This paper proposes a new quantum key distribution(QKD) protocol, namely the pseudo-random bases entangled photon based QKD (PRB-EPQKD) protocol. The latest mainly focuses on three properties, including the security of the protocol, the secure key size and the maximum communication distance between legitimate communication users (Alice and Bob). To achieve this, we first consider a spontaneous-parametric-down (SPDC) photon source located in a low-earth-orbit (LEO) type satellite capable of producing and distributing entangled photons pairs to Alice and Bob. Secondly, we assume that Alice's and Bob's photons state measurement bases are identically generated via a pseudo-random number generator (PRNG), namely the quantum logistic map (QLM). Finally, we also assume that in addition to their photons states, Alice and Bob intentionally share a set of decoy states at each pulse with randomly selected intensity, and with the goal to detect the presence of the eavesdropper (Eve). Under these considerations, the secure key rate upper bound is evaluated applying the Gottesman-Lo-Lutkenhaus-Preskill's (GLLP) formula, for two different implementations, namely the non-decoy states and the infinite active decoy states based QKD. It is observed a significant improvement in the secure key size and the communication distance as well, compared to existing protocols, since we realize that under daylight, downlinks satellite conditions, a kindly selected light source, and good crystal's properties, the maximum communication distance can reach up to 70000 km. In addition, using the combined type-I and type-II SPDC photons source as our entangled photons pairs generator, significantly improved the photon mean number and render our protocol more robust against photon number division attack and against attenuation-induced atmospheric propagation. Furthermore, the protocol is more secure as compared to existing ones, given that any eavesdropper must crack simultaneously the chaotic system used as PRNG and the QKD system, before getting any useful information as regards to the measurement bases used by Alice and Bob, and thus the secure key.