# Modern ZK Crypto - Session 2 Exercises
Today's exercises are hands-on coding exercises. Try writing the following circuits on your own. If you get stuck, you can check the solutions.
If you need to look up circom language features or syntax, take a look at the [circom docs](https://docs.circom.io/circom-language/signals/). I recommend trying to build these circuits in [zkREPL](zkrepl.dev), for fast iteration.
I recommend doing these exercises in order, as later circuits may build on previous ones.
## References
### Field Size
All signals in circom are field elements in the prime field of order
```
r = 21888242871839275222246405745257275088548364400416034343698204186575808495617
```
This is a 254-bit prime known as the BabyJubJub prime. It's the curve order for BN254, a pairing-friendly elliptic curve used by Ethereum and (formerly) ZCash. You can read more about BN254 in Jonathan Wang's excellent document [here](https://hackmd.io/@jpw/bn254).
### Recommendations and Common Issues
If you are testing your circuits using zkREPL's input comment functionality (particularly if you are using negative numbers), be aware that you'll need to surround numbers in quotations marks, and you'll need to write them as non-negative residues. This is due to the interpretation of the signal input JSON as a Javascript object, with biginteger values.
For example, testing an input signal `x` with the value `-1` should look like this:
```
"x": "21888242871839275222246405745257275088548364400416034343698204186575808495616"
```
Forgetting these quotation marks is a common source of failing tests.
## Exercises
### Num2Bits
- Parameters: `nBits`
- Input signal(s): `in`
- Output signal(s): `b[nBits]`
The output signals should be an array of bits of length `nBits` equivalent to the binary representation of `in`. `b[0]` is the least significant bit.
[Solution](https://github.com/iden3/circomlib/blob/master/circuits/bitify.circom#L25)
### IsZero
- Parameters: none
- Input signal(s): `in`
- Output signal(s): `out`
Specification: If `in` is zero, `out` should be `1`. If `in` is nonzero, `out` should be `0`. This one is a little tricky!
[Solution](https://github.com/iden3/circomlib/blob/master/circuits/comparators.circom#L24)
### IsEqual
- Parameters: none
- Input signal(s): `in[2]`
- Output signal(s): `out`
Specification: If `in[0]` is equal to `in[1]`, `out` should be `1`. Otherwise, `out` should be `0`.
[Solution](https://github.com/iden3/circomlib/blob/master/circuits/comparators.circom#L37)
### Selector
- Parameters: `nChoices`
- Input signal(s): `in[nChoices]`, `index`
- Output: `out`
Specification: The output `out` should be equal to `in[index]`. If `index` is out of bounds (not in [0, nChoices)), `out` should be `0`.
[Solution](https://github.com/darkforest-eth/circuits/blob/master/perlin/QuinSelector.circom)
### IsNegative
NOTE: Signals are residues modulo p (the Babyjubjub prime), and there is no natural notion of “negative” numbers mod p. However, it is pretty clear that that modular arithmetic works analogously to integer arithmetic when we treat `p-1` as `-1`. So we define a convention: "Negative" is by convention considered to be any residue in (p/2, p-1], and nonnegative is anything in [0, p/2)
- Parameters: none
- Input signal(s): `in`
- Output signal(s): `out`
Specification: If `in` is negative according to our convention, `out` should be `1`. Otherwise, `out` should be `0`. You are free to use the [CompConstant circuit](https://github.com/iden3/circomlib/blob/master/circuits/compconstant.circom), which takes a constant parameter `ct`, outputs `1` if `in` (a binary array) is strictly greater than `ct` when interpreted as an integer, and `0` otherwise.
[Solution](https://github.com/iden3/circomlib/blob/master/circuits/sign.circom#L23)
- **Understanding check**: Why can’t we just use LessThan or one of the comparator circuits from the previous exercise?
### LessThan
- Parameters: none
- Input signal(s): `in[2]`. Assume that it is known ahead of time that these are at most $2^{252} - 1$.
- Output signal(s): `out`
Specification: If `in[0]` is strictly less than `in[1]`, `out` should be `1`. Otherwise, `out` should be `0`.
- **Extension 1**: If you know that the input signals are at most 2^k - 1 (k ≤ 252), how can you reduce the total number of constraints that this circuit requires? Write a version of this circuit parametrized in `k`.
- **Extension 2**: Write LessEqThan (tests if in[0] is ≤ in[1]), GreaterThan, and GreaterEqThan
[Solution (with extension 1)](https://github.com/iden3/circomlib/blob/master/circuits/comparators.circom#L89)
### IntegerDivide
NOTE: This circuit is pretty hard!
- Parameters: `nbits`. Use `assert` to assert that this is at most 126!
- Input signal(s): `dividend`, `divisor`
- Output signal(s): `remainder`, `quotient`
Specification: First, check that the `dividend` and `divisor` are at most `nbits` in bitlength. Next, compute and constrain `remainder` and `quotient`.
- **Extension**: How would you modify the circuit to handle negative dividends?
[Solution](https://github.com/darkforest-eth/circuits/blob/master/perlin/perlin.circom#L44) (ignore the second parameter SQRT_P; that is extraneous)

For this week's exercise, you will write a spec for one of the following proposed applications. If you are working on a project, you're automatically "exempt," as you'll already be building one of these! In your spec, you should: Outline precisely the ZK circuit(s) you'll be using. You can use our template from class if you'd like. Describe the business logic of the major components of the application: the server (what it stores, what it verifies, etc.) smart contracts (if applicable), clients (what they store, what they prove, etc.), and any additional services (for example, an auxiliary server that stores, verifies, and serves Merkle groups). WhaleChat

6/14/2023ZK Passwords In this exercise, you will implement a ZK-based password verifier! Instructions can be found in the README of the scaffolding repo. You are aiming to produce something that looks like this video demo. You can start with the scaffolding repo, or you can roll your own from scratch. If you use the scaffold repo, make sure to check out circom-starter, the base repo which this is based on! Extension: make proving happen in-browser. We’ll go over this during next week’s Monday optional session. You may also find this circom and snarkjs diagram useful.

6/6/2023This week's exercises are hands-on coding exercises. Try writing the following circuits on your own. If you get stuck, you can check the solutions. We'll go over these circuits in the second Optional Session in Week 2. If you need to look up circom language features or syntax, take a look at the circom docs. I recommend trying to build these circuits in zkREPL, for fast iteration. I recommend doing these exercises in order, as later circuits may build on previous ones. References Field Size All signals in circom are treated as numbers modulo this big prime:

6/1/2023Zupass Official Repo: https://github.com/proofcarryingdata/zupass The README provides a starting point for devs. Usage Examples for Devs: https://consumer-client.onrender.com This page contains examples for devs on how to integrate Zupass into various application flows, such as authentication, proof generation/consumption. Code for these examples is in the official Zupass repo. Example Application: Zuzalu Confessions: https://confessions-client.onrender.com/ This is a fully-functional anonymous message board built on Zupass, demonstrating authentication and identity-hiding proofs. For development/demonstration/reference purposes only! Zuzalu Confessions Reference Code: https://github.com/proofcarryingdata/zuzalu-confessions

4/11/2023
Published on ** HackMD**