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Eq(af), timing attack!

The MirageOS project is aware about security issues. This goal is strong when, at the beginning, the idea of a small unikernel can ensure (by intuition) a small attack surface. By this way, we want to follow as possible as we can improvements into security stacks such as TLS.

Of course, we are not a huge team and some of us don't have a strong knowledge about security. This topic is highly complex and it's easy to think that we are secure - and, by facts, we are not. However, it gives to us an opportunity to learn and improve what we can find about this topic and go on our way to try to provide the best as we can.

This article wants to show a little project which wants to solve a security issue, the timing attack. We will see what we did on this purpose to be able to improve mirage-crypto.

A timing attack!

It's clearly not an usual attack for me and I did not (yet!) understand in which way it's possible to use this side-channel attack over a complex black-box such as an SMTP service. However, the first lesson about security is to agree with the fact that when you can imagine this attack (even if, by your technical means, it's not possible), someone else into our world has the ability to use this attack.

The point is not to try to think how this kind of attack is possible but to assert that this attack is possible.

The timing attack is in this case where we use the time to try to introspect data such as a password. The idea is simple, we can take this simple equal function:

let equal s1 s2 =
  let res = ref true in
  let idx = ref 0 in
  if String.length s1 <> String.length s2
  then false
    ( while !idx < String.length s1 && !res
      do res := s1.[!idx] = s2.[!idx] ; incr idx done ; !res )

If we are in the case where we want to compare the given password by the user and the password stored into our database (or their hashes), we will use this function - and allow the user to enter into a secured area.

However, we can see that the time spent by equal depends on given inputs s1 and s2.

The worst case.

Imagine that the atomic operation s1.[!idx] (or s2.[!idx]) spend 1 second (like 1 CPU tick). So, for each iteration into our loop, we will spend 2 seconds while !res still is true. That means when we meet 2 different bytes, we leave the loop1.

Now, imagine we have these values:

equal "aabb" "aaaa" ;;

We can easily infer that this function will spend 6 seconds (2 seconds for first characters, 2 seconds for second characters, 2 seconds for third characters and we leave the loop). And about equal values such as:

equal "toto" "toto" ;;

We will spend 8 seconds (and return true). The time needed to compute the equal function depends on given inputs. By this way, if we observe time needed to be authenticated from a login page, we can infer values given into the equal function.

Finally, from that fact, imagine that s1 is the given password by us and s2 is the password stored into our database, we can infer just with the time if we are close to be equal to the password stored into our database.

1: In C, the case appears when we return false directly which is a jump and leaves the loop as we do in OCaml.

A smart brute-force attack.

So now we can imagine a tool which will try all possibilities. It will record times spent for each random inputs. Then, when it appears that from an input t0, time spent differs (is lower than) from an input t1, we can imply that our equal function go a bit further when it scanned our input t1 (than t0).

By this fact, we can imply that t1 is more close to the expected password than t0 and we can redo the same operation but with a new base t1 - and by this way, discard all others next (and useless) possibilities with t0 as a base.

Eq(af) gives to you an example of this attack. Of course, we plugged into our operations an Unix.sleep to be able to see a real difference when we call our equal function. Finally, the result is pretty good:

$ dune exec attack/attack.exe
Random: [|253;164;168;66;47;219;88;152;128;242;216;123;|].

The second line is the expected hash, the third line is the hash found only by an introspection of the time. The first line is the seed given by the random to be able to replicate the result. This tool does not work all the time!

Too deep to be dangerous?

Of course, into the real world, our equal function is much much much more fast than some seconds. However, even if it can be hard to track how long equal spend, it's not impossible. I mean, it's a bit hard but a way exists to know how long any function spend.

This way is the micro-benchmark! Of course, when you want to compare speed of your functions, you introspect time! So if we can do that usually to check which function is faster than the other, we can apply the idea of the benchmark on two call of the same equal function but with different inputs.

Eq(af) does the test for you

So, our distribution comes with a little benchmark tool which wants to see how long equal spend when we give to it 2 different values and 2 equal values. Then, we extrapolate results.

We run this function 1 time, 2 times, 3 times, ... to 3000 times and we record times for each run. From that, we are able to plot a curve. So about expected results:



So from this curve, we apply a linear-regression to see if it's a line or not. From that, we can get a ~R²~ which is a regression coefficient and it tells to us if it's a true line (where R² >= 0.99) or not.

As you can see, we have some noises (some points are not strictly aligned) - it's because the inherent volatility when we want to record the time. It's why we need to apply a linear-regression to delete as much as possible this noise. An other factor about that is the context where you execute your function, the scheduler can decide to stop our equal function and do something else (but our equal function still spends his time). Finally, it still is hard to track the time.

How to fix it?

The easiest way to avoid this side-channel attack is to spend exactly the same time for any inputs. At the end, even if we encounter different bytes, we will continue to scan inputs.

let equal s1 s2 =
  if String.length s1 <> String.length s2 then false
    ( let res = ref true in
      for idx = 0 to String.length s1 - 1
      do res := s1.[idx] = s2.[idx] && !res done ;
      !res )

Into the OCaml side!

However, into the OCaml world, the deal can be much more complex that what we can imagine. Even if it's easy to translate an OCaml code to an assembly code, the compiler can emit some jump and produce a non constant-time (or a non branch-less) function at the end.

In our example, the only idea to use None and Some instead false and true into our code implies a call to the garbage collector - and a jump in the emitted assembly. This case appear when we want to implement compare and keep somewhere (into our int option) the subtraction of different bytes while the loop.

Into Eq(af), we did a huge introspection about the assembly generated to ensure that we don't have any jump and any assembly instructions will be executed regardless inputs - and only from that we can say that our function is constant-time2.

2: constant-time is not a good word where it can have an other signification when we talk about complexity and algorithm. And it's not really true when the time still depends on the length of the given input.


Eq(af) provides a quite easy function and anybody can re-implement it. But it comes with tools which want to check our assumption, the time spent can not leak any information. Or course, as we showed, it can be hard to check that.

We have different tools to check/prove that where, at this time, we only do a check but we can go further by an introspection of the emitted code by an other tool to see if are surely branch-less.

At the end, Eq(af) is used by some people who are aware about security issues and it's why we decided to use it as a default into digestif to protect any users from this side-channel attack.