The thesis summarizes known results in the field of NP search problems. We discuss the complexity of integer factoring in detail, and we propose new results which place the problem in known classes and aim to separate it from PLS in some sense. Furthermore, we define several new search problems.

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Given polynomials $f,g,h\,\in \mathbb{F}[x_1,\ldots,x_n]$ such that $f=g/h$, where both $g$ and $h$ are computable by arithmetic circuits of size $s$, we show that $f$ can be computed by a circuit of size $\poly(s,\deg(h))$. This solves a special case of division elimination for high-degree circuits (Kaltofen'87 \& WACT'16). The result ... more >>>

We consider the problem of deterministically factoring integers provided with oracle access to important number-theoretic functions such as Euler's Totient function - phi(.) and Carmichael's Lambda function - lambda(.).
We focus on Carmichael numbers - also known as Fermat pseudoprimes. In particular, we obtain the following results:

1. Let N ... more >>>