**VLSI Implementation of Double-Base Scalar Multiplication on a Twisted Edwards Curve with an Efficiently Computable Endomorphism**

*Zhe Liu and Husen Wang and Johann Großschädl and Zhi Hu and Ingrid Verbauwhede*

**Abstract: **The verification of an ECDSA signature requires a double-base scalar multiplication, an operation of the form $k \cdot G + l \cdot Q$ where $G$ is a generator of a large elliptic curve group of prime order $n$, $Q$ is an arbitrary element of said group, and $k$, $l$ are two integers in the range of $[1, n-1]$. We introduce in this paper an area-optimized VLSI design of a Prime-Field Arithmetic Unit (PFAU) that can serve as a loosely-coupled or tightly-coupled hardware accelerator in a system-on-chip to speed up the execution of double-base scalar multiplication. Our design is optimized for twisted Edwards curves with an efficiently computable endomorphism that allows one to reduce the number of point doublings by some 50% compared to a conventional implementation. An example for such a special curve is $-x^2 + y^2 = 1 + x^2y^2$ over the 207-bit prime field $F_p$ with $p = 2^{207} - 5131$. The PFAU prototype we describe in this paper features a ($16 \times 16$)-bit multiplier and has an overall silicon area of 5821 gates when synthesized with a $0.13\mu$ standard-cell library. It can be clocked with a frequency of up to 50 MHz and is capable to perform a constant-time multiplication in the mentioned 207-bit prime field in only 198 clock cycles. A complete double-base scalar multiplication has an execution time of some 365k cycles and requires the pre-computation of 15 points. Our design supports many trade-offs between performance and RAM requirements, which is a highly desirable property for future Internet-of-Things (IoT) applications.

**Category / Keywords: **implementation / elliptic curve cryptosystem, digital signatures

**Date: **received 4 May 2015

**Contact author: **husen wang at esat kuleuven be

**Available format(s): **PDF | BibTeX Citation

**Version: **20150505:192144 (All versions of this report)

**Short URL: **ia.cr/2015/421

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