PB: A Product-Bitmatrix Construction to Reduce the Complexity of XOR Operations of PM-MSR and PM-MBR Codes over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.993899pt" id="M1" height="16.1363pt" version="1.1" viewBox="-0.0657574 -13.1424 51.1082 16.1363" width="51.1082pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g190-72" d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"/></g><g transform="matrix(.017,0,0,-0.017,12.269,0)"><path id="g190-71" 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"/></g><g transform="matrix(.017,0,0,-0.017,21.428,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,27.365,0)"><path id="g113-51" 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"/></g><g transform="matrix(.012,0,0,-0.012,35.602,-7.578)"><path id="g50-120" d="M698 334C698 397 680 451 655 451C627 451 602 424 602 397C602 389 605 383 614 371C622 360 624 340 624 317C624 161 546 47 458 47C419 47 389 69 389 125C389 142 391 162 397 185L457 430L451 435L380 421L315 164C301 107 266 47 222 47C182 47 153 69 153 126C153 137 157 162 161 185C166 214 179 265 197 334C203 358 208 385 208 407C208 433 201 451 176 451C124 451 64 404 24 348L44 320C86 366 113 382 121 382C126 382 128 381 128 374C128 368 126 356 122 341C102 266 88 204 81 164C77 143 73 121 73 107C73 23 131 -12 187 -12C234 -12 281 10 322 45C343 5 383 -12 422 -12C559 -12 698 174 698 334Z"/></g><g transform="matrix(.017,0,0,-0.017,44.964,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg>

<div>Using Bitmatrix transform <span class="nowrap"><svg height="8.98583pt" id="M364" style="vertical-align:-0.3499308pt" version="1.1" viewbox="-0.0498162 -8.6359 35.8836 8.98583" width="35.8836pt" 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="M699 659C666 654 622 640 595 610C579 591 569 572 562 530L552 472C530 342 504 307 403 302V529C403 604 411 617 483 622V650H238V622C310 617 318 604 318 529V302C232 308 194 347 172 471L161 531C153 573 145 592 131 609C103 643 59 654 25 659L15 632C62 611 63 583 73 524L85 452C97 383 116 347 155 316C199 282 255 269 318 267V122C318 46 310 33 233 28V0H489V28C412 33 403 46 403 122V267C470 269 518 283 559 315C599 346 618 378 632 450L646 519C658 580 665 608 709 632L699 659Z"></path></g><g transform="matrix(.013,0,0,-0.013,12.335,0)"><path d="M528 54L331 254L528 455L492 493L294 291L96 493L60 455L257 254L60 54L96 16L294 217L492 16L528 54Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.871,0)"><path d="M962 650H795L470 145L347 650H176L170 622C268 613 275 606 244 503L190 322C150 188 132 126 118 91C102 50 80 33 18 28L12 0H245L251 28C175 35 170 48 174 93C177 128 191 180 220 284L292 542H294C331 392 383 150 409 4H432L774 555H776L714 137C700 40 694 34 612 28L606 0H868L874 28C793 34 784 37 797 137L849 533C859 612 863 616 956 622L962 650Z"></path></g></svg>.</span> Shown in the upper part of this figure is the transformation of PB-MSR codes for the parameters <span class="nowrap"><svg height="10.8649pt" id="M365" style="vertical-align:-1.57657pt" version="1.1" viewbox="-0.0498162 -9.28833 94.3243 10.8649" width="94.3243pt" 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="M495 86L479 114C446 82 419 66 409 66C401 66 401 72 406 97C420 166 436 231 453 297C489 435 454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 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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,-0.013,60.478,0)"><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,-0.013,65.622,0)"><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"></path></g><g transform="matrix(.013,0,0,-0.013,76.469,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,87.732,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g></svg>,</span> and <span class="nowrap"><svg height="8.8423pt" id="M366" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 29.1286 8.8423" width="29.1286pt" 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="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 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454 448 428 448C406 448 384 439 354 422C305 394 222 327 161 247H159L183 345C200 415 194 448 173 448C143 448 82 410 23 351L38 325C64 349 95 371 105 371C111 371 116 365 109 336L25 -4L31 -12C50 -4 77 3 107 9C119 69 132 122 145 168C197 254 321 381 370 381C387 381 393 374 378 305L329 95C309 17 320 -12 345 -12C372 -12 430 19 495 86Z"></path></g><g transform="matrix(.013,0,0,-0.013,10.158,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,21.421,0)"><path d="M456 178V225H360V632H320C217 496 115 347 20 206V178H280V106C280 40 276 34 189 27V0H445V27C364 34 360 39 360 106V178H456ZM280 225H82C149 335 214 431 278 520H280V225Z"></path></g><g transform="matrix(.013,0,0,-0.013,27.661,0)"><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,-0.013,32.804,0)"><path d="M480 416C480 431 465 448 438 448C388 448 312 383 252 330C217 299 188 273 155 237H153L257 680C262 700 263 712 253 712C240 712 183 684 97 674L92 648L126 647C166 646 172 645 163 606L23 -6L29 -12C51 -5 77 2 107 8C115 62 130 128 142 180C153 193 179 220 204 241C231 170 259 106 288 54C317 0 336 -12 358 -12C381 -12 423 2 477 80L460 100C434 74 408 54 398 54C385 54 374 65 351 107C326 154 282 241 263 299C296 332 351 377 403 377C424 377 436 372 445 368C449 366 456 368 462 375C472 386 480 402 480 416Z"></path></g><g transform="matrix(.013,0,0,-0.013,42.975,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,54.238,0)"><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,-0.013,60.478,0)"><path d="M95 130C70 130 46 113 46 88C46 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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></svg>,</span> and <svg height="8.8423pt" id="M368" style="vertical-align:-0.2064009pt" version="1.1" viewbox="-0.0498162 -8.6359 29.1286 8.8423" width="29.1286pt" 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="M578 512C578 619 494 650 387 650H140L134 622C219 615 223 609 210 542L127 119C112 40 104 34 23 28L17 0H235C317 0 387 7 444 33C515 65 564 122 564 195C564 289 495 335 412 352V354C505 373 578 422 578 512ZM486 510C486 422 423 367 314 367H257L294 565C299 591 305 602 315 608C324 614 339 617 362 617C421 617 486 595 486 510ZM466 200C466 100 393 35 296 35C222 35 198 51 212 127L250 333H303C388 333 466 303 466 200Z"></path></g><g transform="matrix(.013,0,0,-0.013,11.445,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z"></path></g><g transform="matrix(.013,0,0,-0.013,22.708,0)"><path d="M153 550H386L412 615L406 623H120L82 318C104 327 142 338 184 338C294 338 347 275 347 187C347 112 305 39 221 39C160 39 119 71 97 89C88 97 80 96 71 90C59 80 50 67 49 57C48 45 52 36 66 23C80 9 123 -12 169 -12C221 -11 288 15 342 59C403 109 431 165 431 225C431 308 366 395 238 395C212 395 165 379 127 364L153 550Z"></path></g></svg> is shown in the bottom half of the illustration. All encoded blocks are stored on the corresponding nodes.</div>

Security and Communication Networks

fig4

Figure 4

Figure 4: PB: A Product-Bitmatrix Construction to Reduce the Complexity of XOR Operations of PM-MSR and PM-MBR Codes over <svg xmlns:xlink="http://www.w3.org/1999/xlink" xmlns="http://www.w3.org/2000/svg" style="vertical-align:-2.993899pt" id="M1" height="16.1363pt" version="1.1" viewBox="-0.0657574 -13.1424 51.1082 16.1363" width="51.1082pt"><g transform="matrix(.017,0,0,-0.017,0,0)"><path id="g190-72" d="M692 302H438V274C537 267 543 260 543 188V103C543 61 531 48 511 37C489 26 458 20 424 20C231 20 146 188 146 333C146 517 258 630 411 630C507 630 582 597 606 474L634 480C627 546 622 601 619 636C586 643 510 665 426 665C230 665 44 552 44 321C44 122 191 -15 411 -15C491 -15 573 7 635 21C629 49 628 81 628 116V202C628 261 632 266 692 274V302Z"/></g><g transform="matrix(.017,0,0,-0.017,12.269,0)"><path id="g190-71" 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"/></g><g transform="matrix(.017,0,0,-0.017,21.428,0)"><path id="g113-41" d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z"/></g><g transform="matrix(.017,0,0,-0.017,27.365,0)"><path id="g113-51" 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"/></g><g transform="matrix(.012,0,0,-0.012,35.602,-7.578)"><path id="g50-120" d="M698 334C698 397 680 451 655 451C627 451 602 424 602 397C602 389 605 383 614 371C622 360 624 340 624 317C624 161 546 47 458 47C419 47 389 69 389 125C389 142 391 162 397 185L457 430L451 435L380 421L315 164C301 107 266 47 222 47C182 47 153 69 153 126C153 137 157 162 161 185C166 214 179 265 197 334C203 358 208 385 208 407C208 433 201 451 176 451C124 451 64 404 24 348L44 320C86 366 113 382 121 382C126 382 128 381 128 374C128 368 126 356 122 341C102 266 88 204 81 164C77 143 73 121 73 107C73 23 131 -12 187 -12C234 -12 281 10 322 45C343 5 383 -12 422 -12C559 -12 698 174 698 334Z"/></g><g transform="matrix(.017,0,0,-0.017,44.964,0)"><path id="g113-42" d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z"/></g></svg> 