a web interface here. You can find examples how to use this crate in examples/.

TODO

Eventually this crate should be used by the WASM crate. This crate used to use rug::Integers which were not WASM compatible, but I replaced them with num_bigint::BigInts, which are WASM compatible but have less features. Especially keep_bits, keep_signed_bits are missing and have a much slower implementation now. The current mba-wasm crate uses generics (u8, u16, ...). When I made the WASM implementation use this crate, it was noticably slower, so I am holding off on pushing it. (Even if keep_bits had a fast implementation in num_bigint, you would also need a lot of operations to support only calculating n bits, because otherwise you are just wasting time on computing something that is not needed. Not even rug supported that.) (The WASM implementation also uses the faster diagonalization solver that I described in my blog post). Realistically, it would be best to just template everything in this crate the same way I do in the WASM implementation, but this comes at the cost of flexibility: The BigInt implementation works with any number of bits (e.g. 5 or 1000 bit integers), and while no one will ever use that, I think it'd be sad to lose that generality. You could also support both with some crazy templating, but I really can't be bothered writing that for a crate that few people will use. If someone wants to implement that, be my guest.

References

[1] Information Hiding in Software with Mixed Boolean-Arithmetic Transforms

[2] Permutation Polynomials Modulo 2^w

[3] Binary Permutation Polynomial Inversion and Application to Obfuscation Techniques

[4] Solving AX = B using the Hermite normal form