4 ms·

How do you pick how many constants and they're relationships?

Someone 5y ago
I think that’s the “You simply model the PRNG in Z3” part. Whether that’s simple depends is debatable, but for example would be if somebody open-sourced the server-side of their poker dealing site.
- ChrisLomont 5y ago
  It's quite simple. Z3 supports every basic integer option out of the box. You literally write the same code as the original prng. Instead of a uint32, or uint64, or float32, etc., you use the Z3 equivalent type. That's it. You define the state as unknown, but same Z3 type as in C or JavaScript. Then you run the function once. Set it to the output you see, repeat a few observed outputs, then tell Z3 to solve for the original unknown state. It's absolutely trivial, taking a few lines of code. I've broken all sorts of code using similar tools.
  - tmoertel 5y ago
    Do you have a worked example of finding a PRNG's internal state that you could share? (Most people are unfamiliar with Z3 and other SMT solvers. Seeing a relevant worked example would be very illustrative.)
    - ChrisLomont 5y ago
      Yeah, I was considering posting one. I've lately switched to Z3 under C# since it's so easy to do things... If I get around to it tonight I'll make a post.
      - ChrisLomont 5y ago
        Here's a simple example finding a hidden seed. It takes around 1, 2, or 3 values in a row and finds the seed. You can adapt this to all sorts of breaking problems, but for some you'll have to learn more about SATs and be careful how you code things. Note the simple PRNG I chose doesn't reveal all the state, yet over time you can deduce all of it. // Code to demonstrate finding the hidden seed of a PRNG with Z3 // Chris Lomont Oct 2021 // Answering this comment on Hacker News https://news.ycombinator.com/item?id=28886698 // Start by getting Z3, a theorem prover from Microsoft, via nuget. See https://github.com/Z3Prover/z3 using Microsoft.Z3; using System; // get a solution ulong Solution(Context ctx, BoolExpr f, BitVecExpr seed) { Solver s = ctx.MkSolver(); s.Assert(f); // assert constraint into solver if (s.Check() == Status.SATISFIABLE) { var m = s.Model; Expr e = m.Evaluate(seed); return ulong.Parse(e.ToString()); } else throw new Exception("Fatal error"); } // simple PRNG, used in many old C libs UInt32 next = 0x1234; UInt32 rand() { next = next \* 214013 + 2531011; return (next / 65536) % 32768; } void Test() { var R = new Random(); next = (uint)R.Next(); var start = next; Console.WriteLine($"Starting seed {next:X8}"); using (var ctx = new Context()) { // 32 bit vector type Sort bv_type = ctx.MkBitVecSort(32); // unknown seed - we will solve for this BitVecExpr seed = ctx.MkBVConst("seed", 32); // constants used in the PRNG BitVecExpr c214 = (BitVecNum)ctx.MkNumeral("214013", bv_type); BitVecExpr c253 = (BitVecNum)ctx.MkNumeral("2531011", bv_type); BitVecExpr c65536 = (BitVecNum)ctx.MkNumeral("65536", bv_type); BitVecExpr c32768 = (BitVecNum)ctx.MkNumeral("32768", bv_type); // track the boolean to satisfy BoolExpr be = ctx.MkTrue(); var next = seed; // start here int pass = 0; while(true) { ++pass; if (pass > 10) break; // the sampled random var r = rand(); BitVecExpr rndBV = (BitVecNum)ctx.MkNumeral(r, bv_type); // make this code using Z3 variables //next = next * 214013 + 2531011; //return (next / 65536) % 32768; next = ctx.MkBVAdd(ctx.MkBVMul(next, c214), c253); var ret = ctx.MkBVURem(ctx.MkBVUDiv(next, c65536), c32768); // add a constraint boolean for this "Z3 ret = returned value" be = ctx.MkAnd(be, ctx.MkEq(ret, rndBV)); // get solution var val = Solution(ctx, be, seed); // todo - can check unique by adding "seed != soln" to the boolean and checking Console.WriteLine($"{pass} : 0x{val:X8}"); if (val == start) break; // done - can also break when shown unique } } } Test();
        tmoertel 5y ago
        Thanks! That example really helps to crystalize the technique :-)
  - tubby12345 5y ago
    Oh gotcha. I thought you meant that you were some how modeling the prng using some reduced set of constraints.
- ChrisLomont 5y ago
  Also, to break unknown/unseen PRNG, model a few popular classes of PRNG, a few common reductions (so for cards the naive val%52 and better rejection sampling), then solve across the states till you get one. Then watch a few more outputs, check that you guessed the source and reduction, and now you've cracked that too. I doubt any poker site uses anything other than a crypto rand for deals because the technique I just explained would eat them alive. They can use fast and insecure PRNG for doing Monte Carlo hand evaluation, though, since that doesn't give you any advantage.
  - downandout 5y ago
    So building on this...many physical casinos use a manual shuffle, especially at higher limits, as bigger players tend to distrust shuffle machines. Even some online casinos with live dealers will shuffle the previous shoe on camera, manually. Since getting a truly random shuffle would take too long, this is a poor method of getting random results. Would it be possible to, at least to some degree, predict the order of the cards using one of these models, given the order of the cards prior to shuffling? I'm thinking something along the lines of being able to say that the odds of a given card being the next one are higher than they should be, not outright predicting the full order of the cards.
    - ChrisLomont 5y ago
      If I recall, something like 7 hand shuffles becomes indistinguishable from random. I'd guess a pro could get this level of mixing in a few shuffles
      - downandout 5y ago
        Yeah, but that's 7 shuffles of a single deck. Most casinos use 6 or 8 decks. They are certainly not shuffling 42+ times.
        ChrisLomont 5y ago
        Shuffles don't multiply like that. Mixing is an exponential process. It's why mixing a little paint or a lot of paint isn't multiplicatively more work.

I think that’s the “You simply model the PRNG in Z3” part. Whether that’s simple depends is debatable, but for example would be if somebody open-sourced the server-side of their poker dealing site.

ChrisLomont 5y ago
It's quite simple. Z3 supports every basic integer option out of the box. You literally write the same code as the original prng. Instead of a uint32, or uint64, or float32, etc., you use the Z3 equivalent type. That's it. You define the state as unknown, but same Z3 type as in C or JavaScript. Then you run the function once. Set it to the output you see, repeat a few observed outputs, then tell Z3 to solve for the original unknown state. It's absolutely trivial, taking a few lines of code. I've broken all sorts of code using similar tools.
- tmoertel 5y ago
  Do you have a worked example of finding a PRNG's internal state that you could share? (Most people are unfamiliar with Z3 and other SMT solvers. Seeing a relevant worked example would be very illustrative.)
  - ChrisLomont 5y ago
    Yeah, I was considering posting one. I've lately switched to Z3 under C# since it's so easy to do things... If I get around to it tonight I'll make a post.
    - ChrisLomont 5y ago
      Here's a simple example finding a hidden seed. It takes around 1, 2, or 3 values in a row and finds the seed. You can adapt this to all sorts of breaking problems, but for some you'll have to learn more about SATs and be careful how you code things. Note the simple PRNG I chose doesn't reveal all the state, yet over time you can deduce all of it. // Code to demonstrate finding the hidden seed of a PRNG with Z3 // Chris Lomont Oct 2021 // Answering this comment on Hacker News https://news.ycombinator.com/item?id=28886698 // Start by getting Z3, a theorem prover from Microsoft, via nuget. See https://github.com/Z3Prover/z3 using Microsoft.Z3; using System; // get a solution ulong Solution(Context ctx, BoolExpr f, BitVecExpr seed) { Solver s = ctx.MkSolver(); s.Assert(f); // assert constraint into solver if (s.Check() == Status.SATISFIABLE) { var m = s.Model; Expr e = m.Evaluate(seed); return ulong.Parse(e.ToString()); } else throw new Exception("Fatal error"); } // simple PRNG, used in many old C libs UInt32 next = 0x1234; UInt32 rand() { next = next \* 214013 + 2531011; return (next / 65536) % 32768; } void Test() { var R = new Random(); next = (uint)R.Next(); var start = next; Console.WriteLine($"Starting seed {next:X8}"); using (var ctx = new Context()) { // 32 bit vector type Sort bv_type = ctx.MkBitVecSort(32); // unknown seed - we will solve for this BitVecExpr seed = ctx.MkBVConst("seed", 32); // constants used in the PRNG BitVecExpr c214 = (BitVecNum)ctx.MkNumeral("214013", bv_type); BitVecExpr c253 = (BitVecNum)ctx.MkNumeral("2531011", bv_type); BitVecExpr c65536 = (BitVecNum)ctx.MkNumeral("65536", bv_type); BitVecExpr c32768 = (BitVecNum)ctx.MkNumeral("32768", bv_type); // track the boolean to satisfy BoolExpr be = ctx.MkTrue(); var next = seed; // start here int pass = 0; while(true) { ++pass; if (pass > 10) break; // the sampled random var r = rand(); BitVecExpr rndBV = (BitVecNum)ctx.MkNumeral(r, bv_type); // make this code using Z3 variables //next = next * 214013 + 2531011; //return (next / 65536) % 32768; next = ctx.MkBVAdd(ctx.MkBVMul(next, c214), c253); var ret = ctx.MkBVURem(ctx.MkBVUDiv(next, c65536), c32768); // add a constraint boolean for this "Z3 ret = returned value" be = ctx.MkAnd(be, ctx.MkEq(ret, rndBV)); // get solution var val = Solution(ctx, be, seed); // todo - can check unique by adding "seed != soln" to the boolean and checking Console.WriteLine($"{pass} : 0x{val:X8}"); if (val == start) break; // done - can also break when shown unique } } } Test();
      - tmoertel 5y ago
        Thanks! That example really helps to crystalize the technique :-)
- tubby12345 5y ago
  Oh gotcha. I thought you meant that you were some how modeling the prng using some reduced set of constraints.
ChrisLomont 5y ago
Also, to break unknown/unseen PRNG, model a few popular classes of PRNG, a few common reductions (so for cards the naive val%52 and better rejection sampling), then solve across the states till you get one. Then watch a few more outputs, check that you guessed the source and reduction, and now you've cracked that too. I doubt any poker site uses anything other than a crypto rand for deals because the technique I just explained would eat them alive. They can use fast and insecure PRNG for doing Monte Carlo hand evaluation, though, since that doesn't give you any advantage.
- downandout 5y ago
  So building on this...many physical casinos use a manual shuffle, especially at higher limits, as bigger players tend to distrust shuffle machines. Even some online casinos with live dealers will shuffle the previous shoe on camera, manually. Since getting a truly random shuffle would take too long, this is a poor method of getting random results. Would it be possible to, at least to some degree, predict the order of the cards using one of these models, given the order of the cards prior to shuffling? I'm thinking something along the lines of being able to say that the odds of a given card being the next one are higher than they should be, not outright predicting the full order of the cards.
  - ChrisLomont 5y ago
    If I recall, something like 7 hand shuffles becomes indistinguishable from random. I'd guess a pro could get this level of mixing in a few shuffles
    - downandout 5y ago
      Yeah, but that's 7 shuffles of a single deck. Most casinos use 6 or 8 decks. They are certainly not shuffling 42+ times.
      - ChrisLomont 5y ago
        Shuffles don't multiply like that. Mixing is an exponential process. It's why mixing a little paint or a lot of paint isn't multiplicatively more work.