This article describes a field-programmable gate array (FPGA) cryptographic architecture, which combines the elliptic curve--based Ed25519 digital signature algorithm and the X25519 key establishment scheme in a single module. Cryptographically, these are high-security elliptic curve cryptography algorithms with short key sizes and impressive execution times in software. Our goal is to provide a lightweight FPGA module that enables them on resource-constrained devices, specifically for Internet of Things (IoT) applications. In addition, we aim at extensibility with customisable countermeasures against timing and differential power analysis side-channel attacks and fault-injection attacks. For the former, we offer a choice between time-optimised versus constant-time execution, with or without Z -coordinate randomisation and base-point blinding; and for the latter, we offer enabling or disabling default-case statements in the Finite State Machine (FSM) descriptions. To obtain compactness and at the same time fast execution times, we make maximum use of the Digital Signal Processing (DSP) slices on the FPGA. We designed a single arithmetic unit that is flexible to support operations with two moduli and non-modulus arithmetic. In addition, our design benefits in-place memory management and the local storage of inputs into DSP slices’ pipeline registers and takes advantage of distributed memory. These eliminate a memory access bottleneck. The flexibility is offered by a micro-code supported instruction-set architecture. Our design targets 7-Series Xilinx FPGAs and is prototyped on a Zynq System-on-Chip (SoC). The base design combining Ed25519 and X25519 in a single module, and its implementation requires only around 11.1K Lookup Tables (LUTs), 2.6K registers, and 16 DSP slices. Also, it achieves performance of 1.6ms for a signature generation and 3.6ms for a signature verification for a 1024-bit message with an 82MHz clock. Moreover, the design can be optimised only for X25519, which gives the most compact FPGA implementation compared to previously published X25519 implementations.