SECRYPT_2019_1_CR (2).rar_Image Encryption_加密解密

<html xmlns="http://www.w3.org/1999/xhtml"> <head> <meta charset="utf-8"> <meta name="generator" content="pdf2htmlEX"> <meta http-equiv="X-UA-Compatible" content="IE=edge,chrome=1"> <link rel="stylesheet" href="https://static.pudn.com/base/css/base.min.css"> <link rel="stylesheet" href="https://static.pudn.com/base/css/fancy.min.css"> <link rel="stylesheet" href="https://static.pudn.com/prod/directory_preview_static/6259288892dc900e62b787bd/raw.css"> <script src="https://static.pudn.com/base/js/compatibility.min.js"></script> <script src="https://static.pudn.com/base/js/pdf2htmlEX.min.js"></script> <script> try{ pdf2htmlEX.defaultViewer = new pdf2htmlEX.Viewer({}); }catch(e){} </script> <title></title> </head> <body> <div id="sidebar" style="display: none"> <div id="outline"> </div> </div> <div id="pf1" class="pf w0 h0" data-page-no="1"><div class="pc pc1 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6259288892dc900e62b787bd/bg1.jpg"><div class="t m0 x1 h2 y1 ff1 fs0 fc0 sc0 ls0 ws0">High-performance Pipelined FPGA Implementation of the Elliptic Curve</div><div class="t m0 x2 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">Cryptography over GF (2</div><div class="t m0 x3 h3 y3 ff3 fs1 fc0 sc0 ls0 ws0">n</div><div class="t m0 x4 h2 y2 ff1 fs0 fc0 sc0 ls0 ws0">)</div><div class="t m0 x5 h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">Salah Harb</div><div class="t m0 x6 h5 y5 ff2 fs3 fc0 sc0 ls0 ws0">a</div><div class="t m0 x7 h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">, M. Omair Ahmad</div><div class="t m0 x8 h5 y5 ff2 fs3 fc0 sc0 ls0 ws0">b</div><div class="t m0 x9 h4 y4 ff2 fs2 fc0 sc0 ls0 ws0">and M. N. S. Swamy</div><div class="t m0 xa h5 y5 ff2 fs3 fc0 sc0 ls0 ws0">c</div><div class="t m0 xb h6 y6 ff3 fs4 fc0 sc0 ls0 ws0">Electrical and Computer Engineering Department, Concordia University, 1440 De maisonnueve, Montreal, Canada</div><div class="t m0 xc h7 y7 ff4 fs4 fc0 sc0 ls0 ws0">{s rb, omair, swamy}@ece.concordia.ca</div><div class="t m0 x1 h8 y8 ff2 fs4 fc0 sc0 ls0 ws0">Keywords:</div><div class="t m0 xd h8 y9 ff2 fs4 fc0 sc0 ls0 ws0">Cryptography, Elliptic Curve Cryptography, FPGA, Pipelining Architecture, Finite Field Operations, Field</div><div class="t m0 xd h8 ya ff2 fs4 fc0 sc0 ls0 ws0">Multiplications, Projective Coordination, Efﬁciency.</div><div class="t m0 x1 h8 yb ff2 fs4 fc0 sc0 ls0 ws0">Abstract:</div><div class="t m0 xd h8 yc ff2 fs4 fc0 sc0 ls0 ws0">In this paper, a high-performance area-efﬁcient hardware design for the Elliptic Curve Cryptography (ECC)</div><div class="t m0 xd h8 yd ff2 fs4 fc0 sc0 ls0 ws0">is presented, targeting the area-constrained high-bandwidth embedded applications. The high-speed design</div><div class="t m0 xd h8 ye ff2 fs4 fc0 sc0 ls0 ws0">is implemented using pipelining architecture. The applied architecture is performed using n-bit data path of</div><div class="t m0 xd h8 yf ff2 fs4 fc0 sc0 ls0 ws0">the ﬁnite ﬁeld GF(2</div><div class="t m0 xe h9 y10 ff3 fs5 fc0 sc0 ls0 ws0">n</div><div class="t m0 xf h8 yf ff2 fs4 fc0 sc0 ls0 ws0">). For the ﬁnite ﬁeld operations, the implementation in the ECC uses the bit-parallel</div><div class="t m0 xd h8 y11 ff2 fs4 fc0 sc0 ls0 ws0">recursive Karatsuba-Ofman algorithm for multiplication and Itoh-Tsuji for inversion. A modiﬁed efﬁcient</div><div class="t m0 xd h8 y12 ff2 fs4 fc0 sc0 ls0 ws0">montgomery ladder algorithm is utilized for the scalar multiplication of a point. The pipelined registers are</div><div class="t m0 xd h8 y13 ff2 fs4 fc0 sc0 ls0 ws0">inserted in ideal locations, where balanced-execution paths among computing components are guaranteed. A</div><div class="t m0 xd h8 y14 ff2 fs4 fc0 sc0 ls0 ws0">Memory-less ﬁnite state machine model is developed to control the instructions of computing the ﬁnite ﬁeld</div><div class="t m0 xd h8 y15 ff2 fs4 fc0 sc0 ls0 ws0">operations efﬁciently. The high-performance design has been implemented using Xilinx Virtex, Kintex and</div><div class="t m0 xd h8 y16 ff2 fs4 fc0 sc0 ls0 ws0">Artix FPGA devices. It can perform a single scalar multiplication in 226 clock cycles within 0.63µs using</div><div class="t m0 xd h8 y17 ff2 fs4 fc0 sc0 ls0 ws0">2780 slices and 360Mhz working frequency on Virtex-7 over GF (2</div><div class="t m0 x10 ha y18 ff2 fs5 fc0 sc0 ls0 ws0">163</div><div class="t m0 x11 h8 y17 ff2 fs4 fc0 sc0 ls0 ws0">). In GF (2</div><div class="t m0 x12 ha y18 ff2 fs5 fc0 sc0 ls0 ws0">233</div><div class="t m0 x13 h8 y17 ff2 fs4 fc0 sc0 ls0 ws0">) and GF (2</div><div class="t m0 x14 ha y18 ff2 fs5 fc0 sc0 ls0 ws0">571</div><div class="t m0 x15 h8 y17 ff2 fs4 fc0 sc0 ls0 ws0">), a scalar</div><div class="t m0 xd h8 y19 ff2 fs4 fc0 sc0 ls0 ws0">multiplication can be computed in 327 and 674 clock cycles within 1.05µs and 2.32µs, respectively. Comparing</div><div class="t m0 xd h8 y1a ff2 fs4 fc0 sc0 ls0 ws0">with previous works, our design requires less number of clock cycles, and operates using less FPGA resources</div><div class="t m0 xd h8 y1b ff2 fs4 fc0 sc0 ls0 ws0">with competitive high working frequencies. Therefore, the proposed design is well suited in the resources-</div><div class="t m0 xd h8 y1c ff2 fs4 fc0 sc0 ls0 ws0">constrained real time cryptosystems like those in online banking services, wearable smart devices and network</div><div class="t m0 xd h8 y1d ff2 fs4 fc0 sc0 ls0 ws0">attached storages.</div><div class="t m0 x1 hb y1e ff1 fs6 fc0 sc0 ls0 ws0">1 INTRODUCTION</div><div class="t m0 x1 hc y1f ff2 fs7 fc0 sc0 ls0 ws0">Elliptic curve cryptosystem (ECC) is a public-key</div><div class="t m0 x1 hc y20 ff2 fs7 fc0 sc0 ls0 ws0">cryptography, which was ﬁrst proposed by Neal</div><div class="t m0 x1 hc y21 ff2 fs7 fc0 sc0 ls0 ws0">Koblitz and Victor Miller in the 1980s (Kocher et al.,</div><div class="t m0 x1 hc y22 ff2 fs7 fc0 sc0 ls0 ws0">1999) , (Miller, 1985). Since then, many studies have</div><div class="t m0 x1 hc y23 ff2 fs7 fc0 sc0 ls0 ws0">been conducted to explore its security levels against</div><div class="t m0 x1 hc y24 ff2 fs7 fc0 sc0 ls0 ws0">other public-key cryptosystems such as El-Gamal,</div><div class="t m0 x1 hc y25 ff2 fs7 fc0 sc0 ls0 ws0">RSA and Digital Signature Algorithm (DSA) (ElGa-</div><div class="t m0 x1 hc y26 ff2 fs7 fc0 sc0 ls0 ws0">mal, 1985), (Rivest et al., 1978), which are based on</div><div class="t m0 x1 hc y27 ff2 fs7 fc0 sc0 ls0 ws0">either the integer factorization or discrete logarithm</div><div class="t m0 x1 hc y28 ff2 fs7 fc0 sc0 ls0 ws0">problems (McGrew et al., 2011). Equivalent secu-</div><div class="t m0 x1 hc y29 ff2 fs7 fc0 sc0 ls0 ws0">rity levels with smaller sizes of keys, ease to imple-</div><div class="t m0 x1 hc y2a ff2 fs7 fc0 sc0 ls0 ws0">ment, and resource savings, are reasons that give the</div><div class="t m0 x1 hc y2b ff2 fs7 fc0 sc0 ls0 ws0">ECC to be very appealing and more dominant be-</div><div class="t m0 x1 hc y2c ff2 fs7 fc0 sc0 ls0 ws0">tween the hardware reconﬁgurable implementations.</div><div class="t m0 x1 hc y2d ff2 fs7 fc0 sc0 ls0 ws0">Moreover, ECC is well suited to be implemented in</div><div class="t m0 x1 hc y2e ff2 fs7 fc0 sc0 ls0 ws0">such resource-constrained embedded systems, since it</div><div class="t m0 x16 ha y2f ff2 fs5 fc0 sc0 ls0 ws0">a</div><div class="t m0 x17 h8 y30 ff2 fs4 fc0 sc0 ls0 ws0">https://orcid.org/0000-0002-5975-6537</div><div class="t m0 x16 ha y31 ff2 fs5 fc0 sc0 ls0 ws0">b</div><div class="t m0 x17 h8 y32 ff2 fs4 fc0 sc0 ls0 ws0">https://orcid.org/0000-0002-2924-6659</div><div class="t m0 x16 ha y33 ff2 fs5 fc0 sc0 ls0 ws0">c</div><div class="t m0 x17 h8 y34 ff2 fs4 fc0 sc0 ls0 ws0">https://orcid.org/0000-0002-3989-5476</div><div class="t m0 x18 hc y1e ff2 fs7 fc0 sc0 ls0 ws0">provides same security levels as in RSA using small</div><div class="t m0 x18 hc y35 ff2 fs7 fc0 sc0 ls0 ws0">keys. ECC has been standardized by IEEE and the</div><div class="t m0 x18 hc y36 ff2 fs7 fc0 sc0 ls0 ws0">National Institute of Standard and Technology (NIST)</div><div class="t m0 x18 hc y37 ff2 fs7 fc0 sc0 ls0 ws0">as a scheme in digital signature and key agreement</div><div class="t m0 x18 hc y38 ff2 fs7 fc0 sc0 ls0 ws0">protocols (for Standardization (ISO), 2000).</div><div class="t m0 x19 hc y39 ff2 fs7 fc0 sc0 ls0 ws0">Generally, most of cryptographic algorithms are</div><div class="t m0 x18 hc y3a ff2 fs7 fc0 sc0 ls0 ws0">implemented in software platforms. Performing an</div><div class="t m0 x18 hc y3b ff2 fs7 fc0 sc0 ls0 ws0">algorithm on a general purpose processor (e.g. CPU)</div><div class="t m0 x18 hc y3c ff2 fs7 fc0 sc0 ls0 ws0">will require most of its resources to compute results</div><div class="t m0 x18 hc y3d ff2 fs7 fc0 sc0 ls0 ws0">of intensive operations because of the large operands</div><div class="t m0 x18 hc y3e ff2 fs7 fc0 sc0 ls0 ws0">used in these very accurate computations. More-</div><div class="t m0 x18 hc y3f ff2 fs7 fc0 sc0 ls0 ws0">over, CPU is not suitable in performing such these</div><div class="t m0 x18 hc y40 ff2 fs7 fc0 sc0 ls0 ws0">algorithms that having the parallel architecture in na-</div><div class="t m0 x18 hc y41 ff2 fs7 fc0 sc0 ls0 ws0">ture. These issues prove that software implementa-</div><div class="t m0 x18 hc y42 ff2 fs7 fc0 sc0 ls0 ws0">tion of encryption algorithms does not provide the</div><div class="t m0 x18 hc y43 ff2 fs7 fc0 sc0 ls0 ws0">required performance. Due to the diversity in the</div><div class="t m0 x18 hc y44 ff2 fs7 fc0 sc0 ls0 ws0">applications, the trade-off between area, speed and</div><div class="t m0 x18 hc y45 ff2 fs7 fc0 sc0 ls0 ws0">power is required. Some applications, such as RFID</div><div class="t m0 x18 hc y46 ff2 fs7 fc0 sc0 ls0 ws0">cards, nodes of wireless sensor networks and cell</div><div class="t m0 x18 hc y47 ff2 fs7 fc0 sc0 ls0 ws0">phones, need a small area and power. Other applica-</div><div class="t m0 x18 hc y48 ff2 fs7 fc0 sc0 ls0 ws0">tions, such as web servers, large bandwidth networks</div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div> </body> </html>

<div id="pf2" class="pf w0 h0" data-page-no="2"><div class="pc pc2 w0 h0"><img class="bi x0 y0 w1 h1" alt="" src="https://static.pudn.com/prod/directory_preview_static/6259288892dc900e62b787bd/bg2.jpg"><div class="t m0 x1 hc y49 ff2 fs7 fc0 sc0 ls0 ws0">and satellite broadcast require very high throughputs.</div><div class="t m0 x1 hc y4a ff2 fs7 fc0 sc0 ls0 ws0">To cover the issues of software implementation and</div><div class="t m0 x1 hc y4b ff2 fs7 fc0 sc0 ls0 ws0">meet trade-offs in numerous applications, the hard-</div><div class="t m0 x1 hc y4c ff2 fs7 fc0 sc0 ls0 ws0">ware platforms have been utilized for implementing</div><div class="t m0 x1 hc y4d ff2 fs7 fc0 sc0 ls0 ws0">the cryptographic algorithms, where high efﬁciency</div><div class="t m0 x1 hc y4e ff2 fs7 fc0 sc0 ls0 ws0">to perform tasks is achieved in different applications.</div><div class="t m0 x1a hc y4f ff2 fs7 fc0 sc0 ls0 ws0">Field Programmable Gate Array (FPGA) is one</div><div class="t m0 x1 hc y50 ff2 fs7 fc0 sc0 ls0 ws0">of the preferable reconﬁgurable hardware platforms</div><div class="t m0 x1 hc y51 ff2 fs7 fc0 sc0 ls0 ws0">(Xilinx, 2018a) which offers ﬂexible and more cus-</div><div class="t m0 x1 hc y52 ff2 fs7 fc0 sc0 ls0 ws0">tomizable methods for performing and evaluating dif-</div><div class="t m0 x1 hc y53 ff2 fs7 fc0 sc0 ls0 ws0">ferent hardware implementations. Because of this</div><div class="t m0 x1 hc y54 ff2 fs7 fc0 sc0 ls0 ws0">fact and since FPGAs have been employed by most</div><div class="t m0 x1 hc y55 ff2 fs7 fc0 sc0 ls0 ws0">of the previous hardware implementations to evalu-</div><div class="t m0 x1 hc y56 ff2 fs7 fc0 sc0 ls0 ws0">ate their performances; the presented ECC hardware</div><div class="t m0 x1 hc y57 ff2 fs7 fc0 sc0 ls0 ws0">implementation in this paper have been performed</div><div class="t m0 x1 hc y58 ff2 fs7 fc0 sc0 ls0 ws0">using Xilinx FPGA devices (Xilinx, 2018b). Scalar</div><div class="t m0 x1 hc y59 ff2 fs7 fc0 sc0 ls0 ws0">point multiplication (SPM) is the main point opera-</div><div class="t m0 x1 hc y5a ff2 fs7 fc0 sc0 ls0 ws0">tion in ECC cryptosystems or protocols such as Ellip-</div><div class="t m0 x1 hc y5b ff2 fs7 fc0 sc0 ls0 ws0">tic Curve Difﬁe-Hellman (ECDH) (Difﬁe and Hell-</div><div class="t m0 x1 hc y5c ff2 fs7 fc0 sc0 ls0 ws0">man, 1976) for key agreements and Elliptic Curve</div><div class="t m0 x1 hc y5d ff2 fs7 fc0 sc0 ls0 ws0">Digital Signature (ECDS) for digital signatures.</div><div class="t m0 x1a hc y5e ff2 fs7 fc0 sc0 ls0 ws0">SPM can be implemented over many ﬁnite ﬁelds</div><div class="t m0 x1 hc y5f ff2 fs7 fc0 sc0 ls0 ws0">under either prime or polynomial ﬁelds. Finite ﬁelds</div><div class="t m0 x1 hc y60 ff2 fs7 fc0 sc0 ls0 ws0">named also Galois Fields (GF), where GF (p) is a</div><div class="t m0 x1 hc y61 ff2 fs7 fc0 sc0 ls0 ws0">prime ﬁeld and GF (2</div><div class="t m0 x1b hd y62 ff3 fs8 fc0 sc0 ls0 ws0">n</div><div class="t m0 x1c hc y61 ff2 fs7 fc0 sc0 ls0 ws0">) is the polynomial ﬁeld. SPM</div><div class="t m0 x1 hc y63 ff2 fs7 fc0 sc0 ls0 ws0">has two point operations, doubling and adding points.</div><div class="t m0 x1 hc y64 ff2 fs7 fc0 sc0 ls0 ws0">Each operation consists ﬁnite ﬁeld operations such</div><div class="t m0 x1 hc y65 ff2 fs7 fc0 sc0 ls0 ws0">as square, addition, multiplier. Figure 1 presents</div><div class="t m0 x1 hc y66 ff2 fs7 fc0 sc0 ls0 ws0">the hierarchical implementation of the ECC proto-</div><div class="t m0 x1 hc y67 ff2 fs7 fc0 sc0 ls0 ws0">col. Polynomial ﬁelds are more suited and efﬁcient to</div><div class="t m0 x1 hc y68 ff2 fs7 fc0 sc0 ls0 ws0">implement on a customizable platform such FPGAs</div><div class="t m0 x1 hc y69 ff2 fs7 fc0 sc0 ls0 ws0">(Wenger and Hutter, 2011a).</div><div class="c x1 y6a w2 he"><div class="t m0 x1d hf y6b ff5 fs9 fc0 sc0 ls0 ws0">ECC</div><div class="t m0 x1e hf y6c ff5 fs9 fc0 sc0 ls0 ws0">Protocols</div><div class="t m0 x1f hf y6d ff5 fs9 fc0 sc0 ls0 ws0">Scalar Point </div><div class="t m0 xb hf y6e ff5 fs9 fc0 sc0 ls0 ws0">Multiplier</div><div class="t m0 x20 hf y6f ff5 fs9 fc0 sc0 ls0 ws0">Point Doubling Point Addition </div><div class="t m0 x21 hf y70 ff5 fs9 fc0 sc0 ls0 ws0">Square Multiplication Inversion</div></div><div class="t m0 x22 h8 y71 ff2 fs4 fc0 sc0 ls0 ws0">Figure 1: ECC Cryptosystem Hierarchy.</div><div class="t m0 x1a hc y72 ff2 fs7 fc0 sc0 ls0 ws0">To gain high performance in today’s high loaded</div><div class="t m0 x1 hc y73 ff2 fs7 fc0 sc0 ls0 ws0">communication networks, utilization of hardware ac-</div><div class="t m0 x1 hc y74 ff2 fs7 fc0 sc0 ls0 ws0">celerators for physical security has created a great de-</div><div class="t m0 x1 hc y75 ff2 fs7 fc0 sc0 ls0 ws0">mand for efﬁcient and high-speed implementations of</div><div class="t m0 x1 hc y76 ff2 fs7 fc0 sc0 ls0 ws0">ECC. Based on this fact, many FPGA implementa-</div><div class="t m0 x1 hc y34 ff2 fs7 fc0 sc0 ls0 ws0">tions of the ECC have been published in the litera-</div><div class="t m0 x18 hc y49 ff2 fs7 fc0 sc0 ls0 ws0">ture, where various ranges of latencies and number of</div><div class="t m0 x18 hc y4a ff2 fs7 fc0 sc0 ls0 ws0">clock cycles are achieved targeting applications that</div><div class="t m0 x18 hc y4b ff2 fs7 fc0 sc0 ls0 ws0">require high/low throughputs. Providing high perfor-</div><div class="t m0 x18 hc y4c ff2 fs7 fc0 sc0 ls0 ws0">mance as well as utilizing efﬁcient area, is a challenge</div><div class="t m0 x18 hc y4d ff2 fs7 fc0 sc0 ls0 ws0">to achieve it in FPGA’s ECC implementations.</div><div class="t m0 x19 hc y4e ff2 fs7 fc0 sc0 ls0 ws0">In this paper, a high-speed area-efﬁcient Xilinx</div><div class="t m0 x18 hc y4f ff2 fs7 fc0 sc0 ls0 ws0">FPGA implementation of the ECC over GF (2</div><div class="t m0 x23 hd y77 ff3 fs8 fc0 sc0 ls0 ws0">n</div><div class="t m0 x24 hc y4f ff2 fs7 fc0 sc0 ls0 ws0">) using</div><div class="t m0 x18 hc y50 ff2 fs7 fc0 sc0 ls0 ws0">the pipelining architecture is proposed. The main tar-</div><div class="t m0 x18 hc y51 ff2 fs7 fc0 sc0 ls0 ws0">get of our work is to develop high performance design</div><div class="t m0 x18 hc y52 ff2 fs7 fc0 sc0 ls0 ws0">that targets systems that have constrained resources</div><div class="t m0 x18 hc y53 ff2 fs7 fc0 sc0 ls0 ws0">such as wearable smart devices, processing engines</div><div class="t m0 x18 hc y54 ff2 fs7 fc0 sc0 ls0 ws0">in image steganographic systems (Dalal and Juneja,</div><div class="t m0 x18 hc y55 ff2 fs7 fc0 sc0 ls0 ws0">2018), (Amirtharajan, 2014) and Internet of Things</div><div class="t m0 x18 hc y56 ff2 fs7 fc0 sc0 ls0 ws0">(IoTs) network processors. This paper is organized as</div><div class="t m0 x18 hc y57 ff2 fs7 fc0 sc0 ls0 ws0">follow: section 2 describes the arithmetic operations</div><div class="t m0 x18 hc y58 ff2 fs7 fc0 sc0 ls0 ws0">of the ECC; section 3 describes our high-performance</div><div class="t m0 x18 hc y59 ff2 fs7 fc0 sc0 ls0 ws0">hardware implementation core for ECC over GF (2</div><div class="t m0 x25 hd y78 ff3 fs8 fc0 sc0 ls0 ws0">n</div><div class="t m0 x26 hc y59 ff2 fs7 fc0 sc0 ls0 ws0">);</div><div class="t m0 x18 hc y5a ff2 fs7 fc0 sc0 ls0 ws0">section 4 shows the results and comparisons; and ﬁ-</div><div class="t m0 x18 hc y5b ff2 fs7 fc0 sc0 ls0 ws0">nally, section 5 concludes this paper.</div><div class="t m0 x18 hb y79 ff1 fs6 fc0 sc0 ls0 ws0">2 ELLIPTIC CURVE</div><div class="t m0 x27 hb y7a ff1 fs6 fc0 sc0 ls0 ws0">CRYPTOGRAPHY (ECC)</div><div class="t m0 x18 hc y7b ff2 fs7 fc0 sc0 ls0 ws0">Elliptic Curves (ECs) are formulated by the so called</div><div class="t m0 x18 hc y7c ff2 fs7 fc0 sc0 ls0 ws0">Weiestrass equations, which can be performed over</div><div class="t m0 x18 hc y7d ff2 fs7 fc0 sc0 ls0 ws0">by normal or polynomial basis. In this paper, we work</div><div class="t m0 x18 hc y7e ff2 fs7 fc0 sc0 ls0 ws0">on the polynomial basis in GF (2</div><div class="t m0 x28 hd y7f ff3 fs8 fc0 sc0 ls0 ws0">n</div><div class="t m0 x29 hc y7e ff2 fs7 fc0 sc0 ls0 ws0">) for its efﬁciency on</div><div class="t m0 x18 hc y80 ff2 fs7 fc0 sc0 ls0 ws0">the hardware platforms (Wenger and Hutter, 2011a).</div><div class="t m0 x18 hc y81 ff2 fs7 fc0 sc0 ls0 ws0">Equation 1 represents the general form of the none-</div><div class="t m0 x18 hc y82 ff2 fs7 fc0 sc0 ls0 ws0">singular curve over GF (2</div><div class="t m0 x2a hd y83 ff3 fs8 fc0 sc0 ls0 ws0">n</div><div class="t m0 x2b hc y82 ff2 fs7 fc0 sc0 ls0 ws0">) (Hankerson et al., 2006).</div><div class="t m0 x3 h10 y84 ff3 fs7 fc0 sc0 ls0 ws0">y</div><div class="t m0 x2c h11 y85 ff2 fs8 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2d h12 y84 ff6 fs7 fc0 sc0 ls0 ws0">+ xy = x</div><div class="t m0 x2b h11 y85 ff2 fs8 fc0 sc0 ls0 ws0">3</div><div class="t m0 x2e h12 y84 ff6 fs7 fc0 sc0 ls0 ws0">+ ax</div><div class="t m0 x2f h11 y85 ff2 fs8 fc0 sc0 ls0 ws0">2</div><div class="t m0 x30 h12 y84 ff6 fs7 fc0 sc0 ls0 ws0">+ b (1)</div><div class="t m0 x19 h12 y86 ff2 fs7 fc0 sc0 ls0 ws0">where a, b ∈ GF (2</div><div class="t m0 x31 hd y87 ff3 fs8 fc0 sc0 ls0 ws0">n</div><div class="t m0 x32 h12 y86 ff2 fs7 fc0 sc0 ls0 ws0">) and b 6= 0. A set of afﬁne</div><div class="t m0 x18 hc y88 ff2 fs7 fc0 sc0 ls0 ws0">points (x, y) satisfying the curve forms a group (Han-</div><div class="t m0 x18 hc y89 ff2 fs7 fc0 sc0 ls0 ws0">kerson et al., 2006) with an identity point of that</div><div class="t m0 x18 hc y8a ff2 fs7 fc0 sc0 ls0 ws0">group. There are two fundamental elliptic curve oper-</div><div class="t m0 x18 hc y8b ff2 fs7 fc0 sc0 ls0 ws0">ations, doubling and adding points. Doubling point</div><div class="t m0 x18 hc y8c ff2 fs7 fc0 sc0 ls0 ws0">is denoted as P</div><div class="t m0 x33 h11 y8d ff2 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m0 x34 hc y8c ff2 fs7 fc0 sc0 ls0 ws0">=2P</div><div class="t m0 x35 h11 y8d ff2 fs8 fc0 sc0 ls0 ws0">0</div><div class="t m0 x36 hc y8c ff2 fs7 fc0 sc0 ls0 ws0">, where P</div><div class="t m0 x37 h11 y8d ff2 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m0 x38 hc y8c ff2 fs7 fc0 sc0 ls0 ws0">is (x</div><div class="t m0 x39 h11 y8d ff2 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3a hc y8c ff2 fs7 fc0 sc0 ls0 ws0">, y</div><div class="t m0 x3b h11 y8d ff2 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m0 x3c hc y8c ff2 fs7 fc0 sc0 ls0 ws0">) and P</div><div class="t m0 x3d h11 y8d ff2 fs8 fc0 sc0 ls0 ws0">0</div><div class="t m0 x18 hc y8e ff2 fs7 fc0 sc0 ls0 ws0">is (x</div><div class="t m0 x9 h11 y8f ff2 fs8 fc0 sc0 ls0 ws0">0</div><div class="t m0 x3e hc y8e ff2 fs7 fc0 sc0 ls0 ws0">, y</div><div class="t m0 x3f h11 y8f ff2 fs8 fc0 sc0 ls0 ws0">0</div><div class="t m0 x40 hc y8e ff2 fs7 fc0 sc0 ls0 ws0">) while point addition is denoted as P</div><div class="t m0 x41 h11 y8f ff2 fs8 fc0 sc0 ls0 ws0">2</div><div class="t m0 x26 hc y8e ff2 fs7 fc0 sc0 ls0 ws0">=</div><div class="t m0 x18 hc y90 ff2 fs7 fc0 sc0 ls0 ws0">P</div><div class="t m0 x42 h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">0</div><div class="t m0 x43 hc y90 ff2 fs7 fc0 sc0 ls0 ws0">+P</div><div class="t m0 x44 h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m0 x45 hc y90 ff2 fs7 fc0 sc0 ls0 ws0">, where P</div><div class="t m0 x10 h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">2</div><div class="t m0 x46 hc y90 ff2 fs7 fc0 sc0 ls0 ws0">is (x</div><div class="t m0 x47 h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">2</div><div class="t m0 x48 hc y90 ff2 fs7 fc0 sc0 ls0 ws0">, y</div><div class="t m0 x2a h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">2</div><div class="t m0 x2b hc y90 ff2 fs7 fc0 sc0 ls0 ws0">), and P</div><div class="t m0 x38 h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">1</div><div class="t m0 x49 h12 y90 ff4 fs7 fc0 sc0 ls0 ws0">6= P</div><div class="t m0 x3a h11 y91 ff2 fs8 fc0 sc0 ls0 ws0">0</div><div class="t m0 x4a hc y90 ff2 fs7 fc0 sc0 ls0 ws0">. All points</div><div class="t m0 x18 hc y92 ff2 fs7 fc0 sc0 ls0 ws0">in the selected curve are represented in afﬁne coor-</div><div class="t m0 x18 hc y93 ff2 fs7 fc0 sc0 ls0 ws0">dinates. Finite ﬁelds operations are involved in the</div><div class="t m0 x18 hc y94 ff2 fs7 fc0 sc0 ls0 ws0">ECC point operations such as addition, square, mul-</div><div class="t m0 x18 hc y95 ff2 fs7 fc0 sc0 ls0 ws0">tiplication and inversion. Dealing with afﬁne coor-</div><div class="t m0 x18 hc y96 ff2 fs7 fc0 sc0 ls0 ws0">dinates requires an inversion ﬁeld operation. Due to</div><div class="t m0 x18 hc y97 ff2 fs7 fc0 sc0 ls0 ws0">the complexity in the inversion operation, a projective</div><div class="t m0 x18 hc y72 ff2 fs7 fc0 sc0 ls0 ws0">coordinate is utilized to avoid it by mapping points</div><div class="t m0 x18 hc y73 ff2 fs7 fc0 sc0 ls0 ws0">in afﬁne (x, y) to be represented in (X, Y, Z) form.</div><div class="t m0 x18 hc y74 ff2 fs7 fc0 sc0 ls0 ws0">Scalar Point Multiplication (SPM) is the main impor-</div><div class="t m0 x18 hc y75 ff2 fs7 fc0 sc0 ls0 ws0">tant operation that dominants the ECC-based cryp-</div><div class="t m0 x18 hc y76 ff2 fs7 fc0 sc0 ls0 ws0">tosystems. SPM is process of adding a point k times,</div><div class="t m0 x18 hc y34 ff2 fs7 fc0 sc0 ls0 ws0">where k is a positive integer and P is a point on a</div></div><div class="pi" data-data='{"ctm":[1.611850,0.000000,0.000000,1.611850,0.000000,0.000000]}'></div></div>