# Eco VDF

## Routines

### `isProbablePrime(n,k)`

* $n,k$-- 256-bit unsigned.
* if $n\in\{2,3,5\}$ return true;
* if $n==1$ or even return false;
* compute $s$ such that $n-1=d2^s$ with odd $d$;
* for $i<k$:
* $h\leftarrow SHA3(previousBlockHash,i)$;
* $a \leftarrow h\bmod{(n-4)}+2$
* $x \leftarrow a^d\bmod{n}$
* If $x\notin\{1,-1\}$
* If for $1\leq j < s$ it holds $x^{2^j}\neq -1 \bmod{n}$
* Return FALSE
* Return TRUE

### `start(x,t,y)`

* $x,t$ are 256-bit and $y$ is a big integer
* pad $y$ with leading 0s to the multiple of 256 bits
* $N$ is the RSA-2048 bit challenge
* compute $x^2$ as big integer
* require that $y<N$ and $y$ (padded) is at least 512 bits long
* require that $x>1$
* check that `isProbablePrime`$(x,10)$ returns TRUE
* Store $x,t,y,z=x^2,w=y$ and `progress==0` for Sender

### `update(d,u)`
* $u$ is big integer
* Check that $x\neq 0,u^2\bmod{n}>1,u>1$
* Compute $r\leftarrow SHA3(x,y,u,d)$
* Update $z \leftarrow z^ru^2\bmod{n}$
* Update $w\leftarrow (u^2)^rw\bmod{n}$
* If $d== t-1$
* Check if $z^4==w$
* Store $S[SHA3(t,x)]\leftarrow SHA3(\underbrace{y}_{2048-bit})$ and delete the state
* Otherwise FAIL

### `isVerified(x,t,y)`
* Check if $S[SHA3(t,x)]\leftarrow SHA3(\underbrace{y}_{2048-bit})$

## Usage

### Inflation usecase

1. Contract is initialized with difficulty $t$.
2. (`commitEntropyVDFSeed`) Alice selects a prime $x$ as the nearest prime to the last blockhash, calling `isProbablePrime` for verification. Contract stores $x$.
3. (`submitEntropyVDF`) Bob supplies $y$ such that `isVerified`$(x,t,y)$ returns true. Contract stores $s=SHA3(y)$.
4. Payout winners are determined based on $s$.