Path Hopping: An MTD Strategy for Long-Term Quantum-Safe Communication

<div>State transition probabilities for case C1. (a) shows the defender’s moves, <span class="nowrap"><svg height="11.9805pt" id="M354" style="vertical-align:-0.2063999pt" version="1.1" viewbox="-0.0498162 -11.7741 27.5838 11.9805" width="27.5838pt" 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="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path></g><g transform="matrix(.013,0,0,-0.013,7.043,0)"><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" id="g113-78"></path></g><g transform="matrix(.0091,0,0,-0.0091,19.85,-5.741)"><path d="M786 370C786 572 637 650 431 650H137L130 618C217 606 220 603 204 519L129 127C115 48 107 39 26 30L18 0H270C410 0 516 20 609 75C722 140 786 240 786 370ZM683 373C683 178 547 40 323 40C298 40 264 40 241 45C213 51 207 64 220 127L295 545C304 582 309 595 320 601C335 610 367 613 406 613C570 613 683 538 683 373Z" id="g50-69"></path></g></svg>,</span> with arrows bellow and attacker’s moves, <span class="nowrap"><svg height="14.1883pt" id="M355" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -11.9202 55.9117 14.1883" width="55.9117pt" 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" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,4.498,0)"><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" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,13.644,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,24.18,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path></g><g transform="matrix(.013,0,0,-0.013,31.354,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g><g transform="matrix(.013,0,0,-0.013,35.852,0)"><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" id="g113-78"></path></g><g transform="matrix(.0091,0,0,-0.0091,48.659,-5.741)"><path d="M722 32C646 41 636 48 622 116L544 496L509 667L472 658L153 123C106 45 93 42 27 32L18 0H259L268 32C178 40 172 48 204 107L272 233H509C523 180 533 137 540 96C548 50 546 39 466 32L457 0H714L722 32ZM502 283H299L448 547H450L502 283Z" id="g50-66"></path></g></svg>,</span> with arrows above. (b) shows the combined transition matrix <span class="nowrap"><svg height="14.1883pt" id="M356" style="vertical-align:-2.2681pt" version="1.1" viewbox="-0.0498162 -11.9202 124.643 14.1883" width="124.643pt" 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="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" id="g113-78"></path></g><g transform="matrix(.013,0,0,-0.013,16.439,0)"><path d="M535 323V373H52V323H535ZM535 138V188H52V138H535Z" id="g117-34"></path></g><g transform="matrix(.013,0,0,-0.013,27.702,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path></g><g transform="matrix(.013,0,0,-0.013,34.745,0)"><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" id="g113-78"></path></g><g transform="matrix(.0091,0,0,-0.0091,47.552,-5.741)"><path d="M786 370C786 572 637 650 431 650H137L130 618C217 606 220 603 204 519L129 127C115 48 107 39 26 30L18 0H270C410 0 516 20 609 75C722 140 786 240 786 370ZM683 373C683 178 547 40 323 40C298 40 264 40 241 45C213 51 207 64 220 127L295 545C304 582 309 595 320 601C335 610 367 613 406 613C570 613 683 538 683 373Z" id="g50-69"></path></g><g transform="matrix(.013,0,0,-0.013,58.017,0)"><path d="M535 230V280H323V490H265V280H52V230H265V-3H323V230H535Z" id="g117-36"></path></g><g transform="matrix(.013,0,0,-0.013,68.553,0)"><path d="M300 -147C201 -63 143 98 143 270S200 602 300 686L282 710C136 610 70 450 70 271V270C70 89 136 -72 282 -170L300 -147Z" id="g113-41"></path></g><g transform="matrix(.013,0,0,-0.013,73.051,0)"><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" id="g113-50"></path></g><g transform="matrix(.013,0,0,-0.013,82.197,0)"><path d="M535 230V280H52V230H535Z" id="g117-33"></path></g><g transform="matrix(.013,0,0,-0.013,92.733,0)"><path d="M529 97L508 118C475 75 449 58 438 58C428 58 421 66 415 104C393 234 374 403 364 496C345 670 307 712 254 712C220 712 174 691 153 669L161 645C176 653 194 658 206 658C237 658 261 640 278 562C287 522 290 483 293 434C223 269 110 105 23 9L32 -12C59 -6 85 0 108 7C152 64 251 252 300 366C307 297 315 221 337 82C346 24 363 -12 393 -12C425 -12 475 13 529 97Z" id="g113-233"></path></g><g transform="matrix(.013,0,0,-0.013,99.907,0)"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" id="g113-42"></path></g><g transform="matrix(.013,0,0,-0.013,104.405,0)"><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" id="g113-78"></path></g><g transform="matrix(.0091,0,0,-0.0091,117.212,-5.741)"><path d="M722 32C646 41 636 48 622 116L544 496L509 667L472 658L153 123C106 45 93 42 27 32L18 0H259L268 32C178 40 172 48 204 107L272 233H509C523 180 533 137 540 96C548 50 546 39 466 32L457 0H714L722 32ZM502 283H299L448 547H450L502 283Z" id="g50-66"></path></g></svg>.</span></div>

Security and Communication Networks

fig1

Figure 1

Figure 1: Path Hopping: An MTD Strategy for Long-Term Quantum-Safe Communication 