Two interesting things in formally verified software happened recently. The big one was the release of SeL4 - a formally verified L4 microkernel. The second was much smaller, but closer to my usual scope: a paper which showed the correctness of sections of a couple of the assembly implementations of Curve25519.

Overflow and carry bugs are subtle and, although with 32-bit words you might hope to be able to do enough random testing to eliminate them, that's much less plausible with 64-bit implementations. The TweetNaCl paper mentions a bug that lived in one of the assembly implementations of ed25519 for years and I've sinned too:

When I wrote curve25519-donna (back when it was the only 64-bit implementation of Curve25519), I first wrote a 32-bit version which mirrored the implementation of the athlon assembly version. This was just to provide a reference for the 64-bit code, but it was included in the repository for education. It was never really supposed to be used, and wasn't constant-time, but I was aware that it was being used by some groups.

Many years later, someone noticed that it was missing a reduction in the final contraction. Values between 2255-19 and 2255 would be output when they should have been reduced. This turned out to be harmless (as best I can tell), but embarrassing none the less.

And Curve25519 is a very nice curve to implement! Take a look at the overflow considerations for this version of P-256. I hope that I got everything right there, but it's very complex. More formal verification of the software would be welcome!

SeL4 has a multi-stage verification using refinement. Refinement is the process of showing that two pieces of code behave identically where one version is more abstract. SeL4's proof refines from an abstract specification to a Haskell implementation to the C implementation. It also has a SAT-solver based proof that the ARMv6 assembly matches the C implementation.

The refinement proofs are done in Isabelle/HOL which, along with Coq, are the major proof systems. (There are several other systems including HOL Light, HOL4 and Mizar.) Proof systems assist in the construction of automated proofs using simple rules of inference and axioms. Although not used for software verification, Metamath is a good introduction to the idea. It clearly explains its axioms (which Isabelle and Coq are very bad at) and gives an idea for the scale of formal proofs with an example of 2+2 = 4.

The best introduction to Isabelle that I know of is the book Concrete Semantics, although I do wish that it would start with Isar style proofs much sooner. If you're doing work in Isabelle I think you need page 57 printed and stuck to the wall and if you're going through that book and exercises I'd recommend peeking at the Isar chapter sooner.

But Isabelle's traditional workflow is to write in its functional language and export to OCaml or Haskell. That's no help if we're seeking to prove things about C code.

Isabelle's theorems are objects in its underlying ML language and so you can write a program in ML that parses C and use the result in Isabelle. That's what SeL4 does. The underlying framework for expressing imperative languages in Isabelle is Schirmer's Simpl and this imposes some odd limitations on the C that can be processed. For one, all the local variables with the same name in a given .c file have to have the same type because the local state of all functions in a .c file is represented in a single struct. For the same reason, it's not possible to take the address of a local variable.

But we can try parsing a very simple C file with SeL4's parser:

int add(int a, int b) { return a+b; }

That fits SeL4's C subset (called StrictC - see l4v/tools/c-parser/doc in the SeL4 source) and results in this Simpl construct (you'll need to read the Simpl paper to really understand it):

test_global_addresses.add_body ≡ TRY Guard SignedArithmetic ⦃-2147483648 ≤ sint ´a + sint ´b ∧ sint ´a + sint ´b ≤ 2147483647⦄ (creturn global_exn_var_'_update ret__int_'_update (λs. a_' s + b_' s));; Guard DontReach {} SKIP CATCH SKIP END

There's clearly an overflow check in there, which is good, but to call the process of starting to prove things about it intimidating to the beginner would be an understatement. That and the oddities of the C subset motivated me to look further.

Dependent types are closely related to this problem so I checked Wikipedia for the dependently typed languages which support imperative programming and that are still active. That's just ATS and F*, according to that table. ATS doesn't deal with overflows and, while F*/F7 is worthy of a look because of miTLS, it's a CLR language and so not applicable here.

Next up, based on searches, is SPARK. SPARK is a subset of Ada designed for verification. There's a commercial company behind it, but versions are published under the GPL. It even comes with an IDE.

SPARK uses an SMT solver to prove correctness, which is very different from a proof in Isabelle. While an Isabelle proof is, by construction, just an application of axioms and rules of inference, an SMT solver is essentially a magic box into which you feed a proposition and out of which (maybe) comes a true/false answer. Trusting in an SMT solver is much, much stronger than not doing so, but it is a step down from a formal proof system. SPARK uses a version of Why3 to abstract over SMT solvers and we'll discuss Why3 more later.

Anyway, here's the same, trivial function in Ada:

function Add(X, Y : in Integer_32) return Integer_32 is begin return X + Y; end Add;

If we ask SPARK to process that then it generates verification conditions (VCs): one for each possible thing that could go wrong - overflow, array indexing etc. For that code it throws an error saying that X + Y might overflow, which is promising! In order to eliminate that, one needs to prove the verification condition that says that the addition is safe by adding preconditions to the function (which then become VCs at each callsite):

function Add(X, Y : in Integer_32) return Integer_32 with Pre => X < 2**30 and X > -2*30 and Y < 2**30 and Y > -2**30;

With that precondition, the function is accepted. I should note that this is SPARK 2014; there are many older versions (SPARK has been going for a long time) and they kept preconditions and the like in comments. Lots of pages about SPARK still talk about the older versions and major projects in SPARK (like Ironsides - a DNS server) still use the older versions.

With that initial success, let's try something a tiny bit more involved. Curve25519 deals with 255-bit integers but CPUs don't. So, in the same way that a 64-int integer could be represented with a pair of 32-bit integers, 255-bit integers are represented in Donna with ten, 32-bit integers. This representation is a little odd because the values aren't spaced at 32-bit multiples, but rather at alternating 26 and 25 bit positions. That also means that the representation is non-unique (setting bit 26 of the first word is equal to setting bit zero of the second) and that's done for performance. See this for more details, but it's not important right now.

Adding these 255-bit integers is done simply by adding the underlying words without carry, with suitable bounds on the inputs so that overflow doesn't happen:

type Ints is array (Integer range 0 .. 9) of Integer_32; function Add (X, Y : in Ints) return Ints with Pre => (for all i in Ints'Range => X(i) < 2**30 and X(i) > -2**30 and Y(i) < 2**30 and Y(i) > -2**30), Post => (for all i in Ints'Range => Add'Result(i) = X(i) + Y(i)); function Add (X, Y : in Ints) return Ints is Sum : Ints; begin for i in Ints'Range loop Sum(i) := X(i) + Y(i); end loop; return Sum; end Add;

That mostly works, but SPARK can't prove that the result is X(i) + Y(i) for all i despite it being exactly what the function says. One really needs to spoon feed the prover: in this case with a loop invariant in the for loop:

for i in Ints'Range loop pragma Loop_Invariant (for all j in Ints'First .. i-1 => Sum(j) = X(j) + Y(j)); Sum(i) := X(i) + Y(i); end loop;

Sadly, that doesn't work either, despite being correct, because SPARK doesn't seem to like i-1 when i might be zero. So, trying again:

for i in Ints'Range loop Sum(i) := X(i) + Y(i); pragma Loop_Invariant (for all j in Ints'First .. i => Sum(j) = X(j) + Y(j)); end loop;

That one works, but now SPARK isn't sure whether uninitialised elements of Sum are being referenced. The property of being initialised isn't part of SPARK's logic and seems that the only way to proceed here is to disable that warning!

But, there's some promise! We would now like to show a higher level property of Add : that the 255-bit integer that the returned array represents is the sum of the integers that the input arrays represent. This means that the logic needs to deal with arbitrary integers and that we need to write a function in the logic that turns an array into an abstract integer.

Enabling arbitrary integers in the logic is possible (although good luck finding how to do it in the documentation: the answer is to put a file called gnat.adc in the gnatprove subdirectory of the project containing pragma Overflow_Mode (General => Strict, Assertions => Eliminated); ). However, I failed at writing functions in the logic without dropping down into code and triggering overflow checks. It appears that ghost functions should be able to do this but, after getting the answer to a documentation bug on StackOverflow, actually trying to use any logic functions, even the identity function, caused the proof not to terminate. SPARK claims to be able to use Isabelle for proofs but I couldn't get that to work at all: strace says that the .thy file isn't ever written.

Stuck, I moved on from SPARK and tried Frama-C and its Jessie plugin. Even if SPARK was working for me, using C (even a subset) has advantages: it's convenient to use in other projects, there exist verified compilers for it and there's an extension to CompCert that allows for static verification of constant-time behaviour (although I've lost the paper!)

So here's the same function in Frama-C:

/*@ requires \valid_range(out, 0, 9); @ requires \valid_range(x, 0, 9); @ requires \valid_range(y, 0, 9); @ requires \forall integer i; 0 <= i < 10 ==> (x[i] > -1000 && x[i] < 1000); @ requires \forall integer i; 0 <= i < 10 ==> (y[i] > -1000 && y[i] < 1000); @ ensures \forall integer i; 0 <= i < 10 ==> (out[i] == x[i] + y[i]); */ void add(int *out, const int *x, const int *y) { /*@ loop invariant i >= 0 && i <= 10 && (\forall integer j; 0 <= j < i ==> out[j] == x[j] + y[j]); @ loop variant 10-i; */ for (int i = 0; i < 10; i++) { out[i] = x[i] + y[i]; } }

Note that this time we not only need a loop invariant to show the postcondition, but we also need a loop variant to show that the for loop terminates: you really need to spoon feed the verification! But Frama-C/Jessie has no problem with functions in the logic:

/*@ logic integer felemvalue (int *x) = x[0] + x[1] * (1 << 26) + x[2] * (1 << 51) + x[3] * (1 << 77) + x[4] * (1 << 102) + x[5] * 340282366920938463463374607431768211456 + x[6] * 11417981541647679048466287755595961091061972992 + x[7] * 766247770432944429179173513575154591809369561091801088 + x[8] * 25711008708143844408671393477458601640355247900524685364822016 + x[9] * 1725436586697640946858688965569256363112777243042596638790631055949824; */

That function maps from an array to the abstract integer that it represents. Note that the terms switched from using shifts to represent the constants to literal numbers. That's because the proof process suddenly became non-terminating (in a reasonable time) once the values reached about 104-bits, but the literals still worked.

With that logic function in hand, the we can easily prove the higher-level concept that we wanted as a post condition of the add function:

@ ensures felemvalue(out) == felemvalue(x) + felemvalue(y);

Flushed with success, I moved onto the next most basic function: multiplication of 255-bit integers. The code that Donna uses is the textbook, polynomial multiplication code. Here's how the function starts:

/* Multiply two numbers: output = in2 * in * * output must be distinct to both inputs. The inputs are reduced coefficient * form, the output is not. * * output[x] <= 14 * the largest product of the input limbs. */ static void fproduct(limb *output, const limb *in2, const limb *in) { output[0] = ((limb) ((s32) in2[0])) * ((s32) in[0]); output[1] = ((limb) ((s32) in2[0])) * ((s32) in[1]) + ((limb) ((s32) in2[1])) * ((s32) in[0]); output[2] = 2 * ((limb) ((s32) in2[1])) * ((s32) in[1]) + ((limb) ((s32) in2[0])) * ((s32) in[2]) + ((limb) ((s32) in2[2])) * ((s32) in[0]); …

Each of the lines is casting a 64-bit number down to 32 bits and then doing a 32×32⇒64 multiplication. (The casts down to 32 bits are just to tell the compiler not to waste time on a 64×64⇒64 multiplication.) In Frama-C/Jessie we need to show that the casts are safe, multiplications don't overflow and nor do the sums and multiplications by two etc. This should be easy. Here are the preconditions that set the bounds on the input.

/*@ requires \valid_range(output, 0, 18); @ requires \valid_range(in, 0, 9); @ requires \valid_range(in2, 0, 9); @ requires \forall integer i; 0 <= i < 10 ==> -134217728 < in[i] < 134217728; @ requires \forall integer i; 0 <= i < 10 ==> -134217728 < in2[i] < 134217728; */

However, Why3, which is the prover backend that Jessie (and SPARK) uses, quickly gets bogged down. In an attempt to help it out, I moved the casts to the beginning of the function and put the results in arrays so that the code was less complex.

As you can see in the Jessie documentation, the Why3 UI allows one to do transforms on the verification conditions to try and make life easier for the prover. Quickly, splitting a requirement becomes necessary. But this throws away a lot of information. In this case, each of the 32×32⇒64 multiplications is assumed to fit only within its bounds and so the intermediate bounds need to be reestablished every time. This means that the equations have to be split like this:

limb t, t2; t = ((limb) in2_32[0]) * in_32[1]; //@ assert -18014398509481984 < t < 18014398509481984; t2 = ((limb) in2_32[1]) * in_32[0]; //@ assert -18014398509481984 < t2 < 18014398509481984; output[1] = t + t2;

That's only a minor problem really, but the further one goes into the function, the harder the prover needs to work for some reason. It's unclear why because every multiplication has the same bounds - they are all instances of the same proof. But it's very clear that the later multiplications are causing more work.

Why3 is an abstraction layer for SMT solvers and a large number are supported (see “External provers” on its homepage for a list). I tried quite a few and, for this task, Z3 is clearly the best with CVC4 coming in second. However, Z3 has a non-commercial license which is very worrying - does that mean that a verified version of Curve25519 that used Z3 has to also have a non-commercial license? So I stuck to using CVC4.

However, about a quarter of the way into the function, both CVC4 and Z3 are stuck. Despite the bounds being a trivial problem, and just instances of the same problem that they can solve at the beginning of the function, somehow either Why3 is doing something bad or the SMT solvers are failing to discard irrelevant facts. I left it running overnight and Z3 solved one more instance after six hours but achieved nothing after 12 hours on the next one. Splitting and inlining the problem further in the Why3 GUI didn't help either.

Like SPARK, Jessie can use Isabelle for proofs too (and for the same reason: they are both using Why3, which supports Isabelle as a backend). It even worked this time, once I added Real to the Isabelle imports. However, in the same way that the C parser in Isabelle was an ML function that created Isabelle objects internally, the Why3 connection to Isabelle is a function ( why3_open ) that parses an XML file. This means that the proof has large numbers of generated variable names ( o33 , o34 , etc) and you have no idea which intermediate values they are. Additionally, the bounds problem is something that Isabelle could have solved automatically, but the assumptions that you can use are similarly autonamed as things like local.H165 . In short, the Isabelle integration appears to be unworkable.

Perhaps I could split up each statement in the original code into a separate function and then write the statement again in the logic in order in order to show the higher level properties, but at some point you have to accept that you're not going to be able to dig this hole with a plastic spoon.

Conclusion

The conclusion is a bit disappointing really: Curve25519 has no side effects and performs no allocation, it's a pure function that should be highly amenable to verification and yet I've been unable to find anything that can get even 20 lines into it. Some of this might be my own stupidity, but I put a fair amount of work into trying to find something that worked.

There seems to be a lot of promise in the area and some pieces work well (SMT solvers are often quite impressive, the Frama-C framework appears to be solid, Isabelle is quite pleasant) but nothing I found worked well together, at least for verifying C code. That makes efforts like SeL4 and Ironsides even more impressive. However, if you're happy to work at a higher level I'm guessing that verifying functional programs is a lot easier going.