Many important algorithms for public-key cryptography rely on computation-intensive arithmetic operations like modular exponentiation on very long integers, typically in the range of 512 and 2048 bits. Modular exponentiation is generally realized through a sequence of modular multiplications and spends the majority of execution time in simple inner loops. Speeding up these performance-critical inner loop operations with custom instructions has, therefore, a significant impact on the total execution time of public-key cryptosystems. In this paper we analyze the performance of instruction set extensions for long integer arithmetic on a SPARC V8 processor. We discuss various implementation options and optimization opportunities for both modular multiplication and exponentiation. In particular, we introduce a partial loop unrolling (PLU) technique for modular multiplication which allows to achieve large performance gains at the cost of a moderate increase in code size, while maintaining the full flexibility of a "rolled-loop" implementation. In addition, we study window methods for modular exponentiation and analyze their impact on performance and memory requirements. Our experimental results, obtained with an FPGA prototype of the LEON-2 SPARC V8 core, show that a full 1024-bit modular exponentiation can be performed in about 12.5 路 106 clock cycles, which is a reasonable value for embedded devices like smart cards or sensor nodes.
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