Poseidon Hashing

Poseidon is an arithmetic-friendly hash function designed for zero-knowledge circuits. Achronyme’s implementation uses BN254 parameters compatible with circomlibjs, so hashes computed in Achronyme match those from the iden3/circom ecosystem.

Basic Hash

Compute a 2-to-1 Poseidon hash:

let h: Field = poseidon(1, 2)
print(h)
// 7853200120776062878684798364095072458815029376092732009249414926327459813530

Both arguments can be integers or field elements. The result is always a field element.

Known Test Vectors

These values match circomlibjs output:

// poseidon(0, 0)
let h0 = poseidon(0, 0)
print(h0)
// 14744269619966411208579211824598458697587494354926760081771325075741142829156

// poseidon(1, 2)
let h1 = poseidon(1, 2)
print(h1)
// 7853200120776062878684798364095072458815029376092732009249414926327459813530

You can use these vectors to verify compatibility with other Poseidon implementations.

Hash Chains

Chain multiple values by feeding the previous hash as input:

let h01 = poseidon(1, 2)
let h012 = poseidon(h01, 3)
let h0123 = poseidon(h012, 4)
print(h0123)

Or use poseidon_many() for the same result with less code:

let h = poseidon_many(1, 2, 3, 4)
print(h)
// Same as poseidon(poseidon(poseidon(1, 2), 3), 4)

poseidon_many requires at least 2 arguments. It left-folds the hash: the first two values are hashed together, then each subsequent value is hashed with the running result.

Input Order Matters

Poseidon is not commutative — swapping inputs produces a different hash:

let h1 = poseidon(1, 2)
let h2 = poseidon(2, 1)
assert(h1 != h2)  // different values

This property is essential for Merkle trees, where left and right children must be distinguished.

Using with Field Elements

For values larger than i60 or for exact BN254 field arithmetic, use 0p field literals:

let a = 0p21888242871839275222246405745257275088548364400416034343698204186575808495616
let b = 0p1
let h = poseidon(a, b)
print(h)

In Circuit Mode

Poseidon works in both VM and circuit mode. In circuits, each hash emits 361 R1CS constraints (360 permutation rounds + 1 capacity initialization):

circuit poseidon_check(expected: Public, a: Witness, b: Witness) {
    let h = poseidon(a, b)
    assert_eq(h, expected)
}

Compile and run:

ach circuit hash.ach --inputs "expected=7853200120776062878684798364095072458815029376092732009249414926327459813530,a=1,b=2"

Constraint Costs

Expression	Constraints
poseidon(a, b)	361
poseidon_many(a, b, c)	722 (2 × 361)
poseidon_many(a, b, c, d)	1,083 (3 × 361)

Building Merkle Roots

Compute a Merkle root by hashing leaves pairwise, then hashing the results:

// 4-leaf tree
let leaf0 = 0p100
let leaf1 = 0p200
let leaf2 = 0p300
let leaf3 = 0p400

// Level 0: hash pairs
let n0 = poseidon(leaf0, leaf1)
let n1 = poseidon(leaf2, leaf3)

// Level 1: root
let root = poseidon(n0, n1)
print(root)

For larger trees, use loops:

// 8-leaf tree
let leaves = [
    0p100, 0p200, 0p300, 0p400,
    0p500, 0p600, 0p700, 0p800
]

// Level 0
let l0_0 = poseidon(leaves[0], leaves[1])
let l0_1 = poseidon(leaves[2], leaves[3])
let l0_2 = poseidon(leaves[4], leaves[5])
let l0_3 = poseidon(leaves[6], leaves[7])

// Level 1
let l1_0 = poseidon(l0_0, l0_1)
let l1_1 = poseidon(l0_2, l0_3)

// Root
let root = poseidon(l1_0, l1_1)
print(root)

In Prove Blocks

Poseidon works inside prove {} for inline proof generation:

let a: Field = 0p1
let b: Field = 0p2
let h: Field = 0p7853200120776062878684798364095072458815029376092732009249414926327459813530

let p = prove(h: Public) {
    assert_eq(poseidon(a, b), h)
}

print(verify_proof(p))  // true

The prover demonstrates knowledge of a and b without revealing them. The verifier only sees h.

circomlibjs Compatibility

Achronyme uses the same Poseidon constants as circomlibjs v0.1.7:

• Curve: BN254 scalar field
• State width: t = 3
• Full rounds: R_f = 8
• Partial rounds: R_p = 57
• S-box: x^5

Hashes computed with poseidon() in Achronyme produce identical outputs to circomlibjs.poseidon([a, b]). This ensures interoperability with circom circuits, snarkjs proofs, and on-chain Poseidon verifiers.