# Cryptographic Primitives

### 1. Baby Jubjub Elliptic Curve

Baby Jubjub is a twisted Edwards elliptic curve specifically designed for efficient zero-knowledge proof systems. It operates over the prime field **Fr** of the BN254 curve, making it highly compatible with zk-SNARK circuits and the Ethereum ecosystem.

Baby Jubjub is defined by the twisted Edwards equation:

```
ax² + y² = 1 + dx²y²
```

Where:

* **Prime field**: p = 21888242871839275222246405745257275088548364400416034343698204186575808495617

Curve parameters:

* **a** = 168700
* **d** = 168696

For more details, see [circomlib implementation](https://github.com/iden3/circomlib/blob/master/circuits/babyjub.circom).

### 2. Poseidon Hash Function

Poseidon is a cryptographic hash function specifically designed for zero-knowledge proof systems. It provides significant efficiency improvements over traditional hash functions like SHA-256 when used in arithmetic circuits.

Poseidon operates directly over prime fields, eliminating the need for expensive bit operations in zk-SNARK circuits:

* Field Operations: Uses only addition and multiplication in **Fr**
* No Bit Manipulation: Avoids costly binary operations
* Constraint Efficiency: Requires significantly fewer R1CS constraints

**Parameters for BN254 Field**

* **Field Prime**: Same as Baby Jubjub (p = 21888242871839275222246405745257275088548364400416034343698204186575808495617)

**Round Function**

Each Poseidon round consists of three operations:

1. **AddRoundConstants**: state\[i] = state\[i] + roundConstant\[i]
2. **SubWords** (S-box): state\[i] = state\[i]^5
3. **MixLayer**: Linear transformation using MDS matrix

For more details, see [circomlib parameters of Poseidon](https://github.com/iden3/circomlib/blob/master/circuits/poseidon_constants.circom).

### 3. zk-SNARK (Groth16)

Hinkal’s privacy logic is expressed as Circom circuits compiled to R1CS and proven with Groth16 over the BN254 field. Groth16 provides constant-size proofs and fast on-chain verification, making it a practical choice for Ethereum.

**Arithmetization**

* Field: **Fr** of BN254 (same prime p as above).
* Model: Rank-1 Constraint System (R1CS).
* Libs: Circom 2.x with circomlib primitives (BabyJubjub ops, Poseidon hash, Merkle path gadgets).

**Trusted Setup** (two-phase; Phase 1 from Polygon)

* Phase 1 (universal “Powers of Tau”): We reuse [Polygon Hermez’s first-stage](https://hermez.s3-eu-west-1.amazonaws.com/powersOfTau28_hez_final_21.ptau) trusted setup as the universal CRS (Common Reference String). This stage is circuit-agnostic and need not be rerun when circuits change.
* Phase 2 (circuit-specific): Starting from the Phase-1 CRS, we run a per-circuit setup to derive the proving key (pk) and verifying key (vk). Any circuit change requires redoing Phase 2 only; the on-chain verifier uses the fixed vk.


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