PRP, PRF and modular arithmetic Are there any arithmetic or mathematical functions that can be used as PRPs or PRFs ?  Since, Conventional block ciphers like AES are that are proven to be good PRPs are not based on mathematics but on other principles like confusion and diffusion etc.
Do you want DDH/RSA-based PRFs? If so, we have them and I will answer. –  xagawa
  
  @xagawa Yes, I want that :-) –  Dingo13


I list the PRFs based on the number-theoretical assumptions.
They are ``arithmetic or mathematical function.''
You can use the Feistel network to obtain (S)PRPs from PRFs in theory.

From the DDH assumption

The Naor-Reingold Pseudorandom Function is a PRF based on the DDH assumption.
Let $\mathbb{G}$ be a finite group of order prime $q$ with a generator $g$,
on which the DDH problem is hard.
Let $\{0,1\}^\ell$ be a domain.
The function is defined as
$$\mathrm{PRF_{NR}}(k, x) = g^{a_0 \prod_{i=1}^{\ell} a_i^{x_i}},$$
where $k = (a_0,a_1\dots,a_{\ell}) \in \mathbb{Z}_q^{\ell}$.
For the proof, see the original article in J. ACM.

Lewko and Waters generalized the NR PRFs in CCS 2009 (and ePrint 2009/486).
Their PRFs are based on the decisional linear assumption (and its weaker variants).

Most efficient one

Under stronger assumptions such as the DHI assumption,
 we have a cute PRF, say,
the Dodis-Yampolskiy PRF proposed as
Verifiable Random Function (VRF).

Let $\{0,1\}^{\ell}$ be the domain of PRF.
Suppose that $2^{\ell}$ is a polynoimal of the security parameter. 
The DY PRF is defined on a symmetric bilinear group as follows:
$$\mathrm{PRF}_{DY}(k,x) = e(g,g)^{1/(x+k)},$$
where $k \in \mathbb{Z}_q$. 

From the factoring assumption

This is a classical result obtained by a combining the Blum-Blum-Shub PRG and the Goldreich-Goldwasser-Micali PRF from any PRG.
Naor, Reingold, and Rosen improved efficiency of factoring-based PRFs.
See the original article available at ECCC TR 2001-064,

From the hardness of lattice problems

Nowadays, the lattice problems are often employed to construct cryptogrphic primitives.
You can find the PRFs from lattices in
Banerjee, Peikert, and Rosen (EUROCRYPT 2012),
Alwen, Krenn, Pietrzak, and Wichs (CRYPTO 2013),
and Banerjee and Peikert (ePrint 2014/074).