# Poseidon hash function

Poseidon is a hash function that can efficiently run in a zk circuit. (See poseidon-hash.info) It is based on the sponge function, with a state composed of field elements and a permutation based on field element operation (addition and exponentiation).

The permutation contains an S-box (exponentiation of a group element), adding constants to the state, and matrix multiplication of the state (multiplications and additions) with an MDS matrix.

Since a field element is around 255-bit, a single field element is enough as the capacity of the sponge to provide around 128-bit security.

Note

We might want to piggy back on the zcash poseidon spec at some point (perhaps by making this an extension of the zcash poseidon spec).

## APIs

We define a base sponge, and a scalar sponge. Both must be instantiated when verifying a proof (this is due to recursion-support baked in Kimchi).

External users of kimchi (or pickles) are most likely to interact with a wrap proof (see the pickles specification). As such, the sponges they need to instantiate are most likely to be instantiated with:

• Poseidon-Fp for base sponge
• Poseidon-Fq for the scalar sponge

### Base Sponge

• new(params) -> BaseSponge. Creates a new base sponge.
• BaseSponge.absorb(field_element). Absorbs a field element by calling the underlying sponge absorb function.
• BaseSponge.absorb_point(point). Absorbs an elliptic curve point. If the point is the point at infinity, absorb two zeros. Otherwise, absorb the x and y coordinates with two calls to absorb.
• BaseSponge.absorb_scalar(field_element_of_scalar_field). Absorbs a scalar.
• If the scalar field is smaller than the base field (e.g. Fp is smaller than Fq), then the scalar is casted to a field element of the base field and absorbed via absorb.
• Otherwise, the value is split between its least significant bit and the rest. Then both values are casted to field elements of the base field and absorbed via absorb (the high bits first, then the low bit).
• BaseSponge.digest() -> field_element. The squeeze function of the underlying sponge function is called and the first field element is returned.
• BaseSponge.digest_scalar() -> field_element_of_scalar_field.
• BaseSponge.challenge // TODO: specify.
• BaseSponge.challenge_fq // TODO: specify.

### Scalar Sponge

• new(params) -> ScalarSponge
• ScalarSponge.absorb(scalar_field_element)
• ScalarSponge.digest() -> scalar_field_element
• ScalarSponge.challenge // TODO: specify

## Algorithms

Note that every operation is done in the field of the sponge.

In this section we define the high-level algorithm behind the permutation and the sponge. The permutation is never used directly by users, it is used only by the sponge function.

### Permutation

In pseudo-code:

def sbox(field_element):
# modular exponentiation
return field_element^7

# apply MDS matrix
def apply_mds(state):
n = [0, 0, 0]
n[0] = state[0] * mds[0][0] + state[1] * mds[0][1] + state[2] * mds[0][2]
n[1] = state[0] * mds[1][0] + state[1] * mds[1][1] + state[2] * mds[1][2]
n[2] = state[0] * mds[2][0] + state[1] * mds[2][1] + state[2] * mds[2][2]
return n

# a round in the permutation
def apply_round(round, state):
# sbox
state[0] = sbox(state[0])
state[1] = sbox(state[1])
state[2] = sbox(state[2])

# apply MDS matrix
state = apply_mds(state)

state[0] += round_constants[round][0]
state[1] += round_constants[round][1]
state[2] += round_constants[round][2]

# the permutation
def permutation(state):
round_offset = 0
if ARK_INITIAL:
constant = round_constants[0]
state[0] += constant[0]
state[1] += constant[1]
state[2] += constant[2]
round_offset = 1

for round in range(round_offset, ROUNDS + round_offset):
apply_round(round, state)


### Sponge

In pseudo-code:

def new():
return {
"state": [0] * RATE, # RATE field elements set to 0
"mode": "absorbing",
"offset": 0,
}

def absorb(sponge, field_element):
# if we're changing mode, reset the offset
if sponge.mode == "squeezing":
sponge.mode = "absorbing"
sponge.offset = 0
# we reached the end of the rate, permute
elif sponge.offset == RATE:
sponge.state = permutation(sponge.state)
sponge.offset = 0

# absorb by adding to the state
sponge.state[sponge.offset] += field_element
sponge.offset += 1

def squeeze(sponge):
# permute when changing mode or when we already squeezed everything
if sponge.mode == "absorbing" or sponge.offset == RATE:
sponge.mode = "squeezing"
sponge.state = permutation(sponge.state)
sponge.offset = 0

result = sponge.state[sponge.offset]
sponge.offset += 1
return result


## Instantiations

We instantiate two versions of Poseidon, one for the field Fp, and one for the field Fq (see the pasta specification).

Both share the following sponge configuration:

• capacity 1
• rate: 2

and the following permutation configuration:

• number of full rounds: 55
• sbox: 7
• ARK_INITIAL: false

### Poseidon-Fp

You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs.

### Poseidon-Fq

You can find the MDS matrix and round constants we use in /poseidon/src/pasta/fp_kimchi.rs.

## Test vectors

We have test vectors contained in /poseidon/src/tests/test_vectors/kimchi.json.

## Pointers to the OCaml code

• our ocaml implementation: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle/random_oracle.mli
• relies on random_oracle_input: https://github.com/minaprotocol/mina/blob/develop/src/lib/random_oracle_input/random_oracle_input.ml
• is instantiated with two types of fields:
• https://github.com/minaprotocol/mina/blob/develop/src/nonconsensus/snark_params/snark_params_nonconsensus.ml