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The Stochastic Computational Element (SCE) uses streams of random bits (stochastic bits streams) to perform computation with conventional digital logic gates. It can guarantee reliable computation using unreliable devices. In stochastic computing, the linear Finite State Machine (FSM) can be used to implement some sophisticated functions, such as the exponentiation and tanh functions, more efficiently than combinational logic. However, a general approach about how to synthesize a linear FSM-based SCE for a target function has not been available. In this paper, we will introduce three properties of the linear FSM used in stochastic computing and demonstrate a general approach to synthesize a linear FSM-based SCE for a target function. Experimental results show that our approach produces circuits that are much more tolerant of soft errors than deterministic implementations, while the area-delay product of the circuits are less than that of deterministic implementations.