In this paper we present a scalar multiplication hardware architecture that computes a constant-time variable-base point multiplication over the Galbraith-Lin-Scott (GLS) family of binary elliptic curves. Our hardware design is especially tailored for the quadratic extension field F 2 2n, with n = 127, which allows us to attain a security level close to 128 bits. We explore extensively the usage of digit-based and Karatsuba multipliers for performing the quadratic field arithmetic associated to GLS elliptic curves and report the area and time performance obtained by these two types of multipliers. Targeting a XILINX KINTEX-7 FPGA device, we report a hardware implementation of our design that achieves a delay of just 3.98μs for computing one scalar multiplication. This allows us to claim the current speed record for this operation at or around the 128-bit security level for any hardware or software realization reported in the literature.