Hey! I'm David, cofounder of zkSecurity and the author of the Real-World Cryptography book. I was previously a crypto architect at O(1) Labs (working on the Mina cryptocurrency), before that I was the security lead for Diem (formerly Libra) at Novi (Facebook), and a security consultant for the Cryptography Services of NCC Group. This is my blog about cryptography and security and other related topics that I find interesting.

A lot of cryptographic protocols can be reduced to computing some value. Perhaps the value obtained is a shared secret, or it allows us to verify that some other values match (if it's 0). Since we're talking about cryptography, computing the value is most likely done by adding and multiplying numbers together.

Addition is often free, but it seems like multiplication is a pain in most cryptographic protocols.

If you're multiplying two known values together, it's OK. But if you want to multiply one known value with another unknown value, then you will most likely have to reach out to the discrete logarithm problem. With that in hand, you can multiply an unknown value with a known value.

This is used, for example, in key exchanges. In such protocols, a public key usually masks a number. For example, the public key X in X = [x] G masks the number x. To multiply x with another number, we do this hidden in the exponent (or in the scalar since I'm using the elliptic curve notation here): [y] X = [y * x] G. In key exchanges, you use this masked result as something useful.

If you're trying to multiply two unknown values together, you need to reach for pairings. But they only give you one multiplication. With masked(x) and masked(y) you can do pairing(masked(x), masked(y)) and obtain something that's akin to locked_masked(x * y). It's locked as in, you can't do these kind of multiplications anymore with it.

Previously I talked about monads, which are just a way to create a "container" type (that contains some value), and let people chain computation within that container. It seems to be a pattern that's mostly useful in functional languages as they are often limited in the ways they can do things.

Little did I know, there's more to monads, or at least monads are so vague that they can be used in all sorts of ways. One example of this is state monads.

A state monad is a monad which is defined on a type that looks like this:

type 'a t = state -> 'a * state

In other word, the type is actually a function that performs a state transition and also returns a value (of type 'a).

When we act on state monads, we're not really modifying a value, but a function instead. Which can be brain melting.

The bind and return functions are defined very differently due to this.

The return function should return a function (respecting our monad type) that does nothing with the state:

let return a = fun state -> (a, state)
let return a state = (a, state) (* same as above *)

This has the correct type signature of val return : 'a -> 'a t (where, remember, 'a t is state -> ('a, state)). So all good.

The bind function is much more harder to parse. Remember the type signature first:

val bind : 'a t -> f:('a -> 'b t) -> 'b t

which we can extend, to help us understand what this means when we're dealing with a monad type that holds a function:

you should probably spend a few minutes internalizing this type signature. I'll describe it in other words to help: bind takes a state transition function, and another function f that takes the output of that first function to produce another state transition (along with another return value 'b).

The result is a new state transition function. That new state transition function can be seen as the chaining of the first function and the additional one f.

OK let's write it down now:

let bind t ~f = fun state ->
(* apply the first state transition first *)
let a, transient_state = t state in
(* and then the second *)
let b, final_state = f a transient_state in
(* return these *)
(b, final_state)

Hopefully that makes sense, we're really just using this to chain state transitions and produce a larger and larger main state-transition function (our monad type t).

How does that look like when we're using this in practice? As most likely when a return value is created, we want to make it available to the whole scope. This is because we want to really write code that looks like this:

let run state =
(* use the state to create a new variable *)
let (a, state) = new_var () state in
(* use the state to negate variable a *)
let (b, state) = negate a state in
(* use the state to add a and b together *)
let (c, state) = add a b state in
(* return c and the final state *)
(c, state)

where run is a function that takes a state, applies a number of state transition on that state, and return the new state as well as a value produced during that computation.
The important thing to take away there is that we want to apply these state transition functions with values that were created previously at different point in time.

Also, if that helps, here are the signatures of our imaginary state transition functions:

val new_var -> unit -> state -> (var, state)
val negate -> var -> state -> (var, state)
val add -> var -> var -> state -> (var, state)

Rewriting the previous example with our state monad, we should have something like this:

let run =
bind (new_var ()) ~f:(fun a ->
bind (negate a) ~f:(fun b -> bind (add a b) ~f:(fun c ->
return c)))

Which, as I explained in my previous post on monads, can be written more clearly using something like a let% operator:

let t =
let%bind a = new_var () in
let%bind b = negate a in
let%bind c = add a b in
return c

And so now we see the difference: monads are really just way to do things we can already do but without having to pass the state around.

It can be really hard to internalize how the previous code is equivalent to the non-monadic example. So I have a whole example you can play with, which also inline the logic of bind and return to see how they successfuly extend the state. (It probably looks nicer on Github).

type state = { next : int }
(** a state is just a counter *)
type 'a t = state -> 'a * state
(** our monad is a state transition *)
(* now we write our monad API *)
let bind (t : 'a t) ~(f : 'a -> 'b t) : 'b t =
fun state ->
(* apply the first state transition first *)
let a, transient_state = t state in
(* and then the second *)
let b, final_state = f a transient_state in
(* return these *)
(b, final_state)
let return (a : int) (state : state) = (a, state)
(* here's some state transition functions to help drive the example *)
let new_var _ (state : state) =
let var = state.next in
let state = { next = state.next + 1 } in
(var, state)
let negate var (state : state) = (0 - var, state)
let add var1 var2 state = (var1 + var2, state)
(* Now we write things in an imperative way, without monads.
Notice that we pass the state and return the state all the time, which can be tedious.
*)
let () =
let run state =
(* use the state to create a new variable *)
let a, state = new_var () state in
(* use the state to negate variable a *)
let b, state = negate a state in
(* use the state to add a and b together *)
let c, state = add a b state in
(* return c and the final state *)
(c, state)
in
let init_state = { next = 2 } in
let c, _ = run init_state in
Format.printf "c: %d\n" c
(* We can write the same with our monad type [t]: *)
let () =
let run =
bind (new_var ()) ~f:(fun a ->
bind (negate a) ~f:(fun b -> bind (add a b) ~f:(fun c -> return c)))
in
let init_state = { next = 2 } in
let c, _ = run init_state in
Format.printf "c2: %d\n" c
(* To understand what the above code gets translated to, we can inline the logic of the [bind] and [return] functions.
But to do that more cleanly, we should start from the end and work backwards.
*)
let () =
let run =
(* fun c -> return c *)
let _f1 c = return c in
(* same as *)
let f1 c state = (c, state) in
(* fun b -> bind (add a b) ~f:f1 *)
(* remember, [a] is in scope, so we emulate it by passing it as an argument to [f2] *)
let f2 a b state =
let c, state = add a b state in
f1 c state
in
(* fun a -> bind (negate a) ~f:f2 a *)
let f3 a state =
let b, state = negate a state in
f2 a b state
in
(* bind (new_var ()) ~f:f3 *)
let f4 state =
let a, state = new_var () state in
f3 a state
in
f4
in
let init_state = { next = 2 } in
let c, _ = run init_state in
Format.printf "c3: %d\n" c
(* If we didn't work backwards, it would look like this: *)
let () =
let run state =
let a, state = new_var () state in
(fun state ->
let b, state = new_var () state in
(fun state ->
let c, state = add a b state in
(fun state -> (c, state)) state)
state)
state
in
let init_state = { next = 2 } in
let c, _ = run init_state in
Format.printf "c4: %d\n" c

When I was younger, I used to play a lot of counter strike (CS). CS was (and still is) this first-person shooter game where a team of terrorists would play against a team of counter terrorists. Victory would follow from successfully bombing a site or killing all the terrorists. Pretty simple. I discovered the game at a young age, and I got hooked right away. You could play with other people in cyber cafes, and if you had a computer at home with an internet connection you could even play with others online. This was just insane! But what really hooked me was the competitive aspect of the game. If you wanted, you could team up with 4 other players and play matches against others. This was all new to me, and I progressively spent more and more hours playing the game. That is, until I ended up last of my class, flunked my last year of highschool, failed the Bacalaureat. I decided to re-prioritize my passion in gaming in order not to fail at the path that was laid in front of me. But a number of lessons stayed with me, and I always thought I should write about them.

My first lesson was how intense competition is. I never had any experience come close to it since then, and miss competition dearly. Once you start competing, and you start getting good at it, you feel the need to do everything to get the advantage. Back then, I would watch every frag movie that came out (even producing some), I would know all the best players of every clan (and regularly play with them), and I would participate in 3 online tournaments a day. I would wake up every day around noon, play the first tournament at 1pm, then practice the afternoon, then play the tournament of 9pm, and then the last one at 1am. If my team lost, I would volunteer to replace a player dropping out from a winning team. Rinse and repeat, every day. There's no doubt in my mind that I must have reached Gladwell's 10,000 hours.

I used the same kind of technique years later when I started my master in cryptography. I thought: I know how to become the best at something, I just have to do it all the time, constantly. I just need to obsess. So I started blogging here, I started subscribing to a number of blogs on cryptography and security and I would read everything I could every single hours of every day. I became a sponge, and severally addicted to RSS feeds. Of course, reading about cryptography is not as easy as playing video games and I could never maintain the kind of long hours I would when I was playing counter strike. I felt good, years later, when I decided to not care as much about new notifications in my RSS feed.

Younger, my dream was to train with a team for a week in a gaming house, which is something that some teams were starting to do. You'd go stay in some house together for a week, and every day practice and play games together. It really sounded amazing. Spend all my hours with people who cared as much as me. It seemed like a career in gaming was possible, as more and more money was starting to pour into esport. Some players started getting salaries, and even coaches and managers. It was a crazy time, and I felt really sad when I decided to stop playing. I knew a life competing in esport didn't make sense for me, but competition really was fullfiling in a way that most other things proved not to be.

An interesting side effect of playing counter strike every day, competitively, for many years, is that I went through many teams. I'm not sure how many, but probably more than 50, and probably less than 100. A team was usually 5 players, or more (if we had a rotation). We would meet frequently to spend hours practicing together, figuring out new strategies that we could use in our next game, doing practice matches against other teams, and so on. I played with all different kind of people during that time, spending hours on teamspeak and making life-long friendships. One of my friend, I remember, would get salty as fuck when we were losing, and would often scream at us over the microphone. One day we decided to have an intervention, and he agreed to stop using his microphone for a week in order not to get kicked out of the team. This completely cured his raging.

One thing I noticed was that the mood of the team was responsible for a lot in our performance. If we started losing during a game, a kind of "loser" mood would often take over the team. We would become less enthusiastic, some team mates might start raging at the incompetence of their peers, or they might say things that made the whole team want to give up. Once this kind of behavior started, it usually meant we were on a one-way road to a loss.
But sometimes, when people kept their focus on the game, and tried to motivate others, we would make incredible come backs. Games were usually played in two halves of 15 rounds. First in 16 would win. We sometimes went from a wooping 0-15 to an insane strike of victories leading us to a 16-15. These were insane games, and I will remember them forever.
The common theme between all of those come back stories were in how the whole team faced adversity, together. It really is when things are not going well that you can judge a team. A bad team will sink itself, a strong team will support one another and focus on doing its best, potentially turning the tide.

During this period, I also wrote a web service to create tournaments easily. Most tournaments started using it, and I got some help to translate it in 8 different european languages. Thousands of tournaments got created through the interface, in all kind of games (not just CS). Years later I ended up open sourcing the tool for others to use. This really made me understand how much I loved creating products, and writing code that others could directly use.

Today I have almost no proof of that time, besides a few IRC screenshots. (I was completely addicted to IRC.)

I'm not a language theorist, and even less of a functional programming person. I was very confused trying to understand monads, as they are often explained in complicated terms and also mostly using haskell syntax (which I'm not interested in learning). So if you're somewhat like me, this might be the good short post to introduce you to monads in OCaml.

I was surprised to learn that monads are really just two functions: return and bind:

module type Monads = sig
type 'a t
val return : 'a -> 'a t
val bind : 'a t -> ('a -> 'b t) -> 'b t
end

Before I explain exactly what these does, let's look at something that's most likely to be familiar: the option type.

An option is a variant (or enum) that either represents nothing, or something. It's convenient to avoid using real values to represent emptiness, which is often at the source of many bugs. For example, in languages like C, where 0 is often used to terminate strings, or Golang, where a nil pointer is often used to represent nothing.

An option has the following signature in OCaml:

type 'a option = None | Some of 'a

Sometimes, we want to chain operations on an option. That is, we want to operate on the value it might contain, or do nothing if it doesn't contain a value.
For example:

let x = Some 5 in
let y = None
match x with
| None -> None
| Some v1 -> (match y with
| None -> None
| Some v2 -> v1 + v2)

the following returns nothing if one of x or y is None, and something if both values are set.

Writing these nested match statements can be really tedious, especially the more there are, so there's a bind function in OCaml to simplify this:

let x = Some 5 in
let y = None in
Option.bind x (fun v1 ->
Option.bind y (fun v2 ->
Some (v1 + v2)
)
)

This is debatably less tedious.

This is where two things happened in OCaml if I understand correctly:

Jane street came up with a ppx called ppx_let to make it easier to write and read such statements

Let's explain the syntax introduced by OCaml first.

To do this, I'll define our monad by extending the OCaml Option type:

module Monad = struct
type 'a t = 'a option
let return x = Some x
let bind = Option.bind
let ( let* ) = bind
end

The syntax introduced by Ocaml is the let* which we can define ourselves. (There's also let+ if we want to use that.)

We can now rewrite the previous example with it:

open Monad
let print_res res =
match res with
| None -> print_endline "none"
| Some x -> Format.printf "some: %d\n" x
let () =
(* example chaining Option.bind, similar to the previous example *)
let res : int t =
bind (Some 5) (fun a -> bind (Some 6) (fun b -> Some (a + b)))
in
print_res res;
(* same example but using the let* syntax now *)
let res =
let* a = Some 5 in
let* b = Some 6 in
Some (a + b)
in
print_res res

Or I guess you can use the return keyword to write something a bit more idiomatic:

let res =
let* a = Some 5 in
let* b = Some 6 in
return (a + b)
in
print_res res

Even though this is much cleaner, the new syntax should melt your brain if you don't understand exactly what it is doing underneath the surface.

But essentially, this is what monads are. A container (e.g. option) and a bind function to chain operations on that container.

Bonus: this is how the jane street's ppx_let syntax works (only defined on result, a similar type to option, in their library):

open Base
open Result.Let_syntax
let print_res res =
match res with
| Error e -> Stdio.printf "error: %s\n" e
| Ok x -> Stdio.printf "ok: %d\n" x
let () =
(* example chaining Result.bind *)
let res =
Result.bind (Ok 5) ~f:(fun a ->
Result.bind (Error "lol") ~f:(fun b -> Ok (a + b)))
in
print_res res;
(* same example using the ppx_let syntax *)
let res =
let%bind a = Ok 5 in
let%bind b = Error "lol" in
Ok (a + b)
in
print_res res

You will need to following dune file if you want to run it (assuming your file is called monads.ml):

The sum check protocol allows a prover to convince a verifier that the sum of a multivariate polynomial is equal to some known value. It’s an interactive protocol used in a lot of zero-knowledge protocols. I assume that you know the Schwartz Zippel lemma in this video, and only give you intuitions about how the protocol works.

Vitalik recently mentioned zkapps at ETHMexico. But what are these zkapps? By the end of this post you will know what they are, and how they are going to change the technology landscape as we know it.

Zkapps, or zero-knowledge applications, are the modern and secure solution we found to allow someone else to compute arbitrary programs, while allowing us to trust the result.
And all of that thanks to a recently rediscovered cryptographic construction called general-purpose zero-knowledge proofs.
With it, no need to trust the hardware to behave correctly, especially if you're not the one running it (cough cough intel SGX).

Today, we're seeing zero-knowledge proofs impacting cryptocurrencies (which as a whole have been a petri dish for cryptographic innovation), but tomorrow I argue that most applications (not just cryptocurrencies) will be directly or indirectly impacted by zero-knowledge technology.

Because I've spent so much time with cryptocurrencies in recent years, auditing blockchains like Zcash and Ethereum at NCC Group, and working on projects like Libra/Diem at Facebook, I'm mostly going to focus on what's happening in the blockchain world in this post.
If you want a bigger introduction to all of these concepts, check my book Real-World Cryptography.

The origin of the story starts with the ancient search for solutions to the problem of verifiable computation; being able to verify that the result of a computation is correct. In other words, that whoever run the program is not lying to us about the result.

Most of the solutions, until today, were based on hardware. Hardware chips were first invented to be "tamper resistant" and "hard to analyze". Chips capable of performing simple cryptographic operations like signing or encryption.
You would typically find them in sim cards, TV boxes, and in credit cards.
While all of these are being phased out, they are being replaced by equivalent chips called "secure enclaves" that can be found in your phone.
On the enterprise side, more recently technologies were introduced to provide programmability. Chips capable of running arbitrary programs, while providing (signed) attestation that the programs were run correctly. These chips would typically be certified by some vendor (for example, Intel SGX) with some claim that it’s hard to tamper with them.
Unfortunately for Intel and others, the security community has found a lot of interest in publishing attacks on their "secure" hardware, and we see new hacks coming up pretty much every year.
It's a game of cat and mouse.

Cryptocurrencies is just another field that's been dying to find a solution to this verifiable computation problem.
The previous hardware solutions I’ve talked about can be found in oracles like town crier, in bridges like the Ethereum-Avalanche bridge, or even at the core of cryptocurrencies like MobileCoin.

Needless to say, I'm not a fan, but I'll be the first to conceive that in some scenarios you just don't have a choice.
And being expensive enough for attackers to break is a legitimate solution.
I like to be able to pay with my smartphone.

But in recent years, an old cryptographic primitive that can solve our verifiable computation problem for real has made a huge comeback. Yes you know which one I'm talking about: general-purpose zero-knowledge proofs (ZKPs).

With it, there is no need to trust the hardware: whoever runs the program can simply create a cryptographic proof to convince you that the result is correct.

ZKPs have been used to solve ALL kind of problems in cryptocurrency:

"I wish we could process many more transactions" -> simply let someone else run a program that verifies all the transactions and outputs a small list of changes to be made to the blockchain (and a proof that the output is correct). This is what zk rollups do.

"I wish we could mask the sender, recipient, and the amount being transacted" -> just encrypt your transaction! And use a zero-knowledge proof to prove that what's encrypted is correct. This is what ZCash has done (and Monero, to some extent).

"It takes ages for me to download the whole Bitcoin ledger..." -> simply have someone else do it for you, and give you the resulting latest state (with a proof that it's correct). That's what Mina does.

"Everybody using cryptocurrency is just blindly trusting some public server (e.g. Infura) instead of running their own nodes" -> use ZKP to make light clients verifiable! This is what Celo does with Plumo.

"Why can't I easily transfer a token from one blockchain to another one?" -> use these verifiable light clients. This is what zkBridge proposes.

There's many more, but I want to focus on zkapps (remember?) in this post.
Zkapps are a new way to implement smart contracts. Smart contracts were first pioneered by Ethereum, to allow user programs to run on the blockchain itself.

To explain smart contracts, I like the analogy of a single supercomputer floating in the sky above us. We're all using the same computer, the one floating in the sky. We all can install our programs on the floating computer, and everyone can execute functions of these programs (which might mutate the state of the program).

The solution found by Ethereum at the time was to implement the concept naively and without using cryptography:

Users can install a program by placing the program's code in a transaction.

Users can execute functions of a program by writing in a transaction the function they want to execute and with what arguments.

Everyone running a node has to run the functions found in users transactions. All of them. In order to get the result (e.g. move X tokens to wallet Y, update the state of the smart contract, etc.)

The last point is the biggest limitation of Ethereum.
We can't have the user provide the result of executing a function, or anyone else really, because we can't trust them.
And so, not only does this mean that everyone is always re executing the same stuff (which is redundant, and slows down the network), but this also means that everything in a smart contract must be public (as everyone must be able to run the function).
There can be no secrets used.
There can be no asynchronous calls or interaction outside of the network while this happens.

This is where zkapps enters the room. Zkapps allow users to run the programs themselves and give everyone else the result (along with a proof).

This not only solves the problem of having everyone re-execute the same smart contract calls constantly, but it also opens up new applications as computations can be non-deterministic: they can use randomness, they can use secrets, they can use asynchronous calls, etc.

More than that, the state of a zkapp can now mostly live off-chain, like real applications before Ethereum used to do. Reducing the size of the entire blockchain (Today, Ethereum is almost 1 terabyte!).
These applications are not limited by the speed of the blockchain, or by the capabilities of the language exposed by the blockchain anymore.

Perhaps, it would be more correct to describe them as mini-blockchains of their own, that can be run as centralized or decentralized applications, similar to L2s or Cosmos zones.

OK. So far so good, but do these zkapps really exist or is it just talk? Well, not yet. But a few days ago, the Mina cryptocurrency released their implementations of zkapps on a testnet. And if the testnet goes well, there is no reason to believe this won't unlock a gigantic number of applications we haven't seen before on blockchains.

You can read the hello world tutorial and deploy your first zkapp in like 5 minutes (I kid you not).
So I highly recommend you to try it. This is the future :)