# Poseidon Parameter Generation

The problem of Poseidon parameter generation is to pick secure choices for the parameters used in the permutation given the field, desired security level in bits, as well as the width of the hash function one wants to instantiate (i.e. 1:1 hash, 2:1 hash, etc.).

Poseidon parameters consist of:

• Choice of S-box: choosing the exponent for the S-box layer where ,
• Round numbers: the numbers of partial and full rounds,
• Round constants: the constants to be added in the AddRoundConstants step,
• MDS Matrix: generating a Maximum Distance Separable (MDS) matrix to use in the linear layer, where we multiply this matrix by the internal state.

Appendix B of the Poseidon paper provides sample implementations of both the Poseidon permutation as well as parameter generation. There is a Python script called calc_round_numbers.py which provides the round numbers given the security level , the width of the hash function , as well as the choice of used in the S-box step. There is also a Sage script, which generates the round numbers, constants, and MDS matrix, given the security level , the width of the hash function , as well as the choice of used in the S-box step.

Since the publication of the Poseidon paper, others have edited these scripts, resulting in a number of implementations being in use derived from these initial scripts. We elected to implement Poseidon parameter generation in Rust from the paper, checking each step, and additionally automating the S-box parameter selection step such that one can provide only the modulus of a prime field and the best will be selected.

Below we describe where we deviate from the parameter selection procedure described in the text of the Poseidon paper.

## Choice of S-Box

The Poseidon paper focuses on the cases where , as well as BLS12-381 and BN254. For a choice of positive , it must satisfy , where is the prime modulus.

For our use of Poseidon on BLS12-377, we wanted to generate a procedure for selecting the optimal for a general curve, which we describe below.

Given the following tree of shortest addition chains: We proceed down the tree from depth 2 to depth 5 (where depth 0 is the root of 1):

1. At a given depth, proceed from right to left.
2. For a given element, check if is satisfied. If yes, we choose it, else continue.

If we get through these checks to depth of 5 without finding a positive exponent for , then we pick , which is well-studied in the original Poseidon paper.

## Round Numbers

We implement the round numbers as described in the original paper. These are the number of rounds necessary to resist known attacks in the literature, plus a security margin of +2 full rounds, and +7.5% partial rounds.

We test our round number calculations with tests from Appendices G and H from the paper which contain concrete instantiations of Poseidon for and their round numbers.

## Round Constants

We do not use the Grain LFSR for generating pseudorandom numbers as described in Appendix F of the original paper. Instead, we use a Merlin transcript to enable parameter generation to be fully deterministic and easily reproducible.

We bind this transcript to the input parameter choices: the width of the hash function , the security level , and the modulus of the prime field . We also bind the transcript to the specific instance, as done with the Grain LFSR in Appendix F, so we bind to the number of full rounds , the number of partial rounds , as well as the choice of S-box exponent .

We generate random field elements from hashes of this complete transcript of all the input parameters and the derived parameters , , and .

## MDS Matrix

We generate MDS matrices using the Cauchy method. However instead of randomly sampling the field as described in the text, we follow the strategy of Keller and Rosemarin 2020, where we deterministically generate vectors and as:

Each element of the matrix is then constructed as:

where .