We study the classical rumor spreading problem, which is used to spread information in an unknown network with $n$ nodes. We present the first protocol for any expander graph $G$ with $n$ nodes and minimum degree $Θ(n)$ such that, the protocol informs every node in $O(\log n)$ rounds with high probability, and uses $O(\log n\log\log n)$ random bits in total. The runtime of our protocol is tight, and the randomness requirement of $O(\log n\log\log n)$ random bits almost matches the lower bound of $Ω(\log n)$ random bits. We further study rumor spreading protocols for more general graphs, and for several graph topologies our protocols are as fast as the classical protocol and use $\tilde{O}(\log n)$ random bits in total, in contrast to $O(n\log^2n)$ random bits used in the well-known rumor spreading push protocol. These results together give us almost full understanding of the randomness requirement for this basic epidemic process. Our protocols rely on a novel reduction between rumor spreading processes and branching programs, and this reduction provides a general framework to derandomize these complex and distributed epidemic processes. Interestingly, one cannot simply apply PRGs for branching programs as rumor spreading process is not characterized by small-space computation. Our protocols require the composition of several pseudorandom objects, e.g. pseudorandom generators, and pairwise independent generators. Besides designing rumor spreading protocols, the techniques developed here may have applications in studying the randomness complexity of distributed algorithms.