The increased power consumption of high-resolution data converters at higher carrier frequencies and larger bandwidths is becoming a bottleneck for communication systems. In this paper, we consider a fully digital base station equipped with 1-bit analog-to-digital (in uplink) and digital-to-analog (in downlink) converters on each radio frequency chain. The base station communicates with multiple single antenna users with individual SINR constraints. We first establish the uplink downlink duality principle under 1-bit hardware constraints under an uncorrelated quantization noise assumption. We then present a linear solution to the multi-user downlink beamforming problem based on the uplink downlink duality principle. The proposed solution takes into account the hardware constraints and jointly optimizes the downlink beamformers and the power allocated to each user. Optimized dithering obtained by adding dummy users to the true system users ensures that the uncorrelated quantization noise assumption is true under realistic settings. Detailed simulations carried out using 3GPP channel models generated from Quadriga show that our proposed solution outperforms state of the art solutions in terms of the ergodic sum and minimum rate especially when the number of users is large. We also demonstrate that the proposed solution significantly reduces the performance gap from non-linear solutions in terms of the uncoded bit error rate at a fraction of the computational complexity.