# SSSP - Cryptography - 400 points - 4 teams solved

> P = NP?
> nc 50216
> [sssp-58ab171bacc3c82fa6704228fb9f1d78.cpp](./sssp-58ab171bacc3c82fa6704228fb9f1d78.cpp)

This challenge is quite similar to a [Tokyo Westerns 2017 CTF problem](https://github.com/ymgve/ctf-writeups/tree/master/tokyowesterns2017/ppc-backpackers_problem). We are tasked with solving 30 different [Subset sum](https://en.wikipedia.org/wiki/Subset_sum_problem) problems, from 11 to 127 integers in size, and the problems are generated with the standard [Mersenne Twister](https://en.wikipedia.org/wiki/Mersenne_Twister) variant MT19937. There are a few key differences:

* Instead of the MT being initialized by a single 32bit seed, the whole 624-integer initial state is initialized from a secure RNG source
* The number of bits in the integers of a problem set varies from 24 to 120 bits
* The list of integers in a problem set isn't sorted, so we know the exact order of the partial PRNG info we receive

Since there isn't a single seed this time, we decided to try a different approach - predict the PRNG state increasingly from the integers of the problem sets, hopefully recovering the full state before the size of the problems become too large.

Instead of the normal 624 integer state, we model the Twister as an unbounded length array, and each entry is an array containing all possible integers at that point, or None if the number of possible integers is too large. For example, if we know 24 of the 32 bits generated by the PRNG at one point, we know that the state value at that point could only be one of 256 different ones.

We then use the fact that `state[x]` is generated on the basis of `state[x-624]`, `state[x-623]`, and `state[x-227]`. `state[x-624]` only contributes a single bit, so worst case it has two possible values. For the other integers, if we know the possible candidates for two out of three in `(state[x-623], state[x-227], state[x])`, we can compute an array of possible candidates for third one. If this narrows down the number of possible candidates from what we had before, we repeat this process for other values of `x` until we don't get any better constraints. We also add some sane limits for the number of states generated and the depth of iteration to stop the process from using too much time.

Until our predictor has "warmed up", we have to solve the Subset sum problems normally. We use the same solver as last time, though we really should have used LLL or something faster than Python. We did get a speed boost from using [pypy](https://pypy.org/), though. After the first few problems, our predictor starts generating results, and we can eliminate some of the values from the problem set.

We encountered a "hump" with problem 14, where our predictor manages to reduce the problem size from 63 numbers to 50, but this is still too much for our solver algorithm to solve within the five second time limit. Instead of searching for another more optimized algorithm, we decided to just chop off the last 7 numbers, reducing the problem size to 43. This means that on average, our solver will fail to find a solution 127 out of 128 times, but by simply running the program enough times, we eventually will get a solution and can continue. After problem 14 it gets easier, and at problem 20 the full PRNG state is known, and we don't have to solve anything at all. We finally get the flag, `hitcon{SSSP = Silly Shik's Superultrafrostified Present}`

Post contest, we realized that there is no randomness in which bits of the PRNG state gets leaked, so in theory we could have done an in-depth analyzis with for example [z3](https://github.com/Z3Prover/z3) offline, then used the results to fill in the bits quickly. This might have reduced the problem sizes even further, and we might have avoided the repeated executions of the script.

Original writeup (https://github.com/ymgve/ctf-writeups/tree/master/hitcon2017quals/crypto-sssp).