ECC.zip_ECCC实现_椭圆曲线密码学入门示例资源-CSDN下载

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pdb：2个

ncb：1个

idb：1个

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134 浏览量 2022-09-23 16:46:42 上传评论收藏 168KB ZIP 举报

在本文中，我们将深入探讨如何使用C语言实现椭圆曲线密码学（ECC，Elliptic Curve Cryptography）中的点乘运算。ECC是一种公钥加密技术，它基于椭圆曲线理论，提供与RSA等传统公钥算法相当的安全性，但使用更小的密钥尺寸，从而在效率和资源需求上具有优势。标题"**ECC.zip_ECC C实现**"指出了这是一个关于使用C语言实现椭圆曲线加密算法的项目。具体来说，是针对椭圆曲线E23，其方程为**y^2 ≡ x^3 + 16x + 10 (mod 23)**，以及曲线上的一点G=(5,10)，我们需要计算点G的所有倍点。理解椭圆曲线的基本概念是必要的。椭圆曲线是由方程y^2 = x^3 + ax + b (mod p)定义的，其中p是素数，a和b是整数。在本例中，p=23，a=16，b=10。点G=(5,10)是这个曲线上的一点，这意味着10^2 ≡ 5^3 + 16*5 + 10 (mod 23)。 ECC中的点乘运算，即求解点G的n倍点，可以使用双线性对数算法（如Double-and-Add或Montgomery Ladder）进行。对于点G的倍数，我们首先定义一个函数`double_point()`来计算一个点的二倍，然后使用`add_point()`函数将点自身加到二倍点上，重复这个过程n次，直到得到G的n倍点。下面是一些基本的C语言实现步骤： 1. 定义数据结构：为了存储椭圆曲线上的点，我们可以创建一个结构体，包含x和y坐标，以及一个标志来表示无穷远点。 ```c typedef struct { int x; int y; int is_infinity; // 0表示有限点，1表示无穷远点 } ECPoint; ``` 2. 实现基本操作： - `double_point(ECPoint *P)`：计算点P的二倍。 - `add_point(ECPoint *P, ECPoint *Q, ECPoint *R)`：计算点P和Q的和，结果存储在R中。 3. 点乘算法：使用Double-and-Add或Montgomery Ladder算法。例如，如果n=12，我们先计算2G，再计算4G（2G+2G），接着8G（4G+4G），最后12G（8G+4G）。 ```c ECPoint multiply(ECPoint *P, int n) { ECPoint R = {0, 0, 1}; // 初始化为无穷远点 while (n > 0) { if (n % 2 == 1) { R = add_point(&R, P); } P = double_point(P); n /= 2; } return R; } ``` 4. 检查结果：计算出的点应当满足椭圆曲线方程，并且如果n是奇数，那么结果应与原点G相加等于G的n倍。在实际应用中，还需要处理边界条件，比如点在无穷远点时的特殊情况，以及点不在曲线上时的错误检查。此外，为了提高效率，通常会使用更高级的算法，如Montgomery Ladder，它在处理多次加法和乘法时有更高的效率。标签"**ecc_c实现**"表明这个项目专注于C语言的实现细节，这可能包括内存管理、优化和错误处理。在实际开发中，还要考虑到代码的可读性、可维护性和安全性。通过这个项目，开发者可以深入了解ECC的工作原理，同时提高C语言编程技能。这种实现对于理解椭圆曲线密码学的核心概念，以及在实际系统中如何运用这些概念，都是极有价值的实践。

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ECC.zip （13个子文件）

ECC

ECC.dsp 3KB

ECC.ncb 41KB

ECC.cpp 2KB

out.txt 15KB

ECC.opt 48KB

ECC.dsw 531B

ECC.plg 733B

Debug

ECC.pdb 353KB

ECC.obj 7KB

vc60.idb 33KB

vc60.pdb 44KB

ECC.ilk 178KB

ECC.exe 176KB

1P = (5,10) 2P = (8,12) 3P = (13,0) 4P = (8,11) 5P = (5,13) 6P = (13,13) 7P = (17,20) 8P = (10,5) 9P = (9,17) 10P = (2,1) 11P = (2,22) 12P = (9,6) 13P = (10,18) 14P = (17,3) 15P = (13,10) 16P = (5,13) 17P = (13,13) 18P = (17,20) 19P = (10,5) 20P = (9,17) 21P = (2,1) 22P = (2,22) 23P = (9,6) 24P = (10,18) 25P = (17,3) 26P = (13,10) 27P = (5,13) 28P = (13,13) 29P = (17,20) 30P = (10,5) 31P = (9,17) 32P = (2,1) 33P = (2,22) 34P = (9,6) 35P = (10,18) 36P = (17,3) 37P = (13,10) 38P = (5,13) 39P = (13,13) 40P = (17,20) 41P = (10,5) 42P = (9,17) 43P = (2,1) 44P = (2,22) 45P = (9,6) 46P = (10,18) 47P = (17,3) 48P = (13,10) 49P = (5,13) 50P = (13,13) 51P = (17,20) 52P = (10,5) 53P = (9,17) 54P = (2,1) 55P = (2,22) 56P = (9,6) 57P = (10,18) 58P = (17,3) 59P = (13,10) 60P = (5,13) 61P = (13,13) 62P = (17,20) 63P = (10,5) 64P = (9,17) 65P = (2,1) 66P = (2,22) 67P = (9,6) 68P = (10,18) 69P = (17,3) 70P = (13,10) 71P = (5,13) 72P = (13,13) 73P = (17,20) 74P = (10,5) 75P = (9,17) 76P = (2,1) 77P = (2,22) 78P = (9,6) 79P = (10,18) 80P = (17,3) 81P = (13,10) 82P = (5,13) 83P = (13,13) 84P = (17,20) 85P = (10,5) 86P = (9,17) 87P = (2,1) 88P = (2,22) 89P = (9,6) 90P = (10,18) 91P = (17,3) 92P = (13,10) 93P = (5,13) 94P = (13,13) 95P = (17,20) 96P = (10,5) 97P = (9,17) 98P = (2,1) 99P = (2,22) 100P = (9,6) 101P = (10,18) 102P = (17,3) 103P = (13,10) 104P = (5,13) 105P = (13,13) 106P = (17,20) 107P = (10,5) 108P = (9,17) 109P = (2,1) 110P = (2,22) 111P = (9,6) 112P = (10,18) 113P = (17,3) 114P = (13,10) 115P = (5,13) 116P = (13,13) 117P = (17,20) 118P = (10,5) 119P = (9,17) 120P = (2,1) 121P = (2,22) 122P = (9,6) 123P = (10,18) 124P = (17,3) 125P = (13,10) 126P = (5,13) 127P = (13,13) 128P = (17,20) 129P = (10,5) 130P = (9,17) 131P = (2,1) 132P = (2,22) 133P = (9,6) 134P = (10,18) 135P = (17,3) 136P = (13,10) 137P = (5,13) 138P = (13,13) 139P = (17,20) 140P = (10,5) 141P = (9,17) 142P = (2,1) 143P = (2,22) 144P = (9,6) 145P = (10,18) 146P = (17,3) 147P = (13,10) 148P = (5,13) 149P = (13,13) 150P = (17,20) 151P = (10,5) 152P = (9,17) 153P = (2,1) 154P = (2,22) 155P = (9,6) 156P = (10,18) 157P = (17,3) 158P = (13,10) 159P = (5,13) 160P = (13,13) 161P = (17,20) 162P = (10,5) 163P = (9,17) 164P = (2,1) 165P = (2,22) 166P = (9,6) 167P = (10,18) 168P = (17,3) 169P = (13,10) 170P = (5,13) 171P = (13,13) 172P = (17,20) 173P = (10,5) 174P = (9,17) 175P = (2,1) 176P = (2,22) 177P = (9,6) 178P = (10,18) 179P = (17,3) 180P = (13,10) 181P = (5,13) 182P = (13,13) 183P = (17,20) 184P = (10,5) 185P = (9,17) 186P = (2,1) 187P = (2,22) 188P = (9,6) 189P = (10,18) 190P = (17,3) 191P = (13,10) 192P = (5,13) 193P = (13,13) 194P = (17,20) 195P = (10,5) 196P = (9,17) 197P = (2,1) 198P = (2,22) 199P = (9,6) 200P = (10,18) 201P = (17,3) 202P = (13,10) 203P = (5,13) 204P = (13,13) 205P = (17,20) 206P = (10,5) 207P = (9,17) 208P = (2,1) 209P = (2,22) 210P = (9,6) 211P = (10,18) 212P = (17,3) 213P = (13,10) 214P = (5,13) 215P = (13,13) 216P = (17,20) 217P = (10,5) 218P = (9,17) 219P = (2,1) 220P = (2,22) 221P = (9,6) 222P = (10,18) 223P = (17,3) 224P = (13,10) 225P = (5,13) 226P = (13,13) 227P = (17,20) 228P = (10,5) 229P = (9,17) 230P = (2,1) 231P = (2,22) 232P = (9,6) 233P = (10,18) 234P = (17,3) 235P = (13,10) 236P = (5,13) 237P = (13,13) 238P = (17,20) 239P = (10,5) 240P = (9,17) 241P = (2,1) 242P = (2,22) 243P = (9,6) 244P = (10,18) 245P = (17,3) 246P = (13,10) 247P = (5,13) 248P = (13,13) 249P = (17,20) 250P = (10,5) 251P = (9,17) 252P = (2,1) 253P = (2,22) 254P = (9,6) 255P = (10,18) 256P = (17,3) 257P = (13,10) 258P = (5,13) 259P = (13,13) 260P = (17,20) 261P = (10,5) 262P = (9,17) 263P = (2,1) 264P = (2,22) 265P = (9,6) 266P = (10,18) 267P = (17,3) 268P = (13,10) 269P = (5,13) 270P = (13,13) 271P = (17,20) 272P = (10,5) 273P = (9,17) 274P = (2,1) 275P = (2,22) 276P = (9,6) 277P = (10,18) 278P = (17,3) 279P = (13,10) 280P = (5,13) 281P = (13,13) 282P = (17,20) 283P = (10,5) 284P = (9,17) 285P = (2,1) 286P = (2,22) 287P = (9,6) 288P = (10,18) 289P = (17,3) 290P = (13,10) 291P = (5,13) 292P = (13,13) 293P = (17,20) 294P = (10,5) 295P = (9,17) 296P = (2,1) 297P = (2,22) 298P = (9,6) 299P = (10,18) 300P = (17,3) 301P = (13,10) 302P = (5,13) 303P = (13,13) 304P = (17,20) 305P = (10,5) 306P = (9,17) 307P = (2,1) 308P = (2,22) 309P = (9,6) 310P = (10,18) 311P = (17,3) 312P = (13,10) 313P = (5,13) 314P = (13,13) 315P = (17,20) 316P = (10,5) 317P = (9,17) 318P = (2,1) 319P = (2,22) 320P = (9,6) 321P = (10,18) 322P = (17,3) 323P = (13,10) 324P = (5,13) 325P = (13,13) 326P = (17,20) 327P = (10,5) 328P = (9,17) 329P = (2,1) 330P = (2,22) 331P = (9,6) 332P = (10,18) 333P = (17,3) 334P = (13,10) 335P = (5,13) 336P = (13,13) 337P = (17,20) 338P = (10,5) 339P = (9,17) 340P = (2,1) 341P = (2,22) 342P = (9,6) 343P = (10,18) 344P = (17,3) 345P = (13,10) 346P = (5,13) 347P = (13,13) 348P = (17,20) 349P = (10,5) 350P = (9,17) 351P = (2,1) 352P = (2,22) 353P = (9,6) 354P = (10,18) 355P = (17,3) 356P = (13,10) 357P = (5,13) 358P = (13,13) 359P = (17,20) 360P = (10,5) 361P = (9,17) 362P = (2,1) 363P = (2,22) 364P = (9,6) 365P = (10,18) 366P = (17,3) 367P = (13,10) 368P = (5,13) 369P = (13,13) 370P = (17,20) 371P = (10,5) 372P = (9,17) 373P = (2,1) 374P = (2,22) 375P = (9,6) 376P = (10,18) 377P = (17,3) 378P = (13,10) 379P = (5,13) 380P = (13,13) 381P = (17,20) 382P = (10,5) 383P = (9,17) 384P = (2,1) 385P = (2,22) 386P = (9,6) 387P = (10,18) 388P = (17,3) 389P = (13,10) 390P = (5,13) 391P = (13,13) 392P = (17,20) 393P = (10,5) 394P = (9,17) 395P = (2,1) 396P = (2,22) 397P = (9,6) 398P = (10,18) 399P = (17,3) 400P = (13,10) 401P = (5,13) 402P = (13,13) 403P = (17,20) 404P = (10,5) 405P = (9,17) 406P = (2,1) 407P = (2,22) 408P = (9,6) 409P = (10,18) 410P = (17,3) 411P = (13,10) 412P = (5,13) 413P = (13,13) 414P = (17,20) 415P = (10,5) 416P = (9,17) 417P = (2,1) 418P = (2,22) 419P = (9,6) 420P = (10,18) 421P = (17,3) 422P = (13,10) 423P = (5,13) 424P = (13,13) 425P = (17,20) 426P = (10,5) 427P = (9,17) 428P = (2,1) 429P = (2,22) 430P = (9,6) 431P = (10,18) 432P = (17,3) 433P = (13,10) 434P = (5,13) 435P = (13,13) 436P = (17,20) 437P = (10,5) 438P = (9,17) 439P = (2,1) 440P = (2,22) 441P = (9,6) 442P = (10,18) 443P = (17,3) 444P = (13,10) 445P = (5,13) 446P = (13,13) 447P = (17,20) 448P = (10,5) 449P = (9,17) 450P = (2,1) 451P = (2,22) 452P = (9,6) 453P = (10,18) 454P = (17,3) 455P = (13,10) 456P = (5,13) 457P = (13,13) 458P = (17,20) 459P = (10,5) 460P = (9,17) 461P = (2,1) 462P = (2,22) 463P = (9,6) 464P = (10,18) 465P = (17,3) 466P = (13,10) 467P = (5,13) 468P = (13,13) 469P = (17,20) 470P = (10,5) 471P = (9,17) 472P = (2,1) 473P = (2,22) 474P = (9,6) 475P = (10,18) 476P = (17,3) 477P = (13,10) 478P = (5,13) 479P = (13,13) 480P = (17,20) 481P = (10,5) 482P = (9,17) 483P = (2,1) 484P = (2,22) 485P = (9,6) 486P = (10,18) 487P = (17,3) 488P = (13,10) 489P = (5,13) 490P = (13,13) 491P = (17,20) 492P = (10,5) 493P = (9,17) 494P = (2,1) 495P = (2,22) 496P = (9,6) 497P = (10,18) 498P = (17,3) 499P = (13,10) 500P = (5,13) 501P = (13,13) 502P = (17,20) 503P = (10,5) 504P = (9,17) 505P = (2,1) 506P = (2,22) 507P = (9,6) 508P = (10,18) 509P = (17,3) 510P = (13,10) 511P = (5,13) 512P = (13,13) 513P = (17,20) 514P = (10,5) 515P = (9,17) 516P = (2,1) 517P = (2,22) 518P = (9,6) 519P = (10,18) 520P = (17,3) 521P = (13,10) 522P = (5,13) 523P = (13,13) 524P = (17,20) 525P = (10,5) 526P = (9,17) 527P = (2,1) 528P = (2,22) 529P = (9,6) 530P = (10,18) 531P = (17,3) 532P = (13,10) 533P = (5,13) 534P = (13,13) 53

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