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Abstract. Elliptic curve cryptography (ECC) is preferred for highspeed applications due to the lower computational complexity compared with other public-key cryptographic schemes. As the basic arithmetic, the modular multiplication is the most time-consuming operation in publickey cryptosystems. The existing high-radix Montgomery multipliers performed a single Montgomery multiplication either in approximately 2n clock cycles, or approximately n cycles but with a very low frequency, where n is the number of words. In this paper, we first design a novel Montgomery multiplier by combining a quotient pipelining Montgomery multiplication algorithm with a parallel array design. The parallel design with one-way carry propagation can determine the quotients in one clock cycle, thus one Montgomery multiplication can be completed in approximately n clock cycles. Meanwhile, by the quotient pipelining technique applied in digital signal processing (DSP) blocks, our multiplier works in a high frequency. We also implement an ECC processor for generic curves over GF(p) using the novel multiplier on FPGAs. To the best of our knowledge, our processor is the fastest among the existing ECC implementations over GF(p).

True random number generators (TRNGs) are crucial to the implementations of cryptographic algorithms and protocols. The quality of randomness directly influences the security of cryptographic systems. Oscillator-based sampling is popular in the design of TRNGs due to its nice properties of elegant structure and high speed. However, the credibility of randomness generated from high-speed oscillator-based TRNGs, especially ring oscillator-based (RO-based) ones, is still in controversy. This is mainly because pseudo-randomness is hardly distinguished from true randomness and RO-based TRNGs are susceptible to external perturbations. In this paper, we present a stochastic model to evaluate the entropy of oscillator-based TRNGs, and then deduce the requirement of design parameters (including the sampling interval) for sufficient entropy per random bit, i.e., to ensure true randomness. Furthermore, we design a jitter measuring circuit to verify the theory, and the theoretical results are confirmed by both the simulation and practical experiments. Finally, we apply the stochastic model to analyze the effect of deterministic perturbations, and demonstrate that the randomness of RO-based TRNGs (under deterministic perturbations) can be overestimated and predicting the "random" bits could be possible.