Cache side-channel attacks consist, for a malicious process, to infer the current state of the cache by measuring the time it takes to access the memory, and indirectly gain knowledge about other processes sharing this same physical cache. Because cache side-channel attacks leverage a hardware leakage without requiring any physical access to the devices, they represent very serious threats. Among the runtime detection techniques for cache side-channel attacks, hardware solutions are usually fine-grained and benefit from less performance overhead than software solutions. However, they are not flexible enough to suit the rapid evolution and appearance of software attacks. In this paper we describe REHAD, a novel attack detection architecture that uses reconfigurable hardware. More precisely, it includes a hardware detection module that can be reconfigured by means of a trusted software kernel, to adapt to the level of threats and to detect new attacks. This architecture also benefits from hardware parallelism to fill the frequency gap between reconfigurable hardware and core processor. REHAD has been integrated into the ORCA softcore RISC-V on a FPGA and two common cache side-channel attacks have been successfully detected.