Homomorphic Evaluation of the Integer Arithmetic Operations for Mobile Edge Computing

<div>The optimised addition operations; each colour block represents same position subvectors of <svg height="12.488112pt" id="M405" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.049816 -9.197002 14.630378 12.488112" width="14.630378pt" 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="M791 519C728 519 723 566 704 610C688 656 658 701 552 701C503 701 455 682 409 644C373 681 320 699 250 699C134 699 50 643 50 546C50 515 62 483 87 448C112 414 124 381 124 350C124 310 97 280 68 269L76 245C203 265 266 325 266 412C266 439 254 472 229 511S192 583 192 612C192 648 208 666 241 666C260 666 275 651 287 621C299 592 307 561 310 529C314 498 316 467 316 437V202C300 206 283 208 265 208C177 208 107 137 71 45L92 31C112 59 148 73 199 73C225 73 246 69 263 59C270 56 285 49 290 45L316 29V-205H478V-19H491C583 -19 657 2 712 44C767 87 795 147 795 224C795 254 783 290 759 332C736 374 724 403 724 420C724 449 746 476 791 499V519ZM701 484C620 454 579 406 579 341C579 316 589 281 609 234S639 153 639 131C639 70 615 20 560 20C529 20 502 41 478 82V452C478 515 465 569 440 613C459 638 480 651 504 651C551 651 563 591 589 544C606 512 645 486 701 486V484Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.091,3.132)"><path d="M294 607C294 643 272 671 230 671C172 671 143 618 143 593C143 548 173 524 215 524C261 524 294 565 294 607ZM285 104L263 135C232 107 205 93 193 93C185 93 184 106 191 135L247 358C259 406 254 457 224 457C184 457 113 420 28 348L47 315C76 336 101 349 109 349S115 337 109 314L58 109C33 10 61 -12 96 -12C128 -12 209 18 285 104Z"></path></g></svg> and <span class="nowrap"><svg height="11.9218pt" id="M406" style="vertical-align:-2.724797pt" version="1.1" viewbox="-0.049816 -9.197002 10.577795 11.9218" width="10.577795pt" 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="M723 676C697 631 657 608 602 608C537 604 420 704 307 701C236 701 179 683 136 646S71 562 71 505C71 421 137 363 238 363C248 363 255 363 260 364L265 390C214 403 189 431 189 475C189 501 200 523 221 540S263 566 284 566C339 566 378 547 430 516C457 501 482 490 505 482L470 449C450 431 437 419 431 412C426 405 417 395 405 380C381 351 375 327 375 300C375 264 395 224 435 180C452 161 465 145 473 134C482 123 491 105 500 80C509 56 514 30 514 2C514 -111 464 -172 392 -172C363 -172 336 -157 311 -127S274 -60 274 -16C274 17 283 52 302 91S331 161 331 182C331 238 290 285 209 285C174 285 143 274 115 251C88 228 67 203 54 175L77 164C91 178 108 185 128 185C154 185 167 172 167 147C167 138 159 115 143 78C128 41 120 10 120 -13C120 -74 146 -121 197 -154C248 -188 312 -205 389 -205C474 -205 544 -178 597 -124C651 -71 678 -10 678 59C678 141 638 208 558 260C577 287 596 301 613 301C644 301 669 278 687 232H713V360C696 382 685 397 649 397C631 397 611 386 589 365S548 318 533 289C520 300 510 332 510 352C510 401 523 441 549 473H559C652 473 715 538 748 669L723 676Z"></path></g></svg>,</span> the length of each colour block is L, “r shift” represents ciphertext vector right shift one position, “recrypt” represents ciphertext refresh. This figure shows the degree change of each component of <svg height="12.488112pt" id="M407" style="vertical-align:-3.29111pt" version="1.1" viewbox="-0.049816 -9.197002 14.630378 12.488112" width="14.630378pt" 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="M791 519C728 519 723 566 704 610C688 656 658 701 552 701C503 701 455 682 409 644C373 681 320 699 250 699C134 699 50 643 50 546C50 515 62 483 87 448C112 414 124 381 124 350C124 310 97 280 68 269L76 245C203 265 266 325 266 412C266 439 254 472 229 511S192 583 192 612C192 648 208 666 241 666C260 666 275 651 287 621C299 592 307 561 310 529C314 498 316 467 316 437V202C300 206 283 208 265 208C177 208 107 137 71 45L92 31C112 59 148 73 199 73C225 73 246 69 263 59C270 56 285 49 290 45L316 29V-205H478V-19H491C583 -19 657 2 712 44C767 87 795 147 795 224C795 254 783 290 759 332C736 374 724 403 724 420C724 449 746 476 791 499V519ZM701 484C620 454 579 406 579 341C579 316 589 281 609 234S639 153 639 131C639 70 615 20 560 20C529 20 502 41 478 82V452C478 515 465 569 440 613C459 638 480 651 504 651C551 651 563 591 589 544C606 512 645 486 701 486V484Z"></path></g><g transform="matrix(.0091,0,0,-0.0091,11.091,3.132)"><path d="M294 607C294 643 272 671 230 671C172 671 143 618 143 593C143 548 173 524 215 524C261 524 294 565 294 607ZM285 104L263 135C232 107 205 93 193 93C185 93 184 106 191 135L247 358C259 406 254 457 224 457C184 457 113 420 28 348L47 315C76 336 101 349 109 349S115 337 109 314L58 109C33 10 61 -12 96 -12C128 -12 209 18 285 104Z"></path></g></svg> and <span class="nowrap"><svg height="11.9218pt" id="M408" style="vertical-align:-2.724797pt" version="1.1" viewbox="-0.049816 -9.197002 10.577795 11.9218" width="10.577795pt" 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="M723 676C697 631 657 608 602 608C537 604 420 704 307 701C236 701 179 683 136 646S71 562 71 505C71 421 137 363 238 363C248 363 255 363 260 364L265 390C214 403 189 431 189 475C189 501 200 523 221 540S263 566 284 566C339 566 378 547 430 516C457 501 482 490 505 482L470 449C450 431 437 419 431 412C426 405 417 395 405 380C381 351 375 327 375 300C375 264 395 224 435 180C452 161 465 145 473 134C482 123 491 105 500 80C509 56 514 30 514 2C514 -111 464 -172 392 -172C363 -172 336 -157 311 -127S274 -60 274 -16C274 17 283 52 302 91S331 161 331 182C331 238 290 285 209 285C174 285 143 274 115 251C88 228 67 203 54 175L77 164C91 178 108 185 128 185C154 185 167 172 167 147C167 138 159 115 143 78C128 41 120 10 120 -13C120 -74 146 -121 197 -154C248 -188 312 -205 389 -205C474 -205 544 -178 597 -124C651 -71 678 -10 678 59C678 141 638 208 558 260C577 287 596 301 613 301C644 301 669 278 687 232H713V360C696 382 685 397 649 397C631 397 611 386 589 365S548 318 533 289C520 300 510 332 510 352C510 401 523 441 549 473H559C652 473 715 538 748 669L723 676Z"></path></g></svg>.</span></div>

Wireless Communications and Mobile Computing

fig2

Figure 2

Figure 2: Homomorphic Evaluation of the Integer Arithmetic Operations for Mobile Edge Computing 