Certificateless Public Auditing Protocol with<i> Constant</i> Verification Time

Comparison between certificateless public auditing schemes. Let <svg height="8.8423pt" id="M971" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 23.5612 8.8423" width="23.5612pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M600 480C600 590 528 650 384 650H143L137 622C222 614 225 607 210 531L130 127C113 41 106 36 23 28L17 0H294L300 28C204 36 195 42 212 127L243 284L314 263C327 263 339 263 352 264C465 271 600 337 600 480ZM508 481C508 351 402 304 329 304C289 304 265 311 250 317L295 559C302 594 310 606 323 611C335 616 350 619 367 619C455 619 508 573 508 481Z" id="g113-81"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="617" vert-adv-y="617"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.374,0)"><path d="M483 97L471 123C436 91 401 65 392 65C388 65 384 74 390 106C414 239 444 378 457 429L455 433C444 433 429 436 416 439C392 444 368 448 344 448C281 448 204 415 152 376C71 315 23 205 23 103C23 21 57 -12 85 -12C114 -12 149 6 185 34C231 70 285 119 329 183H331L309 81C292 0 308 -12 326 -12C350 -12 421 24 483 97ZM374 387C370 363 356 291 345 261C315 193 181 50 139 50C124 50 110 71 110 118C110 224 153 331 218 379C238 394 271 402 301 402C329 402 359 394 374 387Z" id="g113-98"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="506" vert-adv-y="506"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,14.345,0)"><path d="M244 607C244 633 228 655 200 655C166 655 146 618 146 594C146 564 166 546 191 546C221 546 244 574 244 607ZM222 91L209 114C184 94 148 66 133 66C127 66 124 73 130 96L201 370C213 416 211 448 191 448C162 448 88 407 29 352L42 328C73 354 104 371 114 371C120 371 119 365 115 345L53 92C32 5 45 -12 68 -12C103 -12 186 50 222 91Z" id="g113-106"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="273" vert-adv-y="273"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,17.894,0)"><path d="M393 379C402 394 400 411 393 422C384 437 365 448 348 448C301 448 237 372 186 285H182L193 335C210 408 205 448 178 448C150 448 80 402 29 344L45 321C80 355 114 373 122 373C128 373 130 365 124 330C106 228 76 98 50 -5L57 -12C82 -5 112 3 132 6L172 203C196 256 234 304 254 329C275 355 293 367 306 367C318 367 330 360 342 348C347 343 355 343 365 350S386 367 393 379Z" id="g113-115"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="422" vert-adv-y="422"></glyph.data></g></svg> be the bilinear pairing computation and <svg height="12.5364pt" id="M972" style="vertical-align:-3.30025pt" version="1.1" viewbox="-0.0498162 -9.23615 28.5377 12.5364" width="28.5377pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M861 0V28C774 35 771 41 768 147L759 509C756 612 762 614 851 622V650H681L449 149L221 650H57V622C148 613 153 609 144 479L130 271C123 166 117 123 111 88C104 46 85 34 26 28V0H259V28C192 35 169 42 167 90C166 130 166 173 170 256L185 541H187L411 7H431L675 555H679L683 147C683 41 680 35 598 28V0H861Z" id="g121-75"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="891" vert-adv-y="891"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,11.152,0)"><path d="M518 50L491 51C452 54 444 60 444 110V444C429 441 405 437 377 434C348 430 314 427 287 426V403L323 397C356 392 365 386 365 333V99C331 66 293 51 257 51C212 51 169 75 169 164V299C169 366 169 413 172 444C156 441 128 437 101 433C75 430 50 427 29 426V403L57 397C82 391 90 386 90 333V137C90 29 147 -12 214 -12C241 -12 262 -4 291 13S342 48 365 65V-6L371 -12C390 -7 415 1 441 8C468 15 496 21 518 24V50Z" id="g121-115"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="531" vert-adv-y="531"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,18.121,0)"><path d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z" id="g121-106"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="253" vert-adv-y="253"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,21.491,3.132)"><path d="M664 534L634 568C565 641 475 676 375 676C172 676 45 530 45 333C45 141 176 -13 374 -13C478 -13 572 30 655 90V329H358V285H489V56C462 37 413 31 380 31C319 31 256 44 213 91V579C254 614 326 632 379 632C490 632 563 586 632 502L664 534ZM611 118C584 102 562 78 533 66V285H611V118ZM169 119C118 180 89 253 89 333C89 410 118 492 169 551V119Z" id="g197-18"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="706" vert-adv-y="706"></glyph.data></g></svg> be the multiplication in group <svg height="9.09021pt" id="M973" style="vertical-align:-0.2194395pt" version="1.1" viewbox="-0.0498162 -8.87077 9.31206 9.09021" width="9.31206pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M664 534L634 568C565 641 475 676 375 676C172 676 45 530 45 333C45 141 176 -13 374 -13C478 -13 572 30 655 90V329H358V285H489V56C462 37 413 31 380 31C319 31 256 44 213 91V579C254 614 326 632 379 632C490 632 563 586 632 502L664 534ZM611 118C584 102 562 78 533 66V285H611V118ZM169 119C118 180 89 253 89 333C89 410 118 492 169 551V119Z" id="g197-18"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="706" vert-adv-y="706"></glyph.data></g></svg>. Let <svg height="12.5364pt" id="M974" style="vertical-align:-3.30025pt" version="1.1" viewbox="-0.0498162 -9.23615 34.1226 12.5364" width="34.1226pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M729 650H468V622C546 615 554 610 554 527V365H213V527C213 610 221 616 297 622V650H38V622C120 616 128 611 128 527V123C128 39 120 34 40 28V0H303V28C221 34 213 40 213 123V322H554V123C554 39 546 34 460 28V0H728V28C647 34 639 39 639 123V527C639 610 647 615 729 622V650Z" id="g121-70"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="766" vert-adv-y="766"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,9.645,0)"><path d="M433 39L423 65C413 59 399 54 387 54C370 54 352 69 352 114V299C352 352 342 392 307 422C285 440 255 449 225 449C168 437 102 399 75 379C56 365 44 353 44 339C44 315 69 296 87 296C101 296 111 303 116 319C124 349 133 371 145 385C156 397 171 404 190 404C241 404 275 364 275 291V274C253 256 180 229 120 209C65 190 39 159 39 110C39 47 88 -12 159 -12C189 -12 237 25 277 52C282 35 288 21 301 8C312 -3 333 -12 348 -12L433 39ZM275 84C256 65 221 48 195 48C164 48 124 73 124 124C124 161 146 180 185 198C206 208 254 229 275 240V84Z" id="g121-95"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="439" vert-adv-y="439"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,15.313,0)"><path d="M319 325C317 349 306 409 297 431C277 440 250 449 209 449C117 449 57 389 57 319C57 243 122 209 182 182C232 159 261 135 261 91C261 48 227 21 190 21C130 21 85 79 68 145L41 140C41 104 51 36 58 22C75 7 121 -12 172 -12C252 -12 337 35 337 126C337 195 286 231 210 262C166 281 126 304 126 348C126 388 152 417 191 417C240 417 274 378 294 318L319 325Z" id="g121-113"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="367" vert-adv-y="367"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,20.032,0)"><path d="M513 0V26C455 31 447 37 447 103V278C447 398 390 449 309 449C255 449 202 412 166 376V712C127 700 67 684 19 677V653C85 647 87 643 87 580V103C87 37 79 31 19 26L18 0H231V26C171 32 166 39 166 103V341C194 373 232 392 270 392C337 392 368 351 368 269V103C368 38 360 32 302 26V0H513Z" id="g121-102"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="534" vert-adv-y="534"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,27.076,3.132)"><path d="M664 534L634 568C565 641 475 676 375 676C172 676 45 530 45 333C45 141 176 -13 374 -13C478 -13 572 30 655 90V329H358V285H489V56C462 37 413 31 380 31C319 31 256 44 213 91V579C254 614 326 632 379 632C490 632 563 586 632 502L664 534ZM611 118C584 102 562 78 533 66V285H611V118ZM169 119C118 180 89 253 89 333C89 410 118 492 169 551V119Z" id="g197-18"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="706" vert-adv-y="706"></glyph.data></g></svg> and <svg height="12.1201pt" id="M975" style="vertical-align:-3.73212pt" version="1.1" viewbox="-0.0498162 -8.38798 27.546 12.1201" width="27.546pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M517 162C503 123 484 88 467 68C445 42 417 34 341 34C291 34 256 34 237 47C219 59 213 81 213 128V317H308C395 317 402 311 415 240H444V431H415C403 364 398 356 307 356H213V584C213 613 215 616 246 616H322C394 616 421 609 435 587C448 566 458 544 467 502L496 506C493 557 488 625 488 650H42V622C120 616 128 612 128 523V125C128 43 120 35 29 28V0H511C520 31 540 125 546 158L517 162Z" id="g121-67"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="568" vert-adv-y="568"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,7.449,0)"><path d="M474 0V26C414 34 401 43 364 100L267 248C300 297 324 332 345 358C381 400 394 405 455 411V437H272V411C316 406 323 401 305 370C287 337 267 306 247 276L188 369C169 397 173 405 215 411V437H16V411C71 404 83 396 114 348L201 212C171 167 144 127 116 92C77 42 66 34 4 26V0H190V26C139 34 136 43 156 77C175 113 198 150 220 183L294 66C311 39 302 31 260 26V0H474Z" id="g121-118"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="472" vert-adv-y="472"></glyph.data></g><g transform="matrix(.013,0,0,-0.013,13.611,0)"><path d="M169 380V459C122 440 66 423 24 416V392C86 384 90 382 90 317V-135C90 -201 81 -207 17 -213V-240H253V-213C176 -207 169 -201 169 -125V6C182 -1 208 -11 238 -12C368 12 487 109 487 260C487 358 421 449 310 449C298 449 279 444 261 433L169 380ZM169 346C196 367 237 389 269 389C341 389 403 329 403 221C403 109 347 37 263 37C228 37 191 53 169 76V346Z" id="g121-110"></path><glyph.data ascent="989" descent="-360" horiz-adv-x="524" vert-adv-y="524"></glyph.data></g><g transform="matrix(.0091,0,0,-0.0091,20.5,3.564)"><path d="M664 534L634 568C565 641 475 676 375 676C172 676 45 530 45 333C45 141 176 -13 374 -13C478 -13 572 30 655 90V329H358V285H489V56C462 37 413 31 380 31C319 31 256 44 213 91V579C254 614 326 632 379 632C490 632 563 586 632 502L664 534ZM611 118C584 102 562 78 533 66V285H611V118ZM169 119C118 180 89 253 89 333C89 410 118 492 169 551V119Z" id="g197-18"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="706" vert-adv-y="706"></glyph.data></g></svg> be the hash-to-point computation and the exponentiation in group <svg height="9.09021pt" id="M976" style="vertical-align:-0.2194395pt" version="1.1" viewbox="-0.0498162 -8.87077 9.31206 9.09021" width="9.31206pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M664 534L634 568C565 641 475 676 375 676C172 676 45 530 45 333C45 141 176 -13 374 -13C478 -13 572 30 655 90V329H358V285H489V56C462 37 413 31 380 31C319 31 256 44 213 91V579C254 614 326 632 379 632C490 632 563 586 632 502L664 534ZM611 118C584 102 562 78 533 66V285H611V118ZM169 119C118 180 89 253 89 333C89 410 118 492 169 551V119Z" id="g197-18"></path><glyph.data ascent="693" descent="-218" horiz-adv-x="706" vert-adv-y="706"></glyph.data></g></svg>, respectively. Let <svg height="9.49473pt" id="M977" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -9.28833 7.34169 9.49473" width="7.34169pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink"><g transform="matrix(.013,0,0,-0.013,0,0)"><path d="M530 686C535 705 530 712 521 712C504 712 448 684 359 674L358 648H393C437 648 439 646 429 593L400 435C372 447 345 448 332 448C286 448 194 414 144 373C68 311 23 203 23 111C23 26 57 -12 91 -12C120 -12 147 3 188 29C227 54 290 102 341 170H343L322 71C308 6 320 -12 341 -12C373 -12 442 27 501 96L485 120C455 91 422 67 408 67C401 67 401 76 404 91C440 294 479 473 530 686ZM387 375L355 241C326 187 200 53 142 53C126 53 109 73 109 130C109 217 154 337 218 381C240 396 265 404 297 404S372 390 387 375Z" id="g113-101"></path><glyph.data ascent="3473" descent="-2876" horiz-adv-x="555" vert-adv-y="555"></glyph.data></g></svg> be the number of the challenged blocks.

Security and Communication Networks

tab3

Table 3

Table 3: Certificateless Public Auditing Protocol with<i> Constant</i> Verification Time 

Security and Communication Networks

Certificateless Public Auditing Protocol with Constant Verification Time

Table 3