@aliashraf

I have updated the paper that I wrote with all the details we have discussed. It summarizes it all together. I will be uploading it soon. As you and empty[g] have provided valuable insights I wish to add you as authors. If you wish to allow this please email me your names and the email I should include and I will update the paper to reflect this.

Regarding the impact of collisions on the homomorphic hashing I included the following in the paper to discuss this:

The presence of collisions has implications for the homomorphic feature of the proposed hash function. Specifically a hash can have more than one secret. This implies that equation 7 can have multiple solutions.

If Alice is able to determine two secrets that have the same hash, she can use the one to generate hashes and sums to send to Bob, but use the other secret to unlock the coins Bob sent to her. This will stop Bob from being able to solve equation 5 correctly and unable to claim his coins. After a time delay Alice will then claim the coins.

This increases the requirement for a large search space. An attacker must not be able to determine any of the secrets that generate the same hash except using brute force.

A large search space is required, but also when choosing n and p to have collisions there should be no obvious double roots to a collision, e.g n = 2, p = 13, s

_{1} = 4, h

_{1} = 3, s

_{2} = 9, h

_{2} = 3 where -4 mod 13 = 9 => -s

_{1} mod p = s

_{2} when n = 2

If you want to find out more about the upper bound of the collisions, please look at Lagrange's Theorem on congruence of polynomials under modulo a prime. I also discuss this in the paper.