This study aimed to improve the output quality and statistical characteristics of random output sequences by combining the output of an 8-bit Fibonacci nonlinear feedback shift register (NLFSR) of degree three with a sequence of keys generated by a coupled logistic-sine system (LSS). Subsequently, the proposed pseudorandom number generator PRNG-LSS-NLFSR was investigated using a simple image-encryption application. Theoretical randomness evaluation through the National Institute of Standards and Technology (NIST) and other statistical test of its application by means of the histogram, entropy analysis, correlation test, and other tests confirm that the proposed PRNG-LSS-NLFSR is simple, has random and nonlinear features, and displays a strong cryptographic security aspect.