In this paper, a novel method is proposed to build an improved 1-D discrete chaotic map called flipped product chaotic system (FPCS) by multiplying the output of one map with the output of a vertically flipped second map. Two variants, each with nine combinations, are shown with trade-off between computational cost and performance. The chaotic properties are explored using the bifurcation diagram, Lyapunov exponent, Kolmogorov entropy, and correlation coefficient. The proposed schemes offer a wider chaotic range and improved chaotic performance compared to the constituent maps and several prior works of similar nature. Wide chaotic window and improved chaotic complexity are two desired characteristics for several security applications as these two characteristics ensure enhanced design space with elevated entropic properties. We present a general Field-Programmable Gate Array (FPGA) design framework for the hardware implementation of the proposed flipped-product schemes and the results show good qualitative agreement with the numerical results from MATLAB simulation. Finally, we present a new Pseudo Random Number Generator (PRNG) using the two variants of the proposed chaotic map and validate their excellent randomness property using four standard statistical tests, namely NIST, FIPS, TestU01, and Diehard.