LDPC codes, LT codes, and digital fountain techniques have received significant attention from both academics and industry in the past few years. By employing the underlying ideas of efficient Belief Propagation (BP) decoding process in LDPC and LT codes, this paper designs the BP-XOR codes and use them to design three classes of secret sharing schemes called BP-XOR secret sharing schemes, pseudo-BP-XOR secret sharing schemes, and LDPC secret sharing schemes. By establishing the equivalence between the edge-colored graph model and degree-two BP-XOR secret sharing schemes, we are able to design novel perfect and ideal $2$-out-of- $n$ BP-XOR secret sharing schemes. By employing techniques from array code design, we are also able to design other $(n,k)$ threshold LDPC secret sharing schemes. In the efficient (pseudo) BP-XOR/LDPC secret sharing schemes that we will construct, only linear number of XOR (exclusive-or) operations on binary strings are required for both secret distribution phase and secret reconstruction phase. For a comparison, we should note that Shamir secret sharing schemes require $O(n\,\log\, n)$ field operations for the secret distribution phase and $O(n^2)$ field operations for the secret reconstruction phase. Furthermore, our schemes achieve the optimal update complexity for secret sharing schemes. By update complexity for a secret sharing scheme, we mean the average number of bits in the participant's shares that needs to be revised when certain bit of the master secret is changed. The extremely efficient secret sharing schemes discussed in this paper could be used for massive data storage in cloud environments achieving privacy and reliability without employing encryption techniques.