Multidivisible Online/Offline Cryptography and Its Application to Signcryptions

<div>Description of adversary <svg height="12.0262pt" id="M422" style="vertical-align:-3.18153pt" version="1.1" viewbox="-0.0498162 -8.84467 16.2027 12.0262" width="16.2027pt" 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="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.115,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg> (or <span class="nowrap"><svg height="12.0262pt" id="M423" style="vertical-align:-3.18153pt" version="1.1" viewbox="-0.0498162 -8.84467 16.2027 12.0262" width="16.2027pt" 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="M837 672C829 675 822 674 813 674C710 674 608 541 510 398C475 447 422 480 349 480C260 480 162 415 162 299C162 199 212 133 284 92C229 39 191 15 134 15C91 15 69 45 69 65H71C78 54 95 53 100 53C128 53 143 80 143 101C143 128 121 149 93 149C60 149 31 121 31 77C31 24 74 -15 138 -15C198 -15 255 22 311 79C360 56 416 43 473 39L455 0H552L569 42C612 49 661 63 693 87L684 103C657 88 617 76 581 69C677 301 753 477 846 658L837 672ZM772 638C701 530 628 379 559 227C558 279 547 330 526 371C609 494 688 609 771 638H772ZM536 199C536 191 536 184 535 176L486 67C430 71 380 84 335 105C395 173 451 257 507 344C527 301 536 251 536 199ZM492 372C431 277 369 183 310 119C245 158 203 217 203 301S275 447 349 447C416 447 461 417 492 372Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.115,3.132)"><path d="M414 144C384 79 371 75 317 75H135L276 221C367 316 408 376 408 465C408 570 327 635 237 635C179 635 131 609 100 575L42 494L67 471C94 510 138 565 205 565C277 565 321 517 321 435C321 348 258 270 195 195C146 137 88 81 33 26V0H411C423 44 433 88 446 135L414 144Z"></path></g></svg>)</span> attacking <svg height="11.576pt" id="M424" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -9.3079 113.376 11.576" width="113.376pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M412 140C382 77 369 73 315 73H129L270 222C362 320 402 379 402 466C402 571 322 635 234 635C177 635 130 609 99 576L42 495L64 475C90 514 133 568 201 568C274 568 318 519 318 435C318 349 255 267 193 193C144 135 87 78 32 23V0H405C417 45 427 89 440 131L412 140Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.738,0)"><path d="M95 130C70 130 46 113 46 88C46 72 54 64 59 64C93 55 121 33 121 -3C121 -41 93 -68 44 -88L55 -117C117 -98 186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,15.881,0)"><path d="M384 0V27C293 34 287 42 287 114V635C232 613 172 594 109 583V559L157 557C201 555 205 550 205 499V114C205 42 199 34 109 27V0H384Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.121,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"></path></g><g transform="matrix(.013,0,0,-0.013,26.619,0)"><path d="M300 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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"></path></g><g transform="matrix(.013,0,0,-0.013,59.197,0)"><path d="M716 650H478V622C549 617 570 605 575 566C579 535 582 493 582 401V326C582 161 537 40 393 40C269 40 205 133 205 294V521C205 611 213 617 291 622V650H33V622C111 615 120 611 120 521V272C120 154 148 80 208 34C252 0 309 -14 368 -14C434 -14 501 7 548 55C605 116 623 201 623 319V401C623 493 626 536 629 563C633 605 656 618 716 622V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.765,0)"><path d="M493 503C489 551 484 614 483 650H43V622C120 616 128 611 128 525V126C128 40 120 34 40 28V0H312V28C221 34 213 40 213 126V307H316C407 307 412 296 424 227H453V420H424C412 355 407 346 316 346H213V584C213 613 216 616 246 616H322C398 616 419 607 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28V0H259V28C192 35 169 42 167 90C166 130 166 173 170 256L185 541H187L411 7H431L675 555H679L683 147C683 41 680 35 598 28V0H861Z"></path></g><g transform="matrix(.013,0,0,-0.013,104.079,0)"><path d="M673 0V28C608 34 594 43 563 129C499 303 432 494 370 665L339 656L132 132C98 43 84 37 20 28V0H238V28C162 35 154 45 175 108C188 150 203 192 219 237H432C454 176 474 122 487 83S491 36 428 28V0H673ZM418 280H234C265 362 296 450 328 535H330L418 280Z"></path></g></svg> (or <span class="nowrap"><svg height="11.576pt" id="M425" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -9.3079 113.376 11.576" width="113.376pt" 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="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><path d="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 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transform="matrix(.013,0,0,-0.013,26.619,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.265,0)"><path d="M517 51L485 54C448 58 441 63 441 115V712C404 700 337 684 285 678V653C357 648 362 645 362 580V437C339 446 309 449 295 449C159 449 38 340 38 201C38 61 143 -12 223 -12C234 -12 261 -6 301 17L362 53V-12C420 9 495 22 517 26V51ZM362 85C338 67 301 51 266 51C201 51 128 109 128 228C128 373 212 411 259 411C296 411 338 395 362 360V85Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.129,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,49.781,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,54.426,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,59.197,0)"><path d="M716 650H478V622C549 617 570 605 575 566C579 535 582 493 582 401V326C582 161 537 40 393 40C269 40 205 133 205 294V521C205 611 213 617 291 622V650H33V622C111 615 120 611 120 521V272C120 154 148 80 208 34C252 0 309 -14 368 -14C434 -14 501 7 548 55C605 116 623 201 623 319V401C623 493 626 536 629 563C633 605 656 618 716 622V650Z"></path></g><g transform="matrix(.013,0,0,-0.013,68.765,0)"><path d="M493 503C489 551 484 614 483 650H43V622C120 616 128 611 128 525V126C128 40 120 34 40 28V0H312V28C221 34 213 40 213 126V307H316C407 307 412 296 424 227H453V420H424C412 355 407 346 316 346H213V584C213 613 216 616 246 616H322C398 616 419 607 436 579C449 559 455 539 464 499L493 503Z"></path></g><g transform="matrix(.013,0,0,-0.013,75.722,0)"><path d="M300 253H71L57 198H286L300 253Z"></path></g><g transform="matrix(.013,0,0,-0.013,80.367,0)"><path d="M135 536C164 536 186 560 186 587C186 617 164 639 136 639C109 639 85 617 85 587C85 560 109 536 135 536ZM252 0V26C188 32 181 38 181 106V451C138 433 90 420 39 412V388C99 379 102 374 102 312V106C102 38 95 32 32 26V0H252Z"></path></g><g transform="matrix(.013,0,0,-0.013,83.851,0)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g><g transform="matrix(.013,0,0,-0.013,92.496,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,104.079,0)"><path d="M673 0V28C608 34 594 43 563 129C499 303 432 494 370 665L339 656L132 132C98 43 84 37 20 28V0H238V28C162 35 154 45 175 108C188 150 203 192 219 237H432C454 176 474 122 487 83S491 36 428 28V0H673ZM418 280H234C265 362 296 450 328 535H330L418 280Z"></path></g></svg>)</span> security of <svg height="12.0627pt" id="M426" style="vertical-align:-3.309361pt" version="1.1" viewbox="-0.0498162 -8.75334 50.0747 12.0627" width="50.0747pt" 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="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.03,0)"><path d="M381 665C170 665 44 498 44 318C44 125 188 -15 369 -15C552 -15 703 117 703 333C703 534 550 665 381 665ZM359 629C491 629 601 517 601 306C601 114 502 21 390 21C249 21 146 158 146 346S248 629 359 629Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.741,0)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.903,0)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,40.699,3.132)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,45.029,3.132)"><path d="M389 0V32C297 38 291 46 291 118V635C234 613 175 595 109 583V556L161 554C203 552 207 547 207 497V118C207 46 201 38 110 32V0H389Z"></path></g></svg> (or <span class="nowrap"><svg height="12.0627pt" id="M427" style="vertical-align:-3.309361pt" version="1.1" viewbox="-0.0498162 -8.75334 50.0747 12.0627" width="50.0747pt" 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="M285 378C315 398 338 416 353 432C373 451 384 474 384 503C384 579 325 635 236 635H235C182 635 136 610 108 579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z"></path></g><g transform="matrix(.013,0,0,-0.013,6.24,0)"><path d="M43 650V622C120 616 128 612 128 526V124C128 39 120 33 34 27V0H270C392 0 492 25 567 83C643 141 690 230 690 350C690 444 655 517 605 565C543 625 450 650 323 650H43ZM213 547C213 587 217 598 226 604C236 612 262 617 304 617C371 617 429 604 474 576C554 529 592 439 592 336C592 176 505 36 319 36C246 36 213 55 213 131V547Z"></path></g><g transform="matrix(.013,0,0,-0.013,16.03,0)"><path d="M381 665C170 665 44 498 44 318C44 125 188 -15 369 -15C552 -15 703 117 703 333C703 534 550 665 381 665ZM359 629C491 629 601 517 601 306C601 114 502 21 390 21C249 21 146 158 146 346S248 629 359 629Z"></path></g><g transform="matrix(.013,0,0,-0.013,25.741,0)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.013,0,0,-0.013,31.903,0)"><path d="M614 175C564 76 510 21 408 21C256 21 146 149 146 336C146 488 235 629 402 629C510 629 570 586 597 480L626 488C620 541 614 582 606 638C578 643 510 665 429 665C206 665 44 527 44 316C44 157 153 -15 402 -15C474 -15 558 5 586 11C604 45 629 119 643 165L614 175Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,40.699,3.132)"><path d="M409 504C401 567 396 607 392 642C354 654 312 665 266 665C137 665 60 583 60 487C60 374 161 325 225 290C300 250 355 215 355 141C355 68 311 21 235 21C131 21 86 122 71 183L41 176C48 128 61 42 68 21C78 16 93 8 118 0C142 -7 175 -15 216 -15C349 -15 438 69 438 174C438 287 344 333 265 374C186 414 138 449 138 522C138 576 172 631 249 631C336 631 363 562 380 499L409 504Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,45.029,3.132)"><path d="M414 144C384 79 371 75 317 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Security and Communication Networks

fig9

Figure 9

Figure 9: Multidivisible Online/Offline Cryptography and Its Application to Signcryptions 