The article explores why cryptographic hash functions remain irreversible despite being composed of simple, reversible operations like addition, multiplication, and modulo. While individual operations can be inverted, chaining them creates complex polynomial equations that become computationally difficult to solve.
A simple iterative function using the formula (x * x + m) % P demonstrates this effect. Each iteration doubles the degree of the resulting polynomial equation. By the 20th step, the equation degree exceeds one million, making it impractical to solve using standard algebraic methods.
While algorithms like Cantor-Zassenhaus exist for solving modular polynomial equations, they are not efficient for high-degree polynomials. The author notes that, similar to SHA-256, there is no formal proof of total irreversibility, but finding a solution becomes computationally infeasible as iterations increase.
The author provides examples in a repository, including implementations for 17 and 25 iterations, to illustrate the rapid growth of equation complexity. The study highlights that the security of such functions relies on the mathematical difficulty of finding roots for high-degree polynomials.
