A User-Defined Location-Sharing Scheme with Efficiency and Privacy in Mobile Social Networks

<table class="table-group" id="tab1"><tr><td><table class="table"><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr class="thead"><td class="align_left">Notation</td><td class="align_center">Description</td></tr><tr><td class="thead-hr" colspan="2"><hr/></td></tr><tr><td class="align_left"><span style="width: 39.5609ptpx;"><svg height="12.7178pt" id="M45" style="vertical-align:-3.42947pt" version="1.1" viewbox="-0.0498162 -9.28833 39.5609 12.7178" width="39.5609pt" 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="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.7,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,17.24,0)"><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,-0.013,22.383,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.325,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.865,0)"><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></svg></span></td><td class="align_center">Point coordinate of user <i>u</i></td></tr><tr><td class="align_left"><span style="width: 44.1671ptpx;"><svg height="12.5794pt" id="M46" style="vertical-align:-3.29107pt" version="1.1" viewbox="-0.0498162 -9.28833 44.1671 12.5794" width="44.1671pt" 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="M360 -163L167 270L360 703H298L106 273V267L298 -163H360Z"></path></g><g transform="matrix(.013,0,0,-0.013,5.369,0)"><path d="M748 650H522L515 622L546 617C580 611 587 604 565 575C518 513 469 451 419 393C376 474 349 534 330 580C318 609 325 612 361 618L383 622L392 650H151L144 622C214 616 224 612 257 543L360 327C270 218 187 124 159 95C106 40 92 34 26 28L17 0H252L259 28L236 31C189 37 188 47 209 78C249 136 308 210 377 294L478 79C494 44 487 37 449 32L418 28L409 0H673L680 28C596 34 591 39 554 116L436 361C526 469 574 521 604 553C659 612 669 614 739 622L748 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,15.249,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,20.789,0)"><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,-0.013,25.932,0)"><path d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,33.043,3.132)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,38.583,0)"><path d="M307 267V273L115 703H53L246 270L53 -163H115L307 267Z"></path></g></svg></span></td><td class="align_center">Grid coordinate of user <i>u</i></td></tr><tr><td class="align_left"><span style="width: 39.5609ptpx;"><svg 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186 -56 186 22C186 91 131 130 95 130Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.383,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.912,-5.741)"><path d="M452 282C452 397 389 451 302 451C268 451 227 441 188 422C95 377 24 281 24 154C24 65 70 -12 173 -12C240 -12 297 16 342 52C418 113 452 201 452 282ZM359 282C359 163 308 66 247 39C237 35 225 32 212 32C151 32 114 79 114 157C114 301 180 380 223 398C240 405 250 408 265 408C315 408 359 369 359 282Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,29.325,3.784)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,34.865,0)"><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></svg></span></td><td class="align_center">Origin of the grid user <i>u</i> located</td></tr><tr><td class="align_left"><span style="width: 41.894ptpx;"><svg height="16.2017pt" id="M48" style="vertical-align:-3.943501pt" version="1.1" viewbox="-0.0498162 -12.2582 41.894 16.2017" width="41.894pt" 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="M536 404C536 423 520 448 491 448C445 448 398 404 308 283L286 338C255 416 242 448 217 448C182 448 138 402 92 341L111 321C149 368 169 378 178 378C188 378 198 363 214 320L254 212C185 117 138 65 107 65C98 65 85 69 82 75C79 82 73 86 65 86C44 86 23 60 23 39C23 7 44 -12 71 -12C119 -12 168 33 265 177L306 61C321 17 347 -12 373 -12C413 -12 465 33 507 96L491 119C459 84 432 60 413 60C395 60 378 92 358 148L321 250C341 279 369 310 389 332C417 363 439 382 456 382C466 382 475 376 481 368C486 361 492 358 496 358C513 358 536 381 536 404Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.765,-5.741)"><path d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.7,3.784)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,18.146,0)"><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,-0.013,23.289,0)"><path d="M556 393C556 426 537 448 514 448C496 448 478 435 471 420C468 414 462 401 466 394C473 382 480 368 480 346C480 268 392 143 338 67H336C329 163 319 253 309 330S286 448 254 448C215 448 160 383 127 337L143 311C167 344 200 373 208 373S222 365 229 320C246 214 264 66 268 -20C219 -83 131 -171 17 -239L25 -261L137 -235C248 -110 273 -79 335 8C384 77 481 215 520 287C544 332 556 363 556 393Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.818,-5.741)"><path d="M620 667C620 684 593 710 552 710C515 710 459 687 409 634C360 582 334 506 320 433H247L215 400L220 388H310L259 80C224 -129 199 -161 185 -174C175 -183 160 -192 141 -192C123 -192 90 -181 75 -165C69 -158 64 -156 54 -164C39 -178 24 -195 24 -208C24 -231 61 -257 94 -257C124 -257 162 -243 209 -194C267 -134 303 -66 343 107C369 220 378 279 393 385L503 397L523 433H404C432 619 464 660 500 660C522 660 542 646 566 621C573 614 583 615 593 621C603 628 620 648 620 667Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,30.231,3.784)"><path d="M528 97L513 125C484 95 440 68 431 68C425 68 421 76 427 107C449 227 476 342 505 451H495L418 424L385 278C338 191 220 62 157 62C145 62 133 76 145 130L200 368C215 434 210 451 183 451C157 451 92 418 24 352L37 324C78 356 106 377 115 377C123 377 122 364 116 340L64 115C58 89 56 68 56 52C56 2 78 -12 103 -12C127 -12 159 -4 190 18C247 59 309 113 366 187H368L351 108C328 1 348 -12 368 -12C403 -12 472 33 528 97Z"></path></g><g transform="matrix(.013,0,0,-0.013,37.198,0)"><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></svg></span></td><td class="align_center">Offset to the ordinate origin of user <i>u</i></td></tr><tr><td class="align_left"><i>G</i><sub><i>u</i></sub></td><td class="align_center">Grid identification number of the user <i>u</i></td></tr><tr><td class="align_left"><i>H</i> (<i>u</i>)</td><td class="align_center">Hilbert value of the cell user <i>u</i> located</td></tr><tr><td class="align_left"><span style="width: 41.0302ptpx;"><svg height="12.5794pt" 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422C95 377 24 281 24 154C24 65 70 -12 173 -12C240 -12 297 16 342 52C418 113 452 201 452 282ZM359 282C359 163 308 66 247 39C237 35 225 32 212 32C151 32 114 79 114 157C114 301 180 380 223 398C240 405 250 408 265 408C315 408 359 369 359 282Z"></path></g><g transform="matrix(.013,0,0,-0.013,19.224,0)"><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,-0.013,24.367,0)"><path d="M640 650H413L406 622L438 617C475 611 483 605 463 571C422 503 346 400 296 333C264 403 215 529 199 575C189 604 191 614 228 618L262 622L271 650H30L23 622C90 615 97 609 119 553L212 321C222 295 223 285 218 259L193 127C176 39 167 33 82 28L76 0H350L356 28C266 36 257 40 275 127L303 266C308 290 314 302 328 322C396 418 454 493 498 546C551 610 565 616 631 622L640 650Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,31.478,3.132)"><path d="M452 282C452 397 389 451 302 451C268 451 227 441 188 422C95 377 24 281 24 154C24 65 70 -12 173 -12C240 -12 297 16 342 52C418 113 452 201 452 282ZM359 282C359 163 308 66 247 39C237 35 225 32 212 32C151 32 114 79 114 157C114 301 180 380 223 398C240 405 250 408 265 408C315 408 359 369 359 282Z"></path></g><g transform="matrix(.013,0,0,-0.013,36.324,0)"><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></svg></span></td><td class="align_center">Starting point of Hilbert curve</td></tr><tr class="table-tr"><td colspan="2"><hr class="tbody-hr"/></td></tr></table></td></tr></table>

Scientific Programming

tab1

Table 1

Table 1: A User-Defined Location-Sharing Scheme with Efficiency and Privacy in Mobile Social Networks 