Design of High Area Efficiency Elliptic Curve Scalar Multiplier Based on Fast Modulo Reduction of Bit Reorganization
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摘要: 针对现有椭圆曲线密码标量乘法器难以兼顾灵活性和面积效率的问题，该文设计了一种基于比特重组快速模约简的高面积效率标量乘法器。首先，根据椭圆曲线标量乘的运算特点，设计了一种可实现乘法和模逆两种运算的硬件复用运算单元以提高硬件资源使用率，并采用Karatsuba-Ofman算法提高计算性能。其次，设计了基于比特重组的快速模约简算法，并实现了支持secp256k1, secp256r1和SCA-256(SM2标准推荐曲线)快速模约简计算的硬件架构。最后，对点加和倍点的模运算操作调度进行了优化，提高乘法与快速模约简的利用率，降低了标量乘计算所需的周期数量。所设计的标量乘法器在55 nm CMOS工艺下需要275 k个等效门，标量乘运算速度为48309次/s，面积时间积达到5.7。Abstract: To solve the problem that existing elliptic curve cryptography scalar multipliers are difficult to balance flexibility and area efficiency, a scalar multiplier with high area efficiency based on bit reorganization fast modular reduction is designed. Firstly, according to the operation characteristics of elliptic curve scalar multiplication, a hardware multiplexing operation unit that can realize two operations of multiplication and modular inversion is designed to improve the utilization rate of hardware resources, and the Karatsuba-Ofman algorithm is used to improve the calculation performance. Secondly, a fast modular reduction algorithm based on bit reorganization is designed, and a hardware architecture supporting secp256k1, secp256r1 and SCA-256 (SM2 standard recommended curve) fast modular reduction calculation is implemented. Finally, the scheduling of modular operations for point addition and point doubling is optimized to improve the utilization of multiplication and fast modular reduction, and reduce the number of cycles required for scalar multiplication calculations. The designed scalar multiplier requires 275 k equivalent gates in 55 nm CMOS technology, the scalar multiplication operation speed is 48309 times/s, and the area-time product reaches 5.7.
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算法1 NAF标量乘算法 输入：$ k,P $ 输出：$ Q = kP $ 步骤1：计算${\text{NAF} }\left( k \right) = \left\{ { {k_{m - 1} }, \cdots ,{k_0} } \right\}$ 步骤2：$ Q \leftarrow \infty $ 步骤3：$ {\text{for }}i \leftarrow m - 1{\text{ down to }}0{\text{ do}} $ $ Q \leftarrow 2Q $ $ {\text{if }}{k_i} = 1{\text{ then}} $ $ Q \leftarrow Q + P $ $ {\text{if }}{k_i} = - 1{\text{ then}} $ $ Q \leftarrow Q - P $ 步骤4：返回$ Q $ 
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算法2 改进的Karatsuba算法操作调度 输入：$ A={a}_{1}{2}^{128}+{a}_{0},\;B={b}_{1}{2}^{128}+{b}_{0} $ 输出：$ C = A \cdot B $ 1：$ s = {a_1}\cdot{b_1} $；$ v\left[ {128:0} \right] = {\text{ }}{a_0} + {a_1} $；
$ v\left[ {257:129} \right] = {b_0} + {b_1} $；2：$ r = {a_0}\cdot{b_0} $；$ {c_1} = r + s $；$u = {\text{～}}{c_1} + 1$； 3：$ v = v\left[ {128:0} \right] \cdot v\left[ {257:129} \right] $；$ {c_1} = v + u $；
$ {c_2} = \left\{ {s,r\left[ {255:128} \right]} \right\} + {c_1} $；4：返回$ C = \{ {c_2},r\left[ {127:0} \right]\} $
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算法3 secp256k1快速模约简算法 输入：$ C = {c_{127}}{2^{508}} + {c_{126}}{2^{504}} + \cdots + {c_1}{2^4} + {c_0} $, $0 \le C < {p^2}$ 输出：$ C\text{mod} p $ (1) $ {s_1} = ({c_{63}}, \cdots ,{c_0}) $； $ {s_2} = ({c_{127}}, \cdots ,{c_{64}}) $； $ {s_3} = ({c_{126}}, \cdots ,{c_{64}},{c_{127}}) $； $ {s_4} = (2'h0,{c_{124}}, \cdots ,{c_{64}},2'h0,{c_{127}},4'h0) $； $ {s_5} = ({c_{119}}, \cdots ,{c_{64}},{c_{127}}, \cdots ,{c_{120}}) $； $ {s_6} = (214'h0,{c_{127}},2'h0,{c_{127}}, \cdots ,{c_{120}},4'h0) $； $ {s_7} = (214'h0,{c_{127}}, \cdots ,{c_{120}},{c_{126}},{c_{126}},2'h0) $； $ {s_8} = (192'h0,{c_{127}}, \cdots ,{c_{120}},12'h0,{c_{127}},2'h0,$
${c_{127}},10'h0) $；$ {s_9} = (218'h0,{c_{126}},18'h0,{c_{126}},{c_{127}},8'h0) $； $ {s_{10}} = (220'h0,{c_{127}},18'h0,{c_{127}},10'h0) $； $ {s_{11}} = ({c_{125}},254'h0) $； $ {s_{12}} = (2'h0,{c_{125}}, \cdots ,{c_{64}},{c_{127}},2'h0) $； $ {s_{13}} = (218'h0,{c_{127}}, \cdots ,{c_{120}},6'h0) $； $ {s_{14}} = (218'h0,{c_{127}},18'h0,{c_{127}},2'h0,{c_{127}},6'h0) $； $ {s_{15}} = (240'h0,{c_{127}},{c_{126}},8'h0) $； $ {s_{16}} = ({c_{126}},254'h0) $； $ {{\text{z}}_1} = {s_1} + {s_2} + {s_3} + {s_4} + {s_5} + {s_6} + {s_7} + {s_8} + {s_9} + {s_{10}} + $
${s_{11}} - {s_{12}} - {s_{13}} - {s_{14}} - {s_{15}} - {s_{16}} $；$ = ({r_{65}}, \cdots ,{r_0}) $ (2) $ {t_1} = ({r_{63}}, \ldots ,{r_0}) $； $ {t_2} = \left( {216'h0,{\text{ }}{r_{65}},{r_{64}},20'h0,{\text{ }}{r_{65}},{r_{65}},{r_{64}}} \right) $； $ {t_3} = \left( {235'h0,{\text{ }}{r_{65}},{r_{64}},2'h0,{\text{ }}{r_{64}},4'h0} \right) $； $ {t_4} = \left( {242'h0,{\text{ }}{r_{65}},{r_{64}},6'h0} \right) $； $ {z_2} = {\text{ }}{t_1} + {t_2} + {t_3} - {t_4} $ (3)若$({z_1} \ge 0)$，返回$ \left( {{z_2}\text{mod} p} \right) $； 若$ ( - p < {z_1} < 0) $，返回$ \left( {{z_1} + p} \right) $； 若$( - 2p < {z_1} \le - p)$，返回$ \left( {{z_1} + 2p} \right) $； 若$( - 3p < {z_1} \le- 2p)$，返回$ \left( {{z_1} + 3p} \right) $；
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表 3 256位素数域下椭圆曲线标量乘法器的性能比较
方案 工艺(nm) 曲线 主频(MHz) 周期 面积(K·Gates/LUTs) 单次时间(μs) 吞吐量(次/s) AT 本文 55 secp256k1/r1, SCA-256 500 10366 275 20.7 48309 5.7 文献[22] 55 secp256r1 300 7955 306 27 37712 8.1 文献[9] 55 256 136 458200 189 1450 690 274 文献[10] 65 256 546.5 397300 447 730 1370 326.3 文献[11] 65 secp256r1 188 2300 3500 12.5 80000 43.7 本文 130 secp256k1/r1, SCA-256 243 10366 314 42.5 23529 13.3 文献[6] 130 SCA-256 250 14240 280 57 17544 15.9 文献[12] 130 256 150 610000 57.1 4070 246 232 文献[14] 130 SCA-256 214 45500 208 211 4740 44.2 文献[13] 180 secp256r1 185 36800 144.8 71 14084 28.9 本文 Virtex-7 secp256k1/r1, SCA-256 37 10366 47.4 279 3584 13.2 文献[24] Virtex-7 256 177.7 262.7 32.7 1471 679 48.5 文献[25] Virtex-7 256 72.9 215.9 96.9 2957 338 286.8
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