## Introduction:

What follows is an information-theoretic lower-bound on the Computational Complexity of integer factorisation that is experimentally verifiable using automated program synthesis. This analysis implies that the RSA function is a one-way function and therefore $$\text{P} \neq \text{NP}$$.

## The average information gained from integer factorisation:

From the Prime Number Theorem, we may deduce that the average distance between consecutive primes $$P$$ and $$Q$$ is on the order of $$\sim \log P$$, and according to the Cramér random model the probability of observing a prime in $$[P+1,Q]$$ is on the order of $$\sim \frac{1}{\log P}$$ so the information gained from observing a prime in $$[P+1, P + \log P]$$ is on the order of:

$$H(Q| Q \sim P + \log P) \approx - \sum_{i=1}^{\log P} \frac{1}{\log P} \log \big(\frac{1}{\log P}\big) = \log \log P \approx \log \log N$$

and given that almost all integers are incompressible, for large $$N$$:

$$N = P \cdot Q \implies K_U(N) \sim \log N = \log P + \log Q$$

where $$P$$ uniquely determines $$Q$$ and vice versa so the information gained from observing $$P$$ is on the order of $$\sim \log N$$.

It follows that, by an application of the AM-GM inequality, the average information gained from observing a prime in $$[1,P]$$ or $$[P+1, P + \log P]$$ is on the order of:

$$\sqrt{\log N \cdot \log \log N} \leq \frac{\log N + \log \log N}{2}$$

This allows us to bound the search space $$\mathcal{L}$$(for $$P$$ and $$Q$$) by:

$$\mathcal{L} \geq \sqrt{\log N \cdot \log \log N}$$

which has units of bits, or the logarithm of distinct states(i.e. integers) considered by the search algorithm(i.e. integer factorisation).

## The typical probability $$q$$ of collapsing the entropy $$\mathcal{L}$$:

Now, if we define the typical probability $$q \in (0,1)$$ of collapsing the entropy $$\mathcal{L}$$(i.e. finding $$P$$ or $$Q$$), we have $$q \approx e^{-\mathcal{L}}$$ so the optimal number of mathematical operations will typically be on the order of:

$$\frac{1}{q} \geq e^{\sqrt{\log N \cdot \log \log N}}$$

which may be viewed as a lower-bound on the typical number of states(i.e. integers) considered by an optimal integer factorisation algorithm running on a classical computer.

## Discussion:

Due to a natural correspondence between integer factorisation and unbounded Koopman operators, a rigorous experimental investigation of this lower-bound is reducible to neural program synthesis for approximating the eigenfunctions of a suitable Koopman operator. In fact, I have proposed a challenge in neural program synthesis for precisely this purpose, Koopman Open, which begins on the 13th of September 2021.

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