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The reward functions are essential ingredients of consensus mechanism in Blockchain, which may bias the wealth distributions due to various incentives. Generally, constant reward function in proof of stakes (PoS) may incur the phenomenon of compounding, where rich get richer. That is, the wealth distribution is not so equitable in proof of stakes than that in proof of works (PoW).In the sequel, geometric reward function is proposed as an alternative choice to circumvent this problem. However, it's not so desirable since no parties have incentives to participate in the consensus mechanism, which does not capture the concern of incentive compatability. In this paper, we tailor a new bonus reward function by adding random salts to the geometric reward function. The new reward function is a tradeoff between equitablity and incentive compatibility. We conclude that the quitability of the new reward function is optimal compared with others. Beyond that, we present Gini coefficients to fine-evaluate euqitability of reward functions. We propose a new metric (aka. reward ratio) to quantify the level of incentive compability. Our simulation results show that the new reward function performs better than others in both incentive compatibility and anti-compounding.

The increasing availability of cloud computing allows more and more mobile devices to outsource expensive computations. Among these computations, bilinear pairing is very fundamental and frequently-used by many modern cryptographic protocols. Currently, the most efficient outsourcing algorithm of bilinear pairings requires about 5 point additions in G1 and G2 and 4 multiplications in GT under the one-malicious version of a two-untrusted-program assumption. And the result of the algorithm is checkable with a probability about 1/2. In this paper, we improve the stateof-the-art by proposing two new outsourcing algorithms for bilinear pairings. One is a more efficient outsourcing algorithm under the same assumption with the same checkability. The other is more flexible under a two-untrustedprogram assumption with improved checkability. Both algorithms are better suited to various applications where on-line computations are strictly limited due to the lack of available computing resources.