Kyber-768 | مرکز عملیات رمزنگاری و امنیت شبکه (ML-KEM)

شبیه‌ساز عملیاتی و محاسباتی رمزنگاری کایبر (Kyber-768 / ML-KEM-768)

پیاده‌سازی دقیق ریاضیات مشبکه مدولار بر اساس NIST FIPS 203 و کتاب A Gentle Introduction to Lattice-Based Cryptography

PARAMETERS: q=3329 | n=256 | k=3 | eta1=2 | eta2=2

مرحله اول - تولید کلید (Key Generation)

$t = A \cdot s + e \pmod{3329}$

در استاندارد Kyber-768 (ML-KEM-768)، ابعاد ماژول برابر با $$k=3$$ و درجه چندجمله‌ای‌ها برابر با $$n=256$$ در پیمانه $$q=3329$$ می‌باشد. باب ماتریس تصادفی $A \in R_q^{3 \times 3}$ را به همراه بردارهای خطای کوچک $s \in S_{\eta_1=2}^3$ (کلید خصوصی) و $e \in S_{\eta_1=2}^3$ از توزیع دوجمله‌ای مرکزیت‌یافته $$CBD_2$$ تولید می‌کند. سپس بردار کلید عمومی $t = As + e \pmod{3329}$ محاسبه می‌گردد.

$$ t = A \cdot s + e \pmod{3329} $$

۱. ماتریس تصادفی $A \in R_q^{3 \times 3}$ (شامل ۹ چندجمله‌ای درجه ۲۵۶ در $\mathbb{Z}_{3329}$ )

Row / Col	Column 1	Column 2	Column 3
Row 1	$\mathbf{A_{ 1, 1 }} = 1085 - 294x + 1387x^2 - 122x^3 + 129x^4 + 997x^5 - 508x^6 + 1021x^7 - 1420x^8 - 204x^9 + 72x^10 + 1079x^11 + 1589x^12 - 1598x^13 - 735x^14 - 756x^15 - 43x^16 + 950x^17 + 1258x^18 + 356x^19 - 532x^20 - 1553x^21 + 701x^22 + 489x^23 - 34x^24 - 646x^25 - 813x^26 - 109x^27 + 1306x^28 - 1163x^29 - 476x^30 - 1539x^31 + 1224x^32 - 1415x^33 + 1635x^34 + 347x^35 + 694x^36 - 1190x^37 + 355x^38 - 908x^39 + 1042x^40 - 486x^41 - 1136x^42 - 973x^43 + 1202x^44 - 1333x^45 + 1549x^46 + 850x^47 + 1481x^48 + 377x^49 - 1341x^50 + 880x^51 - 1440x^52 + 1597x^53 - 708x^54 - 726x^55 - 1602x^56 - 79x^57 - 1480x^58 - 1110x^59 + 298x^60 - 118x^61 - 938x^62 - 574x^63 + 863x^64 + 1625x^65 + 1122x^66 - 1585x^67 + 230x^68 - 571x^69 + 1230x^70 - 850x^71 + 1189x^72 - 181x^73 - 1617x^74 - 758x^75 - 1430x^76 - 774x^77 + 1398x^78 + 1337x^79 - 630x^80 + 532x^81 - 1004x^82 + 1385x^83 + 1055x^84 + 386x^85 - 1428x^86 - 72x^87 + 261x^88 + 1634x^89 + 433x^90 - 66x^91 - 960x^92 - 626x^93 + 3x^94 - 1466x^95 - 840x^96 - 860x^97 + 255x^98 + 962x^99 - 778x^100 - 1520x^101 + 1455x^102 - 1270x^103 - 639x^104 + 1538x^105 - 1641x^106 + 1370x^107 + 189x^108 + 1079x^109 - 1354x^110 + 862x^111 - 296x^112 + 1585x^113 + 540x^114 - 1414x^115 - 688x^116 - 1118x^117 + 765x^118 - 569x^119 + 250x^120 - 1606x^121 - 1491x^122 + 1464x^123 + 522x^124 + 576x^125 - 438x^126 + 96x^127 + 498x^128 + 1601x^129 - 746x^130 + 937x^131 - 150x^132 - 616x^133 + 327x^134 + 829x^135 + 1325x^136 - 427x^137 + 117x^138 - 1074x^139 - 122x^140 - 1285x^141 + 452x^142 - 1586x^143 + 1388x^144 + 878x^145 - 1499x^146 + 181x^147 + 1002x^148 - 1304x^149 + 1398x^150 - 689x^151 - 975x^152 + 371x^153 - 1030x^154 - 1662x^155 + 1213x^156 - 749x^157 - 930x^158 + 1578x^159 - 1126x^160 - 415x^161 - 853x^162 - 604x^163 - 826x^164 + 1204x^165 - 1032x^166 + 243x^167 + 393x^168 + 1548x^169 - 273x^170 - 86x^171 - 78x^172 + 45x^173 - 1496x^174 + 662x^175 + 526x^176 - 91x^177 + 907x^178 - 1225x^179 + 1096x^180 - 393x^181 - 1499x^182 + 22x^183 + 197x^184 + 697x^185 - 922x^186 - 600x^187 + 972x^188 + 264x^189 + 1634x^190 + 823x^191 - 116x^192 + 1344x^193 - 1236x^194 + 1388x^195 + 141x^196 + 105x^197 - 1141x^198 - 285x^199 - 1362x^200 - 164x^201 - 1490x^202 - 318x^203 + 637x^204 + 1112x^205 + 558x^206 - 1001x^207 - 451x^208 - 156x^209 + 1455x^210 + 1075x^211 + 759x^212 + 1115x^213 - 1535x^214 - 939x^215 + 898x^216 - 79x^217 - 1503x^218 + 1186x^219 + 695x^220 + 138x^221 - 1531x^222 + 1294x^223 - 185x^224 - 1533x^225 - 1291x^226 - 15x^227 - 1444x^228 + 752x^229 + 1630x^230 - 653x^231 - 1312x^232 - 173x^233 - 805x^234 - 1593x^235 + 1059x^236 - 522x^237 + 860x^238 + 567x^239 + 1411x^240 - 756x^241 + 33x^242 - 627x^243 - 323x^244 - 1431x^245 - 1267x^246 + 323x^247 - 1074x^248 + 802x^249 - 1608x^250 + 162x^251 + 850x^252 + 1337x^253 - 1309x^254 + 1593x^255$	$\mathbf{A_{ 1, 2 }} = 961 + 459x - 385x^2 - 1622x^3 + 812x^4 + 1582x^5 - 799x^6 + 957x^7 + 754x^8 + 1342x^9 + 295x^10 - 269x^11 + 1492x^12 - 1233x^13 - 1479x^14 - 1475x^15 - 809x^16 + 353x^17 - 204x^18 - 1077x^19 - 578x^20 + 1480x^21 - 387x^22 + 928x^23 - 780x^24 - 86x^25 + 544x^26 - 802x^27 + 1396x^28 - 719x^29 - 1041x^30 - 304x^31 + 379x^32 - 1510x^33 - 934x^34 - 628x^35 + 61x^36 + 1334x^37 - 1210x^38 + 665x^39 - 1475x^40 + 1660x^41 - 857x^42 + 297x^43 + 773x^44 - 197x^45 + 739x^46 + 351x^47 - 1592x^48 - 1062x^49 + 1374x^50 - 48x^51 - 1002x^52 + 300x^53 + 1060x^54 + 932x^55 + 540x^56 - 701x^57 + 1216x^58 - 1268x^59 - 403x^60 + 1490x^61 + 96x^62 - 118x^63 + 1420x^64 + 1102x^65 + 1170x^66 - 1425x^67 - 1233x^68 - 734x^69 + 293x^70 - 1299x^71 + 384x^72 + 1254x^73 + 786x^74 + 524x^75 - 234x^76 - 1350x^77 + 1464x^78 - 546x^79 - 951x^80 - 1227x^81 - 1432x^82 + 202x^83 - 586x^84 + 474x^85 + 207x^86 - 611x^87 - 1365x^88 + 1312x^89 - 52x^90 - 585x^91 + 305x^92 - 425x^93 + 1056x^94 - 421x^95 - 1434x^96 + 188x^97 + 1553x^98 - 819x^99 - 923x^100 + 695x^101 + 1353x^102 - 426x^103 - 126x^104 - 1450x^105 - 292x^106 - 1532x^107 - 1617x^108 + 1084x^109 - 754x^110 + 1357x^111 - 340x^112 + 1166x^113 + 424x^114 + 1436x^115 + 665x^116 - 379x^117 - 1253x^118 + 651x^119 + 1035x^120 - 1058x^121 + 1237x^122 - 32x^123 - 1398x^124 - 1192x^125 - 746x^126 - 1637x^127 + 1370x^128 + 1322x^129 + 1094x^130 + 448x^131 + 106x^132 + 256x^133 - 579x^134 - 1264x^135 + 1490x^136 - 6x^137 - 923x^138 - 1110x^139 + 1020x^140 - 65x^141 + 313x^142 - 410x^143 - 94x^144 - 1348x^145 + 503x^146 - 278x^147 - 1252x^148 + 25x^149 - 849x^150 - 1453x^151 - 419x^152 + 1022x^153 + 545x^154 - 1250x^155 - 1606x^156 - 1301x^157 - 688x^158 + 345x^159 + 1472x^160 - 361x^161 - 982x^162 + 243x^163 - 351x^164 + 1208x^165 - 122x^166 - 288x^167 - 151x^168 + 1662x^169 - 732x^170 - 1205x^171 - 767x^172 - 1602x^173 + 795x^174 + 870x^175 + 1106x^176 - 1102x^177 - 860x^178 + 725x^179 + 987x^180 + 829x^181 + 160x^182 - 579x^183 + 1647x^184 + 686x^185 - 881x^186 + 584x^187 - 382x^188 + 375x^189 - 311x^190 + 372x^191 - 575x^192 - 1455x^193 + 971x^194 + 466x^195 + 1110x^196 - 502x^197 - 19x^198 - 27x^199 + 855x^200 + 1518x^201 - 1551x^202 - 1517x^203 - 682x^204 + 414x^205 - 707x^206 + 777x^207 - 1384x^208 + 32x^209 + 1467x^210 + x^211 - 1390x^212 + 1200x^213 - 787x^214 + 649x^215 - 293x^216 - 981x^217 + 375x^218 + 366x^219 - 724x^220 + 1036x^221 - 1459x^222 - 1260x^223 - 940x^224 - 490x^225 + 1048x^226 + 331x^227 - 328x^228 - 1103x^229 - 663x^230 + 80x^231 + 660x^232 - 1259x^233 + 428x^234 + 1232x^235 + 1234x^236 - 1546x^237 - 96x^238 + 454x^239 + 1464x^240 - 1577x^241 + 691x^242 + 1299x^243 + 380x^244 + 1642x^245 + 1187x^246 + 812x^247 - 1373x^248 - 752x^249 + 1193x^250 + 1292x^251 + 1268x^252 - 553x^253 - 881x^254 + 1621x^255$	$\mathbf{A_{ 1, 3 }} = -897 - 738x + 1441x^2 - 1663x^3 + 1229x^4 - 379x^5 + 293x^6 + 1572x^7 - 1598x^8 + 1609x^9 + 701x^10 - 807x^11 + 1224x^12 - 390x^13 + 1445x^14 - 1276x^15 + 704x^16 + 1040x^17 - 922x^18 - 1426x^19 - 722x^20 - 806x^21 - 659x^22 + 1633x^23 - 272x^24 + 659x^25 - 350x^26 + 172x^27 + 1661x^28 + 172x^29 + 1108x^30 - 788x^31 - 668x^32 + 1387x^33 + 154x^34 + 400x^35 - 176x^36 + 626x^37 + 1013x^38 - 846x^39 - 751x^40 + 1356x^41 - 1129x^42 + 299x^43 + 1524x^44 + 1468x^45 - 915x^46 - 1322x^47 - 1146x^48 - 1031x^49 - 1357x^50 + 155x^51 - 18x^52 - 391x^53 - 81x^54 + 972x^55 + 171x^56 + 631x^57 + 1389x^58 + 245x^59 - 411x^60 + 371x^61 - 1224x^62 - 625x^63 + 121x^64 - 1110x^65 - 1610x^66 - 651x^67 + 1182x^68 - 856x^69 - 861x^70 + 900x^71 + 86x^72 + 926x^73 + 287x^74 + 883x^75 - 287x^76 - 1173x^77 + 325x^78 + 159x^79 + 1512x^80 - 1524x^81 + 147x^82 - 1287x^83 + 1649x^84 - 1415x^85 + 1551x^86 - 1139x^87 - 1370x^88 - 1277x^89 + 557x^90 - 1122x^91 - 695x^92 + 367x^93 + 1189x^94 + 1343x^95 + 1620x^96 - 439x^97 - 292x^98 + 1291x^99 - 936x^100 - 784x^101 - 1272x^102 + 31x^103 - 1075x^104 + 1626x^105 - 1569x^106 + 965x^107 - 697x^108 + 1331x^109 + 1612x^110 + 607x^111 + 1024x^112 - 1161x^113 - 579x^114 - 1608x^115 + 1218x^116 - 48x^117 + 148x^118 - 214x^119 + 1436x^120 - 1326x^121 + 79x^122 - 264x^123 - 246x^124 + 911x^125 - 1102x^126 + 828x^127 - 1222x^128 - 1645x^129 - 75x^130 - 453x^131 + 461x^132 + 60x^133 - 823x^134 + 25x^135 + 1385x^136 + 1098x^137 - 564x^138 - 305x^139 + 558x^140 - 639x^141 - 745x^142 - 87x^143 + 1371x^144 - 41x^145 + 1470x^146 + 597x^147 + 1555x^148 - 1231x^149 - 618x^150 - 1084x^151 + 613x^152 + 1554x^153 - 4x^154 - 469x^155 - 715x^156 - 652x^157 + 1004x^158 - 743x^159 + 346x^160 - 371x^161 - 1330x^162 + 988x^163 + 252x^164 + 1043x^165 + 845x^166 + 1633x^167 - 1126x^168 - 1518x^169 + 928x^170 - 686x^171 - 700x^172 + 128x^173 + 1621x^174 - 363x^175 + 424x^176 + 536x^177 - 741x^178 + 749x^179 - 1591x^180 - 422x^181 + 31x^182 - 1218x^183 - 1460x^184 + 835x^185 - 1284x^186 + 1355x^187 - 1032x^188 - 148x^189 + 267x^190 - 1360x^191 + 1218x^192 + 1066x^193 - 978x^194 + 2x^195 + 1631x^196 + 720x^197 + 647x^198 - 351x^199 - 637x^200 + 325x^201 + 1659x^202 - 1256x^203 + 1196x^204 - 975x^205 - 178x^206 - 738x^207 - 976x^208 + 1549x^209 - 1436x^210 + 635x^211 + 1252x^212 - 1426x^213 + 162x^214 + 1044x^215 + 1262x^216 + 1276x^217 + 1126x^218 + 246x^219 + 317x^220 - 1367x^221 + 566x^222 - 1089x^223 + 1624x^224 + 1064x^225 + 95x^226 + 134x^227 - 1550x^228 + 493x^229 + 1658x^230 - 416x^231 + 1331x^232 + 33x^233 - 783x^234 + 883x^235 + 1421x^236 + 255x^237 + 478x^238 + 1270x^239 - 159x^240 - 925x^241 - 1374x^242 + 763x^243 - 94x^244 - 1457x^245 - 829x^246 + 1341x^247 + 1143x^248 - 608x^249 + 1078x^250 + 210x^251 - 920x^252 - 459x^253 - 234x^254 - 1410x^255$
Row 2	$\mathbf{A_{ 2, 1 }} = -1419 - 639x + 581x^2 + 5x^3 + 1633x^4 - 434x^5 - 59x^6 - 626x^7 + 1554x^8 + 145x^9 - 877x^10 + 253x^11 + 668x^12 + 506x^13 + 362x^14 - 159x^15 - 70x^16 - 1539x^17 + 1350x^18 + 856x^19 - 160x^20 + 1336x^21 + 614x^22 - 1469x^23 - 1005x^24 - 223x^25 - 38x^26 + 1250x^27 - 1599x^28 + 1338x^29 + 834x^30 - 1031x^31 + 36x^32 + 1239x^33 - 203x^34 + 348x^35 + 1042x^36 + 1066x^37 + 1227x^38 - 400x^39 - 76x^40 + 1312x^41 - 134x^42 - 995x^43 - 227x^44 + 138x^45 - 885x^46 + 793x^47 - 154x^48 + 809x^49 + 1119x^50 - 328x^51 + 1106x^52 - 710x^53 + 318x^54 - 844x^55 + 485x^56 + 82x^57 - 867x^58 - 805x^59 + 1131x^60 + 454x^61 - 318x^62 - 1453x^63 - 648x^64 - 688x^65 + 1146x^66 + 1412x^67 - 1515x^68 - 1155x^69 - 1005x^70 - 105x^71 - 1246x^72 + 1013x^73 + 456x^74 - 1174x^75 - 1592x^76 + 1153x^77 + 1269x^78 - 762x^79 - 199x^80 - 917x^81 - 958x^82 - 977x^83 + 474x^84 + 1485x^85 - 997x^86 - 1361x^87 + 665x^88 + 1348x^89 - 422x^90 - 1255x^91 - 275x^92 + 733x^93 + 398x^94 - 78x^95 + 833x^96 + 392x^97 + 1490x^98 + 89x^99 - 303x^100 - x^101 - 142x^102 - 1630x^103 - 1246x^104 + 1078x^105 - 595x^106 + 123x^107 - 1620x^108 + 1453x^109 - 1314x^110 + 1570x^111 + 881x^112 - 762x^113 - 1514x^114 - 1080x^115 - 1084x^116 + 1296x^117 - 794x^118 + 188x^119 + 961x^120 + 629x^121 + 1249x^122 + 1122x^123 - 36x^124 - 269x^125 + 1405x^126 + 1473x^127 + 480x^128 - 1413x^129 - 184x^130 - 774x^131 - 1651x^132 - 474x^133 - 957x^134 - 862x^135 + 136x^136 + 795x^137 + 1266x^138 - 143x^139 + 540x^140 - 1081x^141 - 99x^142 - 1019x^143 + 1046x^144 + 334x^145 + 267x^146 + 1633x^147 - 1400x^148 + 1531x^149 + 909x^150 - 1142x^151 + 49x^152 - 222x^153 - 841x^154 - 1482x^155 - 401x^156 - 1309x^157 - 687x^158 - 1119x^159 + 199x^160 + 438x^161 + 689x^162 + 1148x^163 - 1436x^164 + 373x^165 - 1025x^166 - 856x^167 - 762x^168 - 1578x^169 - 89x^170 - 1305x^171 - 558x^172 + 129x^173 + 780x^174 - 1568x^175 + 1139x^176 + 24x^177 + 999x^178 + 223x^179 - 625x^180 - 1506x^181 - 600x^182 - 1499x^183 - 1102x^184 - 871x^185 + 1160x^186 + 460x^187 - 762x^188 + 1497x^189 + 1339x^190 - 623x^191 + 366x^192 + 310x^193 + 1327x^194 - 137x^195 - 198x^196 - 1535x^197 - 649x^198 + 609x^199 - 1354x^200 - 1606x^201 - 1046x^202 - 525x^203 - 1059x^204 + 357x^205 - 146x^206 + 1310x^207 - 553x^208 + 1020x^209 - 441x^210 - 1315x^211 + 269x^212 - 377x^213 + 574x^214 - 791x^215 + 548x^216 - 634x^217 - 1419x^218 - 401x^219 - 1199x^220 + 562x^221 + 1219x^222 - 216x^223 - 84x^224 + 1568x^225 + 1058x^226 - 845x^227 + 219x^228 - 656x^229 + 632x^230 + 1364x^231 - 925x^232 - 1069x^233 + 421x^234 - 573x^235 + 1318x^236 + 373x^237 - 417x^238 + 1564x^239 - 806x^240 - 1017x^241 - 1215x^242 + 566x^243 - 1349x^244 + 638x^245 + 1194x^246 - 36x^247 + 1171x^248 + 1035x^249 - 1657x^250 + 344x^251 - 1395x^252 + 763x^253 - 570x^254 + 632x^255$	$\mathbf{A_{ 2, 2 }} = -1530 + 180x - 33x^2 - 1419x^3 - 308x^4 + 1347x^5 + 111x^6 - 47x^7 - 1136x^8 - 700x^9 + 480x^10 + 1549x^11 - 708x^12 + 1624x^13 - 448x^14 + 129x^15 + 1327x^16 + 456x^17 + 1238x^18 - 1663x^19 + 1549x^20 - 315x^21 - 540x^22 + 702x^23 + 1290x^24 - 291x^25 - 488x^26 + 1476x^27 + 1336x^28 + 538x^29 - 1006x^30 - 1068x^31 + 341x^32 - 1211x^33 - 774x^34 - 535x^35 - 1110x^36 + 503x^37 + 320x^38 - 328x^39 - 1567x^40 - 601x^41 - 906x^42 - 100x^43 + 1578x^44 + 1550x^45 + 1121x^46 - 52x^47 + 204x^48 - 568x^49 + 187x^50 - 825x^51 - 344x^52 - 813x^53 - 1401x^54 + 1000x^55 - 1024x^56 - 1280x^57 + 218x^58 + 1364x^59 + 479x^60 + 1532x^61 - 319x^62 + 1479x^63 + 35x^64 - 1633x^65 - 12x^66 + 601x^67 + 1243x^68 - 1432x^69 - 400x^70 + 1341x^71 + 1369x^72 - 1150x^73 - 397x^74 - 378x^75 - 1032x^76 - 898x^77 - 70x^78 + 601x^79 + 41x^80 - 26x^81 + 231x^82 + 1059x^83 - 1076x^84 - 1523x^85 - 209x^86 + 112x^87 + 28x^88 + 662x^89 + 238x^90 - 403x^91 + 280x^92 - 531x^93 - 1458x^94 + 867x^95 + 236x^96 - 1041x^97 + 1132x^98 - 1523x^99 - 546x^100 - 831x^101 + 565x^102 + 1097x^103 + 402x^104 - 1564x^105 - 901x^106 - 1074x^107 - 849x^108 - 472x^109 - 1109x^110 - 729x^111 - 1207x^112 - 237x^113 - 1242x^114 + 970x^115 - 331x^116 + 1505x^117 - 1131x^118 + 980x^119 + 710x^120 + 1483x^121 + 1055x^122 + 732x^123 - 1582x^124 - 1216x^125 - 708x^126 - 1443x^127 - 402x^128 + 872x^129 + 715x^130 - 469x^131 - 582x^132 + 1556x^133 + 179x^134 - 25x^135 + 328x^136 + 1225x^137 + 235x^138 - 1389x^139 - 912x^140 - 1246x^141 - 1387x^142 - 1524x^143 - 1613x^144 + 168x^145 - 1017x^146 - 541x^147 - 1274x^148 - 1454x^149 - 1451x^150 - 1650x^151 + 92x^152 + 1097x^153 + 242x^154 + 1469x^155 + 633x^156 + 1062x^157 + 1170x^158 - 147x^159 - 79x^160 + 1228x^161 - 1656x^162 - 1136x^163 - 25x^164 + 188x^165 - 38x^166 + 183x^167 + 602x^168 - 579x^169 + 1406x^170 + 523x^171 + 985x^172 - 1407x^173 + 299x^174 - 204x^175 + 541x^176 - 416x^177 + 322x^178 - 344x^179 + 359x^180 - 424x^181 - 995x^182 - 828x^183 - 453x^184 - 857x^185 + 1654x^186 - 724x^187 + 745x^188 - 33x^189 - 1286x^190 + 1662x^191 + 802x^192 - 853x^193 + 704x^194 + 830x^195 - 1109x^196 + 367x^197 + 276x^198 + 1285x^199 - 749x^200 + 514x^201 - 834x^202 - 1575x^203 - 992x^204 + 762x^205 + 984x^206 - 1240x^207 - 619x^208 + 1096x^209 + 1098x^210 - 345x^211 + 1272x^212 + 855x^213 + 374x^214 - 162x^215 + 689x^216 - 55x^217 + 416x^218 + 414x^219 + 1254x^220 - 1238x^221 + 176x^222 - 715x^223 - 614x^224 + 1033x^225 + 745x^226 + 1421x^227 - 406x^228 - 1538x^229 - 801x^230 - 1404x^231 - 246x^232 - 734x^233 - 1324x^234 + 821x^235 + 401x^236 - 995x^237 - 1144x^238 + 22x^239 - 551x^240 + 1662x^241 + 1180x^242 - 238x^243 + 796x^244 - 1026x^245 - 254x^246 - 1375x^247 + 1136x^248 + 877x^249 - 1301x^250 + 672x^251 - 342x^252 - 188x^253 + 594x^254 - 525x^255$	$\mathbf{A_{ 2, 3 }} = 692 + 735x - 453x^2 + 1010x^3 - 219x^4 + 1109x^5 + 1624x^6 + 729x^7 - 1186x^8 - 265x^9 + 218x^10 + 1245x^11 - 1202x^12 - 916x^13 + 973x^14 + 1044x^15 + 598x^16 - 1139x^17 + 1481x^18 + 1579x^19 - 502x^20 + 1059x^21 - 1057x^22 - 945x^23 + 1603x^24 + 1325x^25 - 1363x^26 - 933x^27 - 781x^28 + 901x^29 + 1351x^30 + 59x^31 + 101x^32 - 1567x^33 + 1421x^34 + 773x^35 + 346x^36 - 374x^37 + 1629x^38 - 898x^39 - 972x^40 - 493x^41 + 596x^42 + 930x^43 + 1414x^44 - 480x^45 + 165x^46 - 998x^47 - 780x^48 - 402x^49 + 39x^50 + 754x^51 + 21x^52 + 1577x^53 + 1111x^54 + 795x^55 - 1012x^56 - 1046x^57 - 879x^58 - 334x^59 - 178x^60 - 986x^61 + 1323x^62 - 924x^63 + 1070x^64 - 945x^65 + 283x^66 + 855x^67 + 929x^68 - 941x^69 - 76x^70 - 530x^71 - 134x^72 - 392x^73 - 1282x^74 + 989x^75 + 611x^76 + 1516x^77 + 252x^78 + 164x^79 - 355x^80 + 1316x^81 - 488x^82 - 587x^83 - 1529x^84 + 609x^85 + 1472x^86 - 1383x^87 + 650x^88 + 1419x^89 + 1346x^90 - 147x^91 - 308x^92 + 1366x^93 - 1168x^94 - 1549x^95 - 517x^96 + 260x^97 + 706x^98 - 72x^99 - 868x^100 - 151x^101 + 879x^102 - 1611x^103 + 24x^104 - 982x^105 + 1640x^106 + 913x^107 - 969x^108 + 1502x^109 + 1328x^110 + 579x^111 - 195x^112 - 714x^113 + 640x^114 + 1187x^115 - 472x^116 - 490x^117 + 283x^118 + 1195x^119 - 220x^120 + 199x^121 + 303x^122 - 654x^123 + 455x^124 - 1373x^125 - 585x^126 - 1214x^127 - 1460x^128 - 353x^129 - 630x^130 - 452x^131 - 1190x^132 + 542x^133 - 525x^134 - 200x^135 + 453x^136 - 343x^137 + 1374x^138 - 310x^139 + 1147x^140 + 1513x^141 - 721x^142 - 629x^143 - 1444x^144 - 558x^145 + 192x^146 + 81x^147 + 1352x^148 - 1246x^149 - 1595x^150 - 1113x^151 - 825x^152 + 305x^153 + 1055x^154 + 1337x^155 - 159x^156 + 841x^157 + 822x^158 - 1071x^159 + 1092x^160 + 401x^161 + 479x^162 - 13x^163 + 67x^164 + 1128x^165 + 1322x^166 + 50x^167 - 41x^168 + 1324x^169 - 1095x^170 + 457x^171 + 1257x^172 + 634x^173 - 1409x^174 - 666x^175 + 1610x^176 + 565x^177 + 14x^178 + 1192x^179 + 300x^180 - 1406x^181 - 916x^182 + 1160x^183 - 1412x^184 - 418x^185 - 1101x^186 - 214x^187 + 1618x^188 - 50x^189 - 821x^190 - 928x^191 - 14x^192 - 822x^193 + 430x^194 - 1409x^195 + 873x^196 - 912x^197 - 945x^198 - 912x^199 - 1198x^200 + 84x^201 - 1143x^202 - 151x^203 - 1069x^204 + 984x^205 + 1535x^206 - 1341x^207 + 999x^208 - 77x^209 + 774x^210 - 400x^211 + 518x^212 + 1275x^213 - 1270x^214 - 259x^215 - 830x^216 - 490x^217 + 1311x^218 - 1041x^219 - 1349x^220 + 683x^221 + 650x^222 + 701x^223 - 395x^224 + 933x^225 - 1000x^226 - 125x^227 - 1316x^228 + 166x^229 + 890x^230 + 169x^231 + 865x^232 - 1513x^233 - 437x^234 + 1366x^235 - 733x^236 + 769x^237 - 279x^238 - 1647x^239 + 703x^240 - 1663x^241 + 1532x^242 + 669x^243 + 580x^244 - 15x^245 + 545x^246 + 379x^247 - 1218x^248 - 392x^249 - 184x^250 - 1079x^251 + 1149x^252 + 481x^253 + 968x^254 - 73x^255$
Row 3	$\mathbf{A_{ 3, 1 }} = -1209 - 871x + 58x^2 + 716x^3 - 830x^4 - 459x^5 - 985x^6 + 608x^7 - 93x^8 - 1599x^9 + 643x^10 + 294x^11 + 554x^12 + 256x^13 + 727x^14 + 1377x^15 + 832x^16 + 1190x^17 - 1069x^18 + 234x^19 - 1576x^20 - 1181x^21 + 1406x^22 - 879x^23 + 373x^24 + 1566x^25 - 932x^26 + 1104x^27 - 1576x^28 - 679x^29 - 380x^30 - 993x^31 - 274x^32 + 28x^33 - 928x^34 + 383x^35 - 1112x^36 + 1218x^37 + 609x^38 + 918x^39 - 250x^40 - 649x^41 + 1413x^42 - 1375x^43 + 477x^44 - 1646x^45 - 1164x^46 + 115x^47 + 532x^48 - 274x^49 - 242x^50 - 1222x^51 - 950x^52 - 1273x^53 - 887x^54 - 104x^55 + 891x^56 + 168x^57 + 1604x^58 + 346x^59 + 1139x^60 + 660x^61 + 1388x^62 - 112x^63 + 479x^64 + 138x^65 - 1217x^66 - 1411x^67 + 16x^68 - 1466x^69 + 174x^70 - 965x^71 + 860x^72 + 1324x^73 + 101x^74 + 481x^75 + 1651x^76 - 1556x^77 + 322x^78 + 118x^79 - 980x^80 - 802x^81 + 1490x^82 - 938x^83 + 515x^84 + 596x^85 + 1422x^86 - 684x^87 + 713x^88 - 1146x^89 + 395x^90 + 1548x^91 - 1282x^92 + 1485x^93 - 1187x^94 + 317x^95 - 1505x^96 - 1084x^97 - 1231x^98 - 1536x^99 - 223x^100 + 1246x^101 - 1238x^102 + 361x^103 - 284x^104 - 856x^105 + 685x^106 + 959x^107 - 1661x^108 + 818x^109 - 38x^110 + 6x^111 + 1564x^112 - 394x^113 + 1486x^114 - 388x^115 + 1008x^116 - 1008x^117 + 894x^118 - 1161x^119 - 376x^120 - 560x^121 + 1111x^122 - 1321x^123 - 1434x^124 - 578x^125 - 1611x^126 - 791x^127 - 85x^128 + 1627x^129 - 168x^130 - 321x^131 - 126x^132 + 1469x^133 + 766x^134 - 651x^135 + 636x^136 - 747x^137 - 224x^138 - 1555x^139 - 1643x^140 + 169x^141 - 427x^142 + 1222x^143 + 1235x^144 - 1651x^145 + 558x^146 + 1165x^147 + 1243x^148 - 927x^149 + 878x^150 - 82x^151 - 97x^152 - 1407x^153 + 59x^154 + 1425x^155 - 639x^156 + 1517x^157 + 821x^158 - 864x^159 - 1316x^160 + 264x^161 - 52x^162 - 153x^163 + 1308x^164 - 115x^165 - 1361x^166 - 455x^167 + 910x^168 + 780x^169 - 1493x^170 - 606x^171 + 1361x^172 + 608x^173 + 1453x^174 + 671x^175 + 1345x^176 - 205x^177 + 1547x^178 - 863x^179 - 1311x^180 + 991x^181 + 601x^182 - 652x^183 - 336x^184 + 1645x^185 + 73x^186 - 643x^187 - 574x^188 - 603x^189 - 1039x^190 - 1431x^191 + 582x^192 + 1427x^193 - 1327x^194 - 1352x^195 - 1402x^196 + 139x^197 + 390x^198 - 1561x^199 + 1581x^200 + 474x^201 - 1199x^202 + 536x^203 - 1456x^204 - 92x^205 + 763x^206 - 371x^207 - 591x^208 - 4x^209 + 714x^210 + 1632x^211 + 180x^212 - 590x^213 + 584x^214 - 1200x^215 + 1167x^216 + 598x^217 + 725x^218 + 883x^219 + 575x^220 - 928x^221 + 806x^222 + 1598x^223 + 903x^224 - 630x^225 - 622x^226 + 896x^227 - 1430x^228 - 1002x^229 + 1421x^230 + 1450x^231 + 652x^232 + 1557x^233 - 1367x^234 + 1541x^235 + 151x^236 + 1459x^237 - 368x^238 + 904x^239 - 1007x^240 - 29x^241 + 1577x^242 - 242x^243 - 292x^244 - 604x^245 + 344x^246 + 197x^247 + 1664x^248 + 53x^249 - 424x^250 - 1548x^251 - 906x^252 - 1594x^253 - 1592x^254 - 641x^255$	$\mathbf{A_{ 3, 2 }} = -716 + 1060x - 1632x^2 + 562x^3 + 750x^4 + 1181x^5 + 903x^6 + 1369x^7 + 1324x^8 + 44x^9 + 1425x^10 + 810x^11 - 129x^12 - 457x^13 - 634x^14 - 184x^15 - 369x^16 + 1086x^17 + 826x^18 + 868x^19 - 536x^20 + 681x^21 + 1090x^22 + 1624x^23 - 1624x^24 - 518x^25 + 848x^26 - 602x^27 + 1543x^28 + 108x^29 - 221x^30 - 859x^31 + 1419x^32 - 510x^33 + 1342x^34 - 1041x^35 - 179x^36 - 678x^37 + 1011x^38 + 541x^39 + 377x^40 + 1236x^41 - 1143x^42 - 1079x^43 + 46x^44 - 89x^45 + 790x^46 - 199x^47 - 259x^48 - 1255x^49 - 1491x^50 - 1514x^51 - 1135x^52 + 1241x^53 + 1407x^54 - 923x^55 + 427x^56 + 242x^57 - 321x^58 + 914x^59 + 973x^60 + 545x^61 + 335x^62 + 203x^63 - 415x^64 - 1040x^65 - 477x^66 + 875x^67 - 435x^68 + 1487x^69 - 447x^70 - 972x^71 + 679x^72 + 408x^73 + 1487x^74 - 1443x^75 + 648x^76 - 255x^77 - 123x^78 + 1209x^79 + 383x^80 - 1436x^81 - 1429x^82 + 1544x^83 - 1619x^84 + 828x^85 + 1041x^86 + 491x^87 - 309x^88 - 209x^89 + 1326x^90 + 549x^91 - 365x^92 + 711x^93 - 385x^94 - 1220x^95 - 1504x^96 - 160x^97 + 1087x^98 + 305x^99 + 264x^100 + 32x^101 - 1469x^102 + 214x^103 + 1621x^104 - 1631x^105 - 1100x^106 + 526x^107 - 232x^108 + 331x^109 - 835x^110 - 562x^111 - 1076x^112 + 1295x^113 - 97x^114 + 369x^115 + 1552x^116 - 608x^117 - 1172x^118 - 489x^119 - 1204x^120 - 1446x^121 - 1396x^122 + 786x^123 + 923x^124 - 598x^125 + 373x^126 + 1297x^127 - 1070x^128 + 1533x^129 + 1005x^130 - 244x^131 + 1369x^132 - 431x^133 + 1424x^134 + 480x^135 - 399x^136 + 714x^137 - 1271x^138 - 547x^139 + 1013x^140 + 970x^141 + 989x^142 + 776x^143 - 1543x^144 + 113x^145 - 757x^146 + 262x^147 + 1055x^148 - 254x^149 - 372x^150 - 890x^151 - 360x^152 - 1640x^153 - 1380x^154 - 119x^155 + 1617x^156 + 848x^157 + 1475x^158 - 74x^159 - 659x^160 - 416x^161 + 1311x^162 - 460x^163 - 1022x^164 - 638x^165 + 1568x^166 - 952x^167 + 382x^168 + 69x^169 + 1449x^170 - 16x^171 - 951x^172 + 572x^173 - 1008x^174 + 811x^175 + 963x^176 - 673x^177 - 223x^178 + 267x^179 - 1479x^180 - 710x^181 + 758x^182 + 637x^183 - 163x^184 - 1602x^185 - 103x^186 - 683x^187 - 550x^188 - 1662x^189 - 1642x^190 - 302x^191 - 260x^192 - 897x^193 + 690x^194 + 1081x^195 - 1048x^196 + 1356x^197 - 572x^198 - 362x^199 - 322x^200 + 1319x^201 - 1197x^202 + 80x^203 + 915x^204 + 1659x^205 + 986x^206 - 560x^207 + 883x^208 + 1256x^209 - 905x^210 - 233x^211 + 1405x^212 + 1433x^213 - 1618x^214 - 1120x^215 - 1648x^216 - 536x^217 + 1078x^218 - 493x^219 + 1472x^220 - 945x^221 + 1512x^222 - 1133x^223 - 609x^224 - 37x^225 - 1456x^226 - 225x^227 + 44x^228 - 447x^229 - 362x^230 + 420x^231 + 1020x^232 - 751x^233 - 960x^234 + 1540x^235 + 1244x^236 + 67x^237 + 31x^238 + 588x^239 + 1625x^240 - 1249x^241 - 559x^242 + 401x^243 + 357x^244 + 1195x^245 + 1231x^246 - 1004x^247 + 284x^248 + 213x^249 + 24x^250 - 897x^251 - 1385x^252 - 244x^253 - 147x^254 + 636x^255$	$\mathbf{A_{ 3, 3 }} = -431 - 21x - 1121x^2 + 1605x^3 + 214x^4 - 402x^5 + 888x^6 - 572x^7 + 824x^8 + 518x^9 - 473x^10 + 1185x^11 - 1448x^12 - 1170x^13 - 227x^14 - 934x^15 + 256x^16 - 1148x^17 - 837x^18 - 390x^19 - 1014x^20 + 351x^21 - 1276x^22 - 1396x^23 - 428x^24 - 468x^25 - 1475x^26 + 331x^27 + 171x^28 + 438x^29 + 1373x^30 - 193x^31 - x^32 - 1143x^33 - 363x^34 + 205x^35 + 395x^36 + 1564x^37 + 1623x^38 - 1155x^39 + 1582x^40 + 493x^41 - 384x^42 - 1361x^43 - 1614x^44 - 573x^45 - 1141x^46 + 1071x^47 + 1494x^48 + 608x^49 - 1113x^50 - 1217x^51 + 8x^52 - 1023x^53 + 260x^54 + 80x^55 - 1244x^56 - 129x^57 - 1306x^58 + 646x^59 - 900x^60 + 169x^61 - 224x^62 - 43x^63 - 974x^64 + 683x^65 + 972x^66 - 357x^67 - 378x^68 + 268x^69 - 1394x^70 - 103x^71 + 285x^72 - 1437x^73 - 112x^74 + 1464x^75 - 124x^76 - 114x^77 - 1126x^78 + 1571x^79 - 1478x^80 + 242x^81 + 415x^82 - 1521x^83 + 120x^84 - 1416x^85 + 1169x^86 + 217x^87 - 832x^88 + 72x^89 - 23x^90 + 418x^91 - 1175x^92 - 920x^93 - 785x^94 - 173x^95 - 1194x^96 + 828x^97 - 1137x^98 + 828x^99 + 1543x^100 + 1129x^101 - 482x^102 + 1249x^103 + 128x^104 + 1635x^105 - 77x^106 + 381x^107 + 1490x^108 + 279x^109 + 695x^110 - 1378x^111 - 43x^112 - 827x^113 + 1603x^114 - 251x^115 + 1607x^116 - 1548x^117 - 539x^118 + 1197x^119 - 693x^120 + 1040x^121 - 1507x^122 - 102x^123 + 1327x^124 + 1145x^125 + 1548x^126 - 446x^127 - 834x^128 + 369x^129 + 1154x^130 + 107x^131 - 206x^132 - 615x^133 + 1073x^134 + 26x^135 + 336x^136 - 1556x^137 + 1152x^138 + 926x^139 + 1141x^140 + 1394x^141 - 1159x^142 - 294x^143 + 1621x^144 - 865x^145 - 1314x^146 + 260x^147 - 256x^148 + 204x^149 - 1087x^150 - 1570x^151 + 172x^152 - 1262x^153 - 1076x^154 + 1214x^155 + 464x^156 + 806x^157 + 11x^158 - 436x^159 + 1562x^160 + 1514x^161 - 1489x^162 + 1534x^163 + 1303x^164 + 163x^165 - 687x^166 + 1063x^167 + 882x^168 + 1627x^169 + 864x^170 - 2x^171 + 234x^172 - 1636x^173 - 1033x^174 - 225x^175 - 1365x^176 - 315x^177 - 1531x^178 - 1428x^179 + 1656x^180 - 782x^181 + 90x^182 - 1144x^183 + 514x^184 + 851x^185 - 1002x^186 + 276x^187 - 1091x^188 - 1453x^189 + 1549x^190 - 201x^191 - 1048x^192 - 1562x^193 + 1391x^194 + 1338x^195 - 1112x^196 + 161x^197 + 1579x^198 - 1473x^199 - 1106x^200 - 734x^201 - 739x^202 - 1048x^203 + 414x^204 - 1145x^205 - 683x^206 - 1074x^207 + 104x^208 - 416x^209 + 341x^210 - 1560x^212 + 193x^213 + 1326x^214 - 1364x^215 + 1085x^216 - 287x^217 + 541x^218 + 1004x^219 + 85x^220 - 1391x^221 + 816x^222 + 1399x^223 + 514x^224 + 217x^225 - 365x^226 + 925x^227 - 218x^228 - 396x^229 + 1125x^230 - 582x^231 + 266x^232 + 1461x^233 + 69x^234 - 1432x^235 - 1231x^236 - 1411x^237 - 424x^238 - 1119x^239 - 960x^240 + 1581x^241 + 1124x^242 - 111x^243 + 1216x^244 + 701x^245 + 1467x^246 - 1066x^247 + 1119x^248 - 1414x^249 + 279x^250 + 710x^251 + 298x^252 + 799x^253 - 788x^254 + 1060x^255$

بردار کلید خصوصی $s \in CBD_2^3$

$s_{ 1 } = -1 + x - x^2 - x^3 + x^4 - x^5 - x^8 - 2x^10 - x^11 - x^12 - 2x^13 + x^14 - x^16 + 2x^17 - x^18 + 2x^19 - 2x^21 - x^22 + x^23 - x^26 - x^32 + 2x^36 + x^37 - x^40 - x^41 - x^42 - x^44 - 2x^45 + x^46 - 2x^47 + x^48 + x^49 + x^51 + 2x^52 + x^53 - x^54 + 2x^55 - x^56 + x^58 + x^60 - 2x^61 - x^63 + x^66 + x^68 + x^69 - 2x^70 + x^71 - x^72 - x^76 + x^77 - x^78 + x^79 - x^82 + 2x^86 + x^89 + x^90 - x^92 - x^93 + x^94 + 2x^96 - 2x^97 - x^98 - x^102 - x^103 - x^104 + x^105 - x^106 - x^107 - x^108 - x^109 - x^111 + x^114 - x^115 + x^117 - 2x^118 + x^119 - x^121 - 2x^125 + x^126 + x^128 - x^129 - x^131 - 2x^132 + x^133 - x^134 - x^136 - x^137 + 2x^138 - x^140 + x^141 + x^143 + x^144 - x^146 - 2x^147 - x^149 - x^151 + x^153 - x^154 + x^157 + 2x^158 + 2x^159 - x^160 - x^163 - x^164 - x^167 - x^168 - x^170 + x^171 + x^173 - x^174 + 2x^178 + 2x^180 + x^182 + x^183 - x^184 - x^188 + x^189 + x^190 + x^193 + x^194 - x^195 + x^196 - x^197 + x^199 + x^200 + x^201 - x^202 - x^205 + x^207 - x^208 + x^210 - x^211 - x^213 + x^214 + x^215 - x^216 - 2x^218 - x^219 + x^221 + x^222 - x^223 + x^225 + x^226 - x^227 - 2x^231 - x^232 + 2x^233 - 2x^235 - 2x^236 + x^237 + x^238 - x^241 + x^242 + x^251 - x^252$

$s_{ 2 } = -1 - x^3 + x^4 + x^6 - x^7 + x^8 + x^10 + x^11 + x^13 - x^14 + 2x^16 + 2x^17 + x^18 - x^19 + x^20 + x^23 - x^25 + x^27 + x^28 + x^29 + x^30 + x^32 - x^33 - x^37 - x^38 - x^39 - 2x^40 + x^42 + x^45 - 2x^46 - x^47 + x^50 + x^51 + 2x^53 + x^54 + x^55 - 2x^56 + 2x^57 + x^59 + x^60 - x^61 + x^62 - x^63 + x^65 - x^66 + x^67 - 2x^69 - 2x^71 + 2x^72 - x^74 - 2x^75 - x^77 - x^78 + x^79 - x^80 - 2x^83 + x^84 - x^85 + x^87 - 2x^88 + x^89 - x^91 - x^92 - x^93 - x^94 + x^95 - x^96 + x^99 - x^100 - x^101 - x^106 + x^107 - x^111 - x^113 + x^114 - x^115 - x^117 + x^119 + x^122 + x^124 - x^125 - x^126 - x^127 + x^128 + x^129 - x^130 - x^131 - 2x^133 - x^134 - 2x^135 + x^139 - x^140 - x^142 + x^147 + x^148 + 2x^149 + x^151 - x^154 - x^155 + x^156 - x^157 - x^158 - 2x^162 - x^163 - x^165 - x^166 - 2x^167 - x^169 - 2x^170 + x^171 - x^173 + x^174 + x^176 + x^181 - x^184 + 2x^187 - x^189 - x^192 + 2x^193 - x^196 + 2x^198 + x^199 + x^200 - x^205 + x^209 - x^210 - x^211 + x^212 - 2x^213 + x^214 + 2x^215 + 2x^216 + x^219 + 2x^221 + 2x^222 + x^223 - x^225 + x^226 - x^227 - x^228 + 2x^230 - x^231 + x^232 + x^233 + x^235 + 2x^237 + x^238 - x^242 + x^244 - 2x^245 + x^247 - x^248 - x^250 + x^252 + x^253$

$s_{ 3 } = -1 - x - x^3 - x^7 - x^10 - x^11 + x^12 + x^13 + x^15 - x^18 + x^19 + 2x^20 - x^22 + x^23 + x^25 - x^27 - x^29 - x^30 - x^31 + x^33 - x^34 + x^36 + x^37 + x^38 + x^40 + x^41 + x^42 - x^44 - x^47 + x^48 + 2x^49 + x^50 + 2x^52 + x^55 - 2x^56 - x^59 - x^61 - x^65 + x^66 - x^67 + x^69 - x^71 + x^72 + x^73 + x^74 - x^75 + x^78 + x^80 + x^81 - x^82 + x^83 + x^85 - x^86 + x^87 + x^90 - x^91 - x^94 + x^95 + x^96 + x^100 - x^101 + 2x^102 - x^103 - x^105 + x^106 - 2x^107 + 2x^108 - 2x^110 + x^112 + x^113 - x^114 - x^115 - x^116 + x^117 + 2x^118 - x^119 - x^121 + x^122 + x^123 - 2x^125 - x^126 - 2x^128 + x^130 - x^134 + 2x^137 + x^141 + x^143 + x^144 - x^146 + x^147 - x^149 - 2x^150 - x^151 - x^152 - x^153 + 2x^155 + x^157 + x^158 - 2x^159 - x^160 + 2x^162 - x^163 + x^164 + x^167 - x^168 - x^170 + x^171 - 2x^172 + x^174 + x^175 - x^176 - x^177 + x^179 + x^180 - 2x^182 + x^185 - x^186 + x^188 - 2x^194 - 2x^195 - x^199 - x^200 + x^202 - x^203 - x^204 + x^206 + x^207 + x^208 + x^209 - x^210 - x^211 - x^212 + 2x^213 + x^216 + x^217 + x^219 + x^220 - x^221 - x^223 + x^224 - x^225 - x^227 + 2x^228 - x^229 + x^230 + x^231 - x^232 + x^233 + x^236 - x^237 - x^238 + x^239 + x^241 + x^242 + 2x^243 - x^245 + x^246 - 2x^247 + x^248 - x^251 - x^253 + x^255$

$\|s\|_\infty = 2 \le \eta_1 = 2$

بردار خطای تصادفی $e \in CBD_2^3$

$e_{ 1 } = -1 - 2x - x^3 - x^4 + x^5 + x^7 + 2x^10 + 2x^12 - 2x^13 + x^15 - x^16 + x^17 - x^18 + 2x^19 - x^20 + x^21 - 2x^22 - x^24 + x^25 - x^32 - x^33 - x^34 + x^35 - x^37 - x^40 + x^42 - 2x^43 - x^44 - 2x^45 - x^46 - x^47 - 2x^48 - x^49 - x^50 - x^54 + x^57 - x^59 + x^60 + 2x^62 - x^63 + x^64 - x^66 - x^69 + x^72 + x^75 - x^76 - x^78 - x^79 - x^80 + x^81 - x^83 + x^84 - x^85 + x^86 + x^87 - x^89 - x^90 + 2x^91 - x^93 - x^95 + x^99 + x^100 + x^101 + 2x^102 - 2x^104 - x^105 + x^107 - 2x^108 + x^109 + 2x^110 + 2x^111 + x^114 - 2x^115 + x^116 + x^117 - x^120 - x^122 - x^125 - 2x^126 - x^127 + x^128 - x^130 - x^131 - x^133 - x^135 - x^136 + x^137 - x^139 + x^140 + x^141 + x^142 - x^143 + x^144 + 2x^145 + x^147 - x^149 - x^150 + x^151 - 2x^152 + x^153 - x^154 - x^156 - x^157 + x^158 - 2x^159 - x^160 + 2x^162 + x^163 - x^164 - x^165 + x^166 + x^167 + x^168 - x^169 - x^170 - x^171 - x^174 - 2x^176 + 2x^179 + x^180 - x^181 - x^182 + 2x^184 + x^185 + x^186 + x^187 - x^189 - x^190 - x^191 - x^192 + x^194 + x^196 + x^197 - x^198 - x^199 - x^200 - x^202 + x^203 - x^204 - x^205 - 2x^207 - x^209 + x^210 - x^212 + x^214 - x^215 + 2x^218 - x^219 + x^220 - x^221 + x^223 - x^224 - x^225 + x^226 + 2x^227 + x^229 + x^230 - x^232 + x^234 - x^235 + x^236 - 2x^237 + x^238 - x^240 + 2x^241 - 2x^242 + x^243 - 2x^244 + x^247 - x^248 + x^249 + 2x^254$

$e_{ 2 } = 2x + x^4 - 2x^5 + x^6 - x^7 + x^8 + x^10 - x^12 + x^13 - 2x^14 + x^16 + x^17 - 2x^19 + x^20 + x^21 + 2x^22 + x^23 - x^24 + x^26 - x^27 + x^28 - x^29 - x^30 - x^31 + x^32 - x^33 + x^36 + x^37 - x^38 - 2x^40 + x^44 - x^45 - 2x^48 - x^50 - x^51 + x^53 - x^54 - x^55 + x^56 - x^58 - 2x^60 + x^61 - x^63 - x^64 + x^66 - x^67 - x^68 + x^69 - x^70 + 2x^72 + x^74 - x^75 + x^77 + x^78 + x^79 - x^81 + 2x^83 + x^84 - x^85 + x^86 + x^90 - x^92 + x^93 + 2x^94 + x^95 + x^96 - 2x^97 - 2x^98 - 2x^99 + x^100 + x^101 - x^104 - x^105 + x^108 - 2x^113 + 2x^116 - x^117 + x^118 + x^119 + 2x^122 + 2x^124 - 2x^125 - x^127 - x^128 - x^129 - x^131 - x^133 + x^137 + x^138 - x^139 - x^141 + 2x^142 - x^143 + x^144 + 2x^146 - x^147 + x^148 + x^149 - x^150 + 2x^151 + x^152 - 2x^153 - x^154 - x^156 - x^157 + x^158 + x^159 + x^160 + x^161 - 2x^162 - x^163 - x^164 + x^165 + x^169 - x^170 - x^173 - x^174 + x^175 + 2x^178 - x^181 + x^182 + x^183 + x^188 + x^190 - x^191 - 2x^193 - x^196 + x^197 - x^198 + x^201 - x^203 + 2x^204 - x^205 - x^206 - x^207 + x^209 - x^210 - x^211 + x^212 - x^214 + 2x^217 - x^218 - x^221 + x^222 + x^223 - x^225 - x^227 - x^230 - x^231 - x^233 + x^235 + x^236 - x^238 + 2x^239 + x^240 - x^245 - x^247 + x^248 - 2x^250 + x^252 + x^253 + x^254$

$e_{ 3 } = -x + x^3 + x^4 - x^8 - x^11 + 2x^19 - x^20 - x^21 - x^23 - x^24 - x^25 - x^26 + x^28 - x^29 - x^31 + x^32 + x^36 + x^37 + x^38 - x^40 + x^41 + x^42 + x^43 - 2x^44 + x^45 + 2x^46 - x^47 - x^48 + x^49 + x^50 + x^52 - x^54 - x^55 + x^58 + x^59 + x^60 - x^62 - x^63 - 2x^64 + 2x^66 + 2x^67 + 2x^68 - 2x^69 + x^70 + x^71 - x^74 - x^75 - x^76 + x^77 - x^78 - x^80 + x^81 + x^82 + x^83 - x^84 - x^85 - x^86 - x^87 + 2x^89 + x^90 - x^92 - x^93 + x^94 - x^96 - x^97 + x^99 - 2x^103 + x^105 - x^107 + x^108 + x^109 - x^110 - x^113 + x^114 - 2x^115 - x^119 - x^120 - x^122 + x^123 - x^126 + 2x^127 - x^128 + x^129 + 2x^130 - x^131 - x^132 - x^134 + x^135 - x^137 - x^138 - 2x^139 + x^140 + 2x^143 - x^144 + 2x^145 + 2x^147 + x^149 + x^150 - x^151 - x^153 - x^156 + x^160 + 2x^161 + x^162 - x^163 + x^164 + 2x^165 + x^166 + x^167 + x^169 - x^170 - x^171 + x^175 + x^176 + x^179 - x^183 - x^186 + x^189 - x^190 + x^192 - x^193 - x^194 - x^195 - x^200 + x^203 + x^204 + x^205 + x^208 - 2x^209 - x^211 + x^214 - x^219 + x^220 - x^221 + x^222 + x^224 - 2x^225 - x^226 - x^227 - x^228 - x^229 + 2x^231 - x^232 - x^233 + x^236 + x^238 + x^240 - 2x^243 - x^244 + x^246 - x^247 - 2x^249 + 2x^250 + x^251 + x^253 + 2x^254$

$\|e\|_\infty = 2 \le \eta_1 = 2$

۲. بردار کلید عمومی محاسبه‌شده $t = As + e \pmod{3329}$

$t_{ 1 } = -1024 + 1607x - 1528x^2 + 546x^3 + 1048x^4 + 941x^5 + 239x^6 - 1185x^7 + 150x^8 + 1249x^9 - 1441x^10 - 1027x^11 - 85x^12 - 1393x^13 + 265x^14 - 1164x^15 - 92x^16 + 1189x^17 + 864x^18 + 1335x^19 + 36x^20 - 1501x^21 + 405x^22 + 872x^23 - 641x^24 - 615x^25 - 449x^26 + 1559x^27 + 21x^28 - 627x^29 + 1324x^30 - 1155x^31 - 1553x^32 + 1231x^33 + 1586x^34 + 351x^35 + 933x^36 - 548x^37 + 173x^38 + 1191x^39 - 1503x^40 + 762x^41 - 457x^42 - 378x^43 - 994x^44 - 908x^45 + 951x^46 + 115x^47 + 1424x^48 + 391x^49 - 1616x^50 - 609x^51 - 495x^52 - 1498x^53 + 505x^54 + 136x^55 + 1083x^56 + 142x^57 + 1504x^58 + 959x^59 - 899x^60 + 1069x^61 + 1495x^62 + 1429x^63 + 1388x^64 - 442x^65 - 929x^66 - 1138x^67 + 343x^68 + 1351x^69 + 922x^70 + 906x^71 - 235x^72 + 1582x^73 - 394x^74 + 1288x^75 - 495x^76 - 731x^77 - 121x^78 - 1308x^79 - 759x^80 - 1554x^81 + 733x^82 - 564x^83 + 39x^84 + 892x^85 + 1587x^86 - 314x^87 + 326x^88 - 755x^89 - 12x^90 + 1426x^91 - 1332x^92 - 1508x^93 + 122x^94 + 696x^95 - 1072x^96 - 155x^97 + 999x^98 - 319x^99 - 1272x^100 + 561x^101 - 1118x^102 - 822x^103 - 482x^104 + 826x^105 + 35x^106 + 545x^107 + 116x^108 - 196x^109 - 261x^110 - 1576x^111 + 647x^112 - 562x^113 - 775x^114 + 549x^115 + 700x^116 - 498x^117 - 392x^118 + 768x^119 + 776x^120 - 1293x^121 + 643x^122 - 381x^123 + 1558x^124 + 377x^125 - 906x^126 - 222x^127 - 194x^128 + 661x^129 - 354x^130 - 1086x^131 + 533x^132 + 478x^133 - 144x^134 - 532x^135 - 334x^136 + 192x^137 + 1155x^138 - 726x^139 - 1081x^140 - 70x^141 + 1664x^142 - 1438x^143 - 1006x^144 + 985x^145 - 521x^146 + 338x^147 + 330x^148 - 798x^149 + 796x^150 + 285x^151 + 1407x^152 - 53x^153 + 706x^154 - 598x^155 + 441x^156 - 211x^157 - 129x^158 + 505x^159 + 1194x^160 + 1566x^161 - 947x^162 + 555x^163 - 748x^164 - 477x^165 + 1562x^166 - 412x^167 + 702x^168 + 981x^169 - 187x^170 + 1493x^171 - 1334x^172 + 1080x^173 - 1400x^174 + 1059x^175 + 1338x^176 + 613x^177 - 1178x^178 - 231x^179 - 1040x^180 - 1260x^181 + 1314x^182 + 351x^183 - 1358x^184 + 1073x^185 - 52x^186 - 265x^187 + 27x^188 + 1454x^189 - 1342x^190 - 346x^191 - 235x^192 - 1003x^193 + 1179x^194 - 1270x^195 + 293x^196 + 1479x^197 + 319x^198 - 124x^199 - 228x^200 + 1422x^201 - 1199x^202 + 412x^203 - 1317x^204 + 773x^205 - 116x^206 + 1617x^207 - 1272x^208 + 948x^209 - 71x^210 - 1266x^211 - 894x^212 - 162x^213 - 699x^214 + 1344x^215 + 279x^216 + 443x^217 + 326x^218 + 969x^219 + 953x^220 - 1630x^221 + 1333x^222 + 662x^223 - 604x^224 - 932x^225 + 1078x^226 - 31x^227 - 380x^228 - 1420x^229 + 905x^230 - 876x^231 + 11x^232 + 416x^233 - 491x^234 - 438x^235 + 1490x^236 - 337x^237 - 536x^238 - 793x^239 + 341x^240 - 224x^241 - 727x^242 - 1004x^243 - 1142x^244 + 780x^245 + 772x^246 - 816x^247 - 290x^248 - 1654x^249 + 194x^250 + 901x^251 - 1626x^252 + 429x^253 - 425x^254 + 867x^255$

$t_{ 2 } = -807 + 1096x - 1277x^2 - 190x^3 - 1270x^4 - 336x^5 - 685x^6 - 44x^7 - 1451x^8 + 1452x^9 + 838x^10 - 922x^11 + 595x^12 + 1120x^13 + 629x^14 - 1432x^15 + 763x^16 - 693x^17 + 514x^18 - 474x^19 + 1416x^20 + 84x^21 - 408x^22 - 460x^23 - 102x^24 - 1154x^25 + 355x^26 - 345x^27 - 975x^28 - 266x^29 - 173x^30 + 1449x^31 + 14x^32 - 189x^33 - 1290x^34 + 1622x^35 + 898x^36 + 1364x^37 + 1287x^38 - 648x^39 + 1365x^40 + 1110x^41 + 1359x^42 - 937x^43 - 807x^44 - 1620x^45 - 1165x^46 - 1217x^47 - 105x^48 + 217x^49 + 904x^50 + 1082x^51 - 1226x^52 - 1260x^53 + 1515x^54 + 1036x^55 + 1425x^56 + 1237x^57 + 1315x^58 - 1072x^59 - 1442x^60 - 698x^61 + 921x^62 + 937x^63 + 487x^64 + 1209x^65 - 1249x^66 + 1448x^67 + 1327x^68 - 873x^69 + 432x^70 + 886x^71 + 603x^72 - 959x^73 + 475x^74 + 69x^75 - 1133x^76 - 247x^77 + 788x^78 - 1277x^79 + 1307x^80 + 931x^81 - 457x^82 - 960x^83 - 1269x^84 + 258x^85 - 1423x^86 - 1228x^87 - 130x^88 + 1380x^89 - 1271x^90 + 821x^91 - 187x^92 + 293x^93 - 753x^94 + 730x^95 + 257x^96 - 163x^97 + 657x^98 - 380x^99 + 480x^100 - 4x^101 - 323x^102 + 1060x^103 + 1502x^104 - 1657x^105 + 680x^106 + 316x^107 - 1037x^108 + 92x^109 + 485x^110 - 1186x^111 + 686x^112 + 785x^113 - 1532x^114 - 961x^115 - 932x^116 + 672x^117 - 731x^118 - 1051x^119 - 1346x^120 + 1312x^121 - 1395x^122 + 148x^123 - 640x^124 + 988x^125 + 534x^126 - 1215x^127 + 660x^128 - 1024x^129 + 92x^130 - 658x^131 - 875x^132 - 1476x^133 + 40x^134 - 200x^135 - 397x^136 + 923x^137 + 1037x^138 + 776x^139 + 12x^140 + 875x^141 - 603x^142 - 1495x^143 + 369x^144 - 623x^145 + 1412x^146 + 1119x^147 - 1041x^148 + 1140x^149 - 641x^150 - 224x^151 + 138x^152 - 315x^153 - 246x^154 - 1391x^155 + 1557x^156 + 1587x^157 + 893x^158 + 212x^159 - 92x^160 + 41x^161 - 259x^162 + 1139x^163 - 462x^164 + 803x^165 + 879x^166 - 360x^167 - 1079x^168 + 1267x^169 - 354x^170 - 740x^171 + 735x^172 + 35x^173 - 305x^174 - 1190x^175 - 1540x^176 - 750x^177 + 763x^178 - 1546x^179 + 1530x^180 - 1146x^181 + 1024x^182 - 22x^183 - 1383x^184 - 1362x^185 - 1258x^186 - 1467x^187 - 644x^188 + 921x^189 + 1204x^190 + 349x^191 - 877x^192 - 1113x^193 - 1534x^194 + 523x^195 + 333x^196 + 271x^197 + 20x^198 + 640x^199 + 1264x^200 + 1289x^201 - 847x^202 + 273x^203 - 1253x^204 + 1285x^205 + 922x^206 + 533x^207 + 1059x^208 - 1365x^209 - 238x^210 - 1594x^211 - 599x^212 + 346x^213 - 407x^214 + 954x^215 + 393x^216 - 680x^217 + 1391x^218 + 1504x^219 - 589x^220 - 1110x^221 + 676x^222 - 1386x^223 - 1336x^224 + 1043x^225 + 1304x^226 - 1280x^227 + 1298x^228 + 681x^229 + 102x^230 - 1239x^231 - 474x^232 - 1181x^233 - 1147x^234 - 985x^235 + 1537x^236 - 986x^237 + 1498x^238 - 93x^239 + 1426x^240 + 61x^241 - 1012x^242 - 577x^243 + 157x^244 + 387x^245 - 1444x^246 + 728x^247 - 234x^248 - 781x^249 - 515x^250 - 1040x^251 - 1011x^252 - 213x^253 - 1030x^254 + 684x^255$

$t_{ 3 } = 1583 + 478x - 381x^2 - 100x^3 + 104x^4 + 1097x^5 + 1546x^6 - 1539x^7 - 190x^8 + 600x^9 + 707x^10 - 367x^11 + 1442x^12 + 895x^13 + 161x^14 - 1596x^15 + 1365x^16 - 597x^17 + 1092x^18 + 1469x^19 + 714x^20 - 134x^21 + 1661x^22 - 430x^23 + 626x^24 + 487x^25 + 48x^26 - 1294x^27 + 395x^28 - 407x^29 - 567x^30 - 1313x^31 - 933x^32 + 1354x^33 - 1081x^34 - 475x^35 + 475x^36 + 573x^37 - 354x^38 - 430x^39 - 1327x^40 + 231x^41 - 933x^42 + 181x^43 + 289x^44 - 1382x^45 + 1606x^46 - 1659x^47 + 1362x^48 - 791x^49 - 509x^50 - 88x^51 - 737x^52 + 1637x^53 + 1169x^54 + 1200x^55 + 909x^56 - 761x^57 - 1301x^58 + 1237x^59 + 924x^60 + 1603x^61 - 536x^62 + 1447x^63 - 277x^64 + 385x^65 + 341x^66 - 310x^67 - 737x^68 + 328x^69 - 484x^70 + 290x^71 + 684x^72 + 886x^73 - 420x^74 + 1306x^75 - 932x^76 - 1190x^77 + 57x^78 + 1079x^79 - 796x^80 + 1502x^81 + 803x^82 - 1441x^83 + 1271x^84 - 96x^85 + 1250x^86 - 673x^87 + 292x^88 - 1125x^89 + 861x^90 + 1515x^91 + 154x^92 - 707x^93 + 261x^94 - 1161x^95 - 1453x^96 - 1083x^97 + 85x^98 - 826x^99 - 1500x^100 - 893x^101 - 1453x^102 + 1044x^103 + 174x^104 - 1636x^105 - 932x^106 + 278x^107 - 240x^108 - 1183x^109 + 1355x^110 - 300x^111 + 1180x^112 - 1318x^113 + 440x^114 + 1000x^115 - 1321x^116 + 725x^117 - 116x^118 + 481x^119 - 1444x^120 - 764x^121 + 1192x^122 - 886x^123 + 799x^124 - 249x^125 - 899x^126 + 510x^127 + 1012x^128 - 1149x^129 - 33x^130 - 367x^131 - 113x^132 + 870x^133 + 1077x^134 + 248x^135 + 864x^136 + 113x^137 + 723x^138 - 623x^139 - 1130x^140 - 271x^141 + 533x^142 - 770x^143 + 736x^144 - 1292x^145 + 496x^146 - 1407x^147 - 1304x^148 + 859x^149 + 507x^150 + 358x^151 + 1259x^152 + 1634x^153 + 1474x^154 - 1237x^155 - 807x^156 + 1093x^157 + 1412x^158 - 1107x^159 - 1514x^160 - 58x^161 - 2x^162 - 144x^163 - 1607x^164 - 268x^165 + 1201x^166 - 767x^167 - 394x^168 - 159x^169 + 1512x^170 - 559x^171 + 868x^172 + 136x^173 + 1426x^174 + 1348x^175 - 614x^176 + 456x^177 - 772x^178 - 59x^179 + 1505x^180 - 520x^181 + 1383x^182 - 598x^183 - 50x^184 - 445x^185 - 1440x^186 + 629x^187 - 974x^188 + 914x^189 - 147x^190 + 669x^191 - 1260x^192 + 662x^193 - 1534x^194 + 677x^195 - 1224x^196 + 1633x^197 + 1390x^198 + 1160x^199 + 1517x^200 + 1180x^201 - 356x^202 - 1243x^203 + 1542x^204 - 293x^205 + 1564x^206 + 543x^207 + 202x^208 - 551x^209 + 186x^210 - 193x^211 + 1208x^212 - 220x^213 + 1344x^214 - 1024x^215 - 667x^216 - 244x^217 - 923x^218 + 1107x^219 - 133x^220 + 409x^221 - 701x^222 + 1169x^223 - 1200x^224 - 1520x^225 - 1573x^226 - 382x^227 + 695x^228 - 374x^229 - 767x^230 - 1260x^231 - 97x^232 + 1436x^233 - 1448x^234 + 1414x^235 + 957x^236 - 1105x^237 - 1478x^238 - 1192x^239 + 286x^240 - 777x^241 + 329x^242 - 1607x^243 - 693x^244 + 193x^245 + 253x^246 + 81x^247 + 124x^248 - 666x^249 - 1043x^250 + 171x^251 + 527x^252 + 422x^253 + 1440x^254 + 1116x^255$

قالب‌بندی کلیدهای خروجی سیستم (Key Pair)

Public Key (PK):

$$ PK = (A_{3 \times 3}, t_{3 \times 1}) $$

Secret Key (SK):

$$ SK = (s_{3 \times 1}) $$

مرحله دوم - رمزگذاری (Encryption)

$u = A^T r + e_1 \pmod{3329}$

متن پیام ورودی باب جهت ارسال به آلیس:

                "Advanced Cryptography Class, Dr. Najafi"
            

Bits Count: 312 | Bits Preview: 0100000101100100011101100110000101101110011000110110010101100100...

۱. بردارهای تصادفی و خطای رمزگذاری ( $$r, e_1, e_2$$ )

Random Vector $r \in CBD_2^3$:

$r_{ 1 } = 1 + x^2 + x^4 - x^5 - x^6 - x^7 - x^8 - 2x^9 + x^10 - x^11 + x^12 + 2x^13 + x^15 + x^16 + 2x^17 - x^19 - x^20 + x^21 + x^22 + x^25 + x^26 + x^27 + x^30 - x^32 + x^33 + x^34 + 2x^35 - x^38 + x^39 + x^43 + x^44 - x^45 + x^48 - x^49 - x^50 - x^51 + x^52 - x^53 + x^54 - x^55 + x^59 + x^60 - 2x^61 - x^62 - x^66 + 2x^67 + 2x^68 + x^69 + x^70 - 2x^71 - x^74 + 2x^75 + x^77 - x^78 + x^79 + x^81 + x^82 - x^83 + x^86 - x^87 + x^88 + x^91 - 2x^93 + x^94 + 2x^95 - x^97 - x^98 + x^99 - x^101 - x^102 - x^103 - 2x^105 + x^106 - x^107 - x^109 + x^110 - x^111 + x^113 + x^114 - x^116 - x^117 - x^118 + 2x^119 - x^121 + 2x^122 - x^127 - x^129 - x^130 + x^131 - x^132 - 2x^133 - x^135 + x^136 - x^137 - 2x^138 + x^139 - x^140 - 2x^141 - x^144 - x^145 + 2x^146 + 2x^147 - x^148 + 2x^152 + x^153 - 2x^154 + 2x^155 + x^156 - x^157 + x^158 - x^159 + x^160 - 2x^161 + x^162 + x^163 + x^164 - x^165 + x^166 - x^167 - x^170 + x^171 + x^172 - x^173 + x^174 + 2x^178 + x^181 + x^182 + 2x^183 + 2x^184 - x^185 + x^186 + x^187 - x^190 - x^192 + x^193 + x^197 + x^198 - 2x^199 - x^200 + 2x^201 - x^202 - x^203 + x^206 - x^207 + x^208 - 2x^212 - x^213 - 2x^214 + x^215 + x^216 + x^217 + x^219 + x^222 + 2x^223 + x^224 - 2x^225 - x^226 + 2x^227 - x^230 + 2x^232 + x^233 + x^234 - 2x^235 - x^237 - x^239 + 2x^240 + 2x^241 - x^244 - 2x^245 - x^247 - x^248 + x^251 - x^252$

$r_{ 2 } = x + x^2 + 2x^3 + 2x^4 - x^6 + 2x^7 - x^9 - 2x^13 - x^14 - x^15 + x^16 + x^17 - x^18 - x^19 + 2x^20 - x^23 - x^24 + x^29 + x^32 - x^33 - x^34 + x^35 + x^38 - x^39 + x^40 + x^41 - 2x^45 - x^46 - x^47 + x^49 + x^50 + x^51 - x^53 - x^54 + x^56 - 2x^57 + x^58 - x^59 + x^60 - 2x^62 + x^63 - x^64 + 2x^65 - x^67 + x^68 + x^70 - x^72 + x^73 + 2x^75 + x^76 - x^77 - x^78 + x^81 + x^83 - x^84 + x^85 - x^88 + x^91 + x^92 + x^95 + x^99 - 2x^100 + x^101 + x^102 - x^103 + 2x^104 - 2x^106 + x^107 + x^109 + x^110 + x^111 - x^112 - x^113 + x^114 - x^115 + x^116 + 2x^117 - 2x^119 - x^120 + 2x^122 + 2x^123 - x^124 + 2x^126 - x^127 - x^129 + x^131 - x^132 + 2x^134 - 2x^135 - x^139 + 2x^142 - 2x^143 - 2x^145 - x^146 + x^147 + x^149 - x^150 - x^151 - x^152 - 2x^154 + x^156 - x^157 + x^158 - x^159 + x^160 - x^162 + x^164 + x^167 - x^168 + x^171 - x^172 + 2x^174 - x^177 - x^178 - x^180 - x^181 + x^182 + x^183 - x^186 + x^187 + 2x^188 - 2x^190 - x^193 - x^194 + x^195 + x^198 - x^199 - x^200 - 2x^202 + x^205 - 2x^206 + 2x^207 - x^208 - x^210 - x^216 + 2x^217 + x^218 + 2x^219 + x^220 + x^221 + x^223 - x^224 - x^225 - 2x^226 - x^228 - x^230 - x^231 - x^234 - x^235 + x^237 + x^240 - x^241 - x^243 + x^245 - x^247 - 2x^249 + x^250 - x^254 - x^255$

$r_{ 3 } = -1 + x + 2x^2 + x^3 - x^4 - x^5 + 2x^6 + 2x^7 - x^8 - x^9 + x^12 + x^13 - x^14 - x^17 + x^22 - 2x^23 + x^24 - x^26 + x^27 - x^28 - x^29 + x^31 - x^32 - x^33 + x^34 - x^35 + x^39 + x^40 + x^42 - x^43 + 2x^44 - x^45 - 2x^47 + x^48 - x^49 - x^50 - x^51 - x^53 + x^54 + x^55 - x^58 + x^60 + x^61 - 2x^62 + x^63 + x^64 + x^65 - x^66 - x^67 + 2x^69 + x^72 - x^73 + x^75 - x^77 - x^82 + 2x^83 + 2x^84 - x^86 - x^87 - x^88 - x^94 + x^95 - x^96 + x^99 + x^101 + x^102 - x^103 + x^104 + x^106 + x^107 + 2x^108 + x^109 + 2x^110 - x^111 + x^113 - x^114 + x^116 - x^117 + x^118 - x^119 + x^120 + x^125 + 2x^126 - x^130 - 2x^131 + x^132 - x^134 + x^136 + x^137 + x^138 - x^139 - x^141 - x^142 + x^143 + 2x^144 - x^145 + x^146 - x^147 - x^149 + x^150 + x^151 - x^156 + 2x^157 + x^160 + x^161 + x^162 - x^163 + x^165 - x^168 - 2x^170 + x^171 - x^173 - x^175 + x^176 - x^177 + x^178 + x^179 - x^181 - x^183 + x^186 - x^193 + x^194 - 2x^196 - x^197 - 2x^198 + x^202 + x^204 + 2x^205 - 2x^207 - x^208 + x^209 + x^210 + x^211 + x^212 - x^213 - x^214 + x^215 + x^216 + x^217 + x^218 + x^221 - 2x^225 + x^226 - x^229 - x^230 + x^232 + x^233 + x^237 - x^241 - x^245 + x^246 - x^247 + x^248 - x^249 - x^253 + 2x^254 - 2x^255$

Error Vector $e_1 \in CBD_2^3$:

$e_{1, 1 } = -2 + x^2 + x^3 + x^4 + x^7 - x^10 - x^11 + x^12 - x^13 - x^15 + x^16 - x^19 + x^20 + 2x^21 + x^22 - x^25 - x^26 + 2x^29 - x^30 - x^32 + x^33 + x^34 - x^35 - x^36 + x^38 + 2x^39 - x^40 - 2x^41 + x^44 - 2x^45 - x^46 - 2x^50 - 2x^52 - x^53 - x^54 - x^56 - x^57 + 2x^61 + x^62 - x^63 + x^64 - x^65 + x^66 - x^67 - x^68 + x^69 - x^70 + x^71 + x^73 - x^74 + x^75 + x^77 + x^78 + 2x^79 + x^80 - x^81 - x^83 + 2x^84 + x^85 + x^86 - x^88 + x^89 - x^91 - x^93 + 2x^95 + x^96 + 2x^97 + x^100 - 2x^101 + 2x^102 + 2x^103 + x^105 - x^106 + x^108 - 2x^109 + x^110 + x^111 - 2x^112 - x^113 + 2x^114 - x^115 - x^116 + x^118 - x^119 - x^120 - x^121 - x^122 + x^124 - 2x^126 + x^128 + x^129 - 2x^130 + x^131 - 2x^132 + x^133 + x^135 - x^138 - x^141 + x^142 - x^144 - x^145 + x^147 + x^148 + x^149 + x^150 - x^151 + x^154 - 2x^155 + 2x^156 - x^157 - x^159 + x^161 - 2x^162 - x^165 - x^166 + x^168 + x^170 + 2x^172 + 2x^174 + 2x^175 - x^176 + x^178 - x^179 - x^180 + x^181 - x^183 - x^184 - x^186 + x^189 + x^192 + x^193 + x^196 + x^201 - x^203 - x^206 - x^208 - 2x^209 - x^210 + x^211 - x^214 + x^216 + 2x^217 + x^218 + x^219 - x^220 - x^222 + x^223 + x^224 - 2x^225 - x^226 + 2x^227 - x^229 + x^230 - x^231 - 2x^232 - x^233 - x^235 + x^236 + x^237 - 2x^238 - x^239 + 2x^240 - 2x^241 - x^242 - x^243 - x^245 + x^246 - 2x^247 + 2x^248 + 2x^251 - x^253 - 2x^254$

$e_{1, 2 } = -x + x^2 - x^4 + x^5 - x^7 + x^8 - x^9 - x^10 + x^11 - x^12 + x^14 - x^15 - x^16 + 2x^17 + x^18 - x^19 - x^20 - x^23 + x^24 - 2x^25 - 2x^26 - 2x^29 + x^31 + x^32 + x^33 + x^34 + x^35 + 2x^37 - x^40 + x^41 + x^42 + x^43 - x^45 - x^46 - x^47 - x^48 - 2x^49 - x^50 + 2x^51 + x^52 + x^53 + 2x^54 - x^55 - x^57 - x^58 - x^59 - x^61 - x^62 - x^63 - x^65 - 2x^67 - x^69 - x^70 + 2x^71 + x^76 - 2x^77 + 2x^78 - x^79 + x^80 - x^81 + x^82 - 2x^83 - x^87 - x^89 + x^93 + x^94 + 2x^95 + x^96 - 2x^97 + x^99 - x^100 + x^101 - x^102 - x^103 - 2x^104 - x^105 - x^106 + x^107 + x^108 + x^109 - 2x^111 - x^113 - x^115 + x^116 - x^117 + x^118 + x^119 + 2x^121 + 2x^122 - x^123 - x^124 + x^125 - x^127 + x^128 + x^133 + x^134 + 2x^135 + x^136 + x^137 + x^139 - x^140 - x^145 - x^146 + x^147 + x^148 + x^149 + x^150 - x^151 + x^152 - x^153 - 2x^156 - x^157 - x^159 + x^160 - x^162 - x^163 + 2x^164 + x^165 + x^168 - x^170 + x^172 - x^173 - 2x^175 - x^176 + x^177 - x^178 + 2x^179 - x^181 - x^183 + x^187 + x^188 + x^189 - 2x^194 - 2x^195 - x^197 - x^201 + x^202 + x^203 + x^204 + 2x^205 + x^207 - x^208 + x^209 - 2x^213 + x^214 + x^216 - x^218 - 2x^219 + 2x^220 - 2x^221 + x^223 - x^224 + x^225 - x^230 + x^231 + x^232 - 2x^233 - x^234 + x^235 - x^237 + x^239 - x^241 + x^242 - x^243 + x^244 - x^246 + x^247 + x^248 + x^249 + x^250 - x^252 + x^253 - x^254$

$e_{1, 3 } = -2 - x + 2x^2 - x^4 - 2x^7 + x^8 - x^9 + x^11 + x^14 + x^18 - 2x^19 + x^21 + x^22 + x^23 - x^26 + x^27 - x^28 - x^30 - x^31 + 2x^33 + x^34 + 2x^36 - x^39 - x^41 + 2x^42 - x^44 - 2x^45 + x^46 + x^47 + x^48 + x^49 - 2x^50 + x^54 - x^55 - x^56 - 2x^57 - x^61 - x^62 + 2x^64 + x^67 - x^68 - x^70 - x^72 + x^73 + x^74 - x^75 + x^78 - x^79 - 2x^84 + x^85 - 2x^87 - x^88 + 2x^89 - x^90 + 2x^93 + x^96 + x^97 + x^98 - x^99 + x^100 - x^101 + x^102 - 2x^104 - x^105 + x^106 - x^107 + x^108 + x^109 + x^111 - x^113 + x^114 + x^115 - 2x^116 - x^118 + x^120 + x^122 + x^123 - x^126 - x^128 + x^135 + x^137 - x^138 - x^139 + 2x^140 - x^141 - 2x^142 - x^143 - x^144 + x^146 - x^149 + x^150 - x^151 - x^153 - 2x^154 + x^155 + x^157 + 2x^158 + x^159 + x^161 - x^162 - x^163 - x^164 + 2x^166 + x^167 + x^168 + x^169 + x^171 - 2x^172 - x^173 - 2x^176 + x^178 - x^182 + x^183 + x^184 + x^185 - x^187 - 2x^188 + x^189 + 2x^191 - x^192 + x^193 - x^196 + x^198 + x^201 - x^203 - x^205 - x^207 + x^209 - x^210 - 2x^212 + 2x^213 + x^214 + x^215 - x^216 + 2x^217 + x^219 + x^220 - 2x^222 + x^225 - x^226 - x^229 - x^231 + 2x^232 - x^233 + x^236 - 2x^237 + x^238 - 2x^239 + x^240 - x^241 + x^243 + x^245 - x^247 + x^248 + x^249 - x^250 + x^251 + x^252 + 2x^253 + x^254$

۲. بخش اول متن رمز شده: بردار $u = A^T r + e_1 \pmod{3329}$

$$ u = A^T \cdot r + e_1 \pmod{3329} $$

$u_{ 1 } = -1324 + 1149x - 1001x^2 + 321x^3 - 248x^4 + 995x^5 + 509x^6 + 6x^7 + 208x^8 + 1513x^9 + 292x^10 + 1253x^11 - 404x^12 + 787x^13 - 1534x^14 - 728x^15 + 870x^16 - 1518x^17 - 27x^18 + 53x^19 + 20x^20 + 650x^21 - 719x^22 + 531x^23 - 1395x^24 + 1055x^25 - 957x^26 + 286x^27 + 1637x^28 + 927x^29 + 1041x^30 - 586x^31 + 1299x^32 - 1116x^33 + 1167x^34 - 1063x^35 - 1541x^36 - 1311x^37 - 1230x^38 + 329x^39 + 713x^40 + 1337x^41 - 1555x^42 + 54x^43 - 226x^44 + 1412x^45 + 1002x^46 - 850x^47 + 18x^48 - 315x^49 + 441x^50 - 826x^51 - 741x^52 + 1014x^53 - 957x^54 + 808x^55 + 131x^56 + 344x^57 + 1228x^58 - 510x^59 + 539x^60 - 171x^61 + 852x^62 - 575x^63 + 196x^64 - 408x^65 - 1286x^66 - 335x^67 - 1156x^68 + 132x^69 + 167x^70 + 943x^71 - 1405x^72 - 572x^73 - 853x^74 - 622x^75 + 1334x^76 - 117x^77 + 566x^78 - 750x^79 + 1422x^80 + 667x^81 + 920x^82 + 125x^83 + 272x^84 - 1639x^85 - 1182x^86 + 1291x^87 - 1097x^88 - 1192x^89 - 1001x^90 + 1656x^91 - 1262x^92 - 505x^93 - 1472x^94 + 1031x^95 - 1362x^96 - 293x^97 + 793x^98 - 304x^99 - 505x^100 + 347x^101 + 85x^102 - 1537x^103 - 1261x^104 + 1509x^105 + 1534x^106 + 354x^107 + 536x^108 + 500x^109 + 1135x^110 - 35x^111 - 1457x^112 - 1385x^113 - 688x^114 + 1315x^115 - 908x^116 + 72x^117 + 113x^118 + 1584x^119 + 413x^120 - 434x^121 + 970x^122 + 1102x^123 + 253x^124 + 542x^125 - 181x^126 - 469x^127 - 379x^128 + 255x^129 - 4x^130 + 386x^131 + 214x^132 + 1457x^133 + 136x^134 + 501x^135 + 518x^136 - 922x^137 - 403x^138 - 186x^139 - 1283x^140 - 811x^141 - 326x^142 - 616x^143 + 954x^144 + 926x^145 + 1213x^146 + 1399x^147 - 693x^148 + 563x^149 + 1483x^150 + 685x^151 + 168x^152 - 1308x^153 - 688x^154 - 714x^155 - 835x^156 + 1057x^157 - 1482x^158 - 1577x^159 + 856x^160 + 830x^161 - 962x^162 + 760x^163 + 1260x^164 - 1482x^165 - 451x^166 - 428x^167 - 806x^168 - 76x^169 - 1299x^170 - 624x^171 - 27x^172 + 1106x^173 + 230x^174 + 1460x^175 + 1470x^176 - 84x^177 - 190x^178 - 1622x^179 + 840x^180 + 960x^181 + 30x^182 - 1643x^183 + 537x^184 - 593x^185 + 1209x^186 - 1070x^187 - 963x^188 + 672x^189 + 973x^190 + 1546x^191 - 857x^192 + 1049x^193 - 463x^194 + 19x^195 + 588x^196 - 1275x^197 - 139x^198 + 100x^199 - 98x^200 - 512x^201 + 1333x^202 + 846x^203 + 1416x^204 + 135x^205 - 1531x^206 + 560x^207 + 583x^208 + 1202x^209 + 35x^210 - 757x^211 + 491x^212 - 1177x^213 - 936x^214 - 191x^215 + 1136x^216 + 118x^217 - 481x^218 + 428x^219 + 1607x^220 + 741x^221 - 183x^222 + 1012x^223 + 588x^224 + 1285x^225 + 70x^226 - 642x^227 + 750x^228 - 715x^229 + 1239x^230 - 1146x^231 - 1159x^232 + 570x^233 - 778x^234 + 693x^235 - 935x^236 - 1230x^237 + 1014x^238 + 296x^239 - 636x^240 - 245x^241 + 272x^242 + 197x^243 - 1416x^244 - 794x^245 + 218x^246 - 420x^247 - 1191x^248 - 562x^249 - 116x^250 - 332x^251 - 1203x^252 + 593x^253 - 1428x^254 - 137x^255$

$u_{ 2 } = 1572 + 1293x - 1458x^2 + 982x^3 - 1558x^4 + 1439x^5 - 52x^6 + 190x^7 - 604x^8 + 1230x^9 - 1205x^10 + 314x^11 - 1116x^12 + 629x^13 - 170x^14 + 1516x^15 + 355x^16 - 962x^17 + 397x^18 - 1123x^19 - 389x^20 - 1483x^21 + 80x^22 - 697x^23 - 1374x^24 + 1566x^25 + 1414x^26 - 6x^27 + 252x^28 + 966x^29 + 285x^30 - 905x^31 - 1580x^32 - 28x^33 - 515x^34 + 1159x^35 + 1198x^36 - 1304x^37 + 888x^38 - 1470x^39 - 110x^40 - 879x^41 - 690x^42 - 1651x^43 + 1049x^44 - 752x^45 - 1212x^46 - 442x^47 + 698x^48 + 1039x^49 + 988x^50 - 988x^51 + 680x^52 + 1393x^53 - 77x^54 + 178x^55 + 972x^56 - 146x^57 + 937x^58 - x^59 - 486x^60 - 584x^61 - 1348x^62 - 114x^63 + 975x^64 + 795x^65 + 877x^66 + 875x^67 - 1478x^68 - 773x^69 + 113x^70 - 1492x^71 + 447x^72 - 1033x^73 - 1323x^74 + 1247x^75 + 291x^76 + 1180x^77 + 468x^78 + 1620x^79 + 107x^80 - 416x^81 + 292x^82 + 466x^83 + 868x^84 + 1115x^85 + 1434x^86 + 160x^87 - 599x^88 - 1145x^89 + 821x^90 - 982x^91 + 287x^92 + 1300x^93 + 505x^94 - 1256x^95 + 440x^96 + 1255x^97 - 750x^98 + 1156x^99 + 21x^100 - 80x^101 - 1578x^102 - 1133x^103 + 1072x^104 - 1078x^105 - 1196x^106 - 1621x^107 + 1440x^108 + 69x^109 - 1018x^110 - 1081x^111 - 286x^112 + 1065x^113 + 562x^114 - 1173x^115 + 1383x^116 + 1313x^117 - 850x^118 - 92x^119 + 1610x^120 + 499x^121 + 638x^122 + 1552x^123 - 929x^124 + 37x^125 - 635x^126 - 1007x^127 - 82x^128 + 325x^129 + 1309x^130 - 1539x^131 + 1288x^132 + 507x^133 + 1222x^134 - 200x^135 + 739x^136 + 981x^137 - 493x^138 + 753x^139 - 958x^140 + 1131x^141 + 1145x^142 - 1468x^143 + 1391x^144 - 454x^145 - 789x^146 + 1014x^147 + 1503x^148 - 31x^149 + 277x^150 + 77x^151 - 770x^152 - 1332x^153 + 1228x^154 + 1514x^155 + 615x^156 + 836x^157 - 558x^158 + 478x^159 - 844x^160 + 1328x^161 + 1474x^162 - 1366x^163 - 520x^164 + 878x^165 + 407x^166 - 235x^167 - 745x^168 - 1245x^169 - 668x^170 + 1553x^171 + 1595x^172 + 462x^173 + 168x^174 - 465x^175 + 832x^176 - 834x^177 - 841x^178 - 1148x^179 - 1347x^180 - 716x^181 + 673x^182 + 131x^183 + 1404x^184 - 548x^185 + 916x^186 + 837x^187 + 1600x^188 + 263x^189 + 918x^190 + 1317x^191 + 479x^192 - 1645x^193 + 129x^194 + 429x^195 - 615x^196 - 566x^197 - 1073x^198 + 512x^199 + 1397x^200 + 569x^201 - 366x^202 - 462x^203 - 1007x^204 - 137x^205 + 1217x^206 - 843x^207 - 484x^208 + 1286x^209 + 482x^210 - 315x^211 - 669x^212 - 700x^213 + 120x^214 + 917x^215 + 618x^216 - 62x^217 - 1207x^218 - 1397x^219 - 2x^220 + 143x^221 - 1230x^222 + 1494x^223 - 77x^224 - 1632x^225 - 226x^226 - 748x^227 - 1207x^229 + 1363x^230 - 1405x^231 + 178x^232 - 129x^233 - 1263x^234 - 1539x^235 + 422x^236 + 1569x^237 + 1004x^238 + 1267x^239 - 106x^240 - 571x^241 + 787x^242 + 492x^243 + 1185x^244 + 1629x^245 + 1521x^246 + 104x^247 + 604x^248 - 1055x^249 + 1576x^250 - 280x^251 - 308x^252 + 1592x^253 - 67x^254 + 1570x^255$

$u_{ 3 } = 1269 - 1457x + 580x^2 - 457x^3 + 344x^4 + 1664x^5 - 150x^6 + 577x^7 - 1164x^8 - 1143x^9 + 370x^10 + 1620x^11 + 1002x^12 - 824x^13 - 1645x^14 - 220x^15 + 202x^16 + 1053x^17 + 1275x^18 + 159x^19 + 1538x^20 - 884x^21 - 1510x^22 + 1138x^23 - 591x^24 - 1226x^25 - 926x^26 - 1137x^27 + 1626x^28 - 1072x^29 - 288x^30 + 873x^31 - 450x^32 - 1188x^33 - 403x^34 + 569x^35 + 208x^36 - 1618x^37 + 1513x^38 + 1484x^39 - 1304x^40 - 270x^41 + 1373x^42 + 1531x^43 + 53x^44 - 1452x^45 + 1524x^46 - 1049x^47 + 215x^48 - 790x^49 - 1602x^50 - 1651x^51 + 3x^52 + 200x^53 + 1067x^54 - 1479x^55 + 1281x^56 - 643x^57 + 467x^58 - 1095x^59 + 885x^60 - 1264x^61 - 1357x^62 - 903x^63 - 353x^64 - 902x^65 - 1574x^66 - 820x^67 - 23x^68 + 879x^69 - 1070x^70 + 1399x^71 + 235x^72 + 305x^73 - 1582x^74 - 402x^75 + 1629x^76 - 158x^77 + 1349x^78 + 1230x^79 - 398x^80 + 963x^81 - 1437x^82 + 1253x^83 - 352x^84 - 738x^85 + 633x^86 + 1288x^87 + 651x^88 + 1066x^89 - 45x^90 + 402x^91 - 361x^92 + 984x^93 + 1065x^94 - 442x^95 + 1247x^96 - 1121x^97 - 1522x^98 - 1454x^99 + 1527x^100 - 1290x^101 - 1298x^102 - 1575x^103 + 1037x^104 + 45x^105 + 803x^106 - 1528x^107 - 1458x^108 - 1526x^109 - 724x^110 + 1168x^111 - 705x^112 + 897x^113 - 581x^114 - 1634x^115 - 1476x^116 + 448x^117 - 897x^118 - 1060x^119 - 1344x^120 - 1352x^121 + 547x^122 - 1383x^123 + 1655x^124 - 282x^125 + 763x^126 - 553x^127 + 291x^128 - 167x^129 + 1153x^130 - 1347x^131 + 1400x^132 - 1618x^133 - 1407x^134 + 1402x^135 - 1425x^136 - 179x^137 + 651x^138 - 721x^139 - 349x^140 - 237x^141 + 1519x^142 + 1395x^143 - 773x^144 + 227x^145 + 1148x^146 + 808x^147 - 1167x^148 + 1304x^149 + 1162x^150 - 681x^151 - 1151x^152 + 960x^153 - 1429x^154 - 1295x^155 + 1063x^156 + 1353x^157 - 1034x^158 - 388x^159 - 1611x^160 - 1348x^161 - 1471x^162 + 224x^163 + 449x^164 - 443x^165 + 990x^166 - 1094x^167 + 1565x^168 - 1312x^169 + 529x^170 + 578x^171 + 606x^172 - 426x^173 + 404x^174 - 170x^175 - 172x^176 - 1487x^177 + 16x^178 + 203x^179 + 1461x^180 - 1662x^181 - 875x^182 + 1558x^183 - 748x^184 - 1343x^185 + 52x^186 + 1501x^187 - 133x^188 + 857x^189 + 9x^190 + 292x^191 - 1000x^192 - 1079x^193 - 388x^194 + 440x^195 + 803x^196 - 1529x^197 - 16x^198 - 1216x^199 + 329x^200 + 1190x^201 - 7x^202 + 1513x^203 - 412x^204 - 913x^205 + 583x^206 - 165x^207 - 812x^208 - 405x^209 + 914x^210 - 203x^211 + 530x^212 + 1443x^213 + 85x^214 + 291x^215 + 1183x^216 + 1594x^217 + 280x^218 + 1660x^219 + 733x^220 + 368x^221 - 1324x^222 + 269x^223 - 781x^224 + 654x^225 + 1422x^226 + 1089x^227 + 1112x^228 + 369x^229 - 1244x^230 - 1522x^231 - 824x^232 + 907x^233 + 637x^234 - 862x^235 - 1026x^236 + 1601x^237 - 1648x^238 + 1394x^239 - 539x^240 - 779x^241 + 428x^242 + 255x^243 - 1599x^244 + 154x^245 - 248x^246 + 306x^247 - 1115x^248 - 372x^249 - 836x^250 - 1644x^251 + 378x^252 - 405x^253 + 953x^254 - 1309x^255$

۳. بخش دوم متن رمز شده: چندجمله‌ای $v = t^T r + e_2 + 1665 \cdot m \pmod{3329}$

$$ v = t^T \cdot r + e_2 + \left\lceil \frac{3329}{2} \right\rceil \cdot m(x) \pmod{3329} $$

بلوک 1 پیام (تا ۲۵۶ بیت):

چندجمله‌ای پیام $m(x)$:

$m(x) = x + x^7 + x^9 + x^10 + x^13 + x^17 + x^18 + x^19 + x^21 + x^22 + x^25 + x^26 + x^31 + x^33 + x^34 + x^36 + x^37 + x^38 + x^41 + x^42 + x^46 + x^47 + x^49 + x^50 + x^53 + x^55 + x^57 + x^58 + x^61 + x^66 + x^73 + x^78 + x^79 + x^81 + x^82 + x^83 + x^86 + x^89 + x^90 + x^91 + x^92 + x^95 + x^97 + x^98 + x^99 + x^105 + x^106 + x^107 + x^109 + x^113 + x^114 + x^116 + x^117 + x^118 + x^119 + x^121 + x^122 + x^125 + x^126 + x^127 + x^129 + x^130 + x^131 + x^134 + x^137 + x^138 + x^143 + x^145 + x^146 + x^147 + x^153 + x^154 + x^156 + x^161 + x^162 + x^163 + x^164 + x^167 + x^170 + x^177 + x^182 + x^183 + x^185 + x^186 + x^188 + x^189 + x^193 + x^194 + x^199 + x^201 + x^202 + x^203 + x^206 + x^207 + x^209 + x^210 + x^211 + x^214 + x^215 + x^218 + x^220 + x^221 + x^226 + x^233 + x^237 + x^241 + x^242 + x^243 + x^246 + x^250 + x^252 + x^253 + x^254$

پیام مقیاس شده $1665 \cdot m(x)$:

$\bar{m}(x) = 1665x + 1665x^7 + 1665x^9 + 1665x^10 + 1665x^13 + 1665x^17 + 1665x^18 + 1665x^19 + 1665x^21 + 1665x^22 + 1665x^25 + 1665x^26 + 1665x^31 + 1665x^33 + 1665x^34 + 1665x^36 + 1665x^37 + 1665x^38 + 1665x^41 + 1665x^42 + 1665x^46 + 1665x^47 + 1665x^49 + 1665x^50 + 1665x^53 + 1665x^55 + 1665x^57 + 1665x^58 + 1665x^61 + 1665x^66 + 1665x^73 + 1665x^78 + 1665x^79 + 1665x^81 + 1665x^82 + 1665x^83 + 1665x^86 + 1665x^89 + 1665x^90 + 1665x^91 + 1665x^92 + 1665x^95 + 1665x^97 + 1665x^98 + 1665x^99 + 1665x^105 + 1665x^106 + 1665x^107 + 1665x^109 + 1665x^113 + 1665x^114 + 1665x^116 + 1665x^117 + 1665x^118 + 1665x^119 + 1665x^121 + 1665x^122 + 1665x^125 + 1665x^126 + 1665x^127 + 1665x^129 + 1665x^130 + 1665x^131 + 1665x^134 + 1665x^137 + 1665x^138 + 1665x^143 + 1665x^145 + 1665x^146 + 1665x^147 + 1665x^153 + 1665x^154 + 1665x^156 + 1665x^161 + 1665x^162 + 1665x^163 + 1665x^164 + 1665x^167 + 1665x^170 + 1665x^177 + 1665x^182 + 1665x^183 + 1665x^185 + 1665x^186 + 1665x^188 + 1665x^189 + 1665x^193 + 1665x^194 + 1665x^199 + 1665x^201 + 1665x^202 + 1665x^203 + 1665x^206 + 1665x^207 + 1665x^209 + 1665x^210 + 1665x^211 + 1665x^214 + 1665x^215 + 1665x^218 + 1665x^220 + 1665x^221 + 1665x^226 + 1665x^233 + 1665x^237 + 1665x^241 + 1665x^242 + 1665x^243 + 1665x^246 + 1665x^250 + 1665x^252 + 1665x^253 + 1665x^254$

خطای تصادفی $e_2 \in CBD_2$:

$e_2 = -2x^2 + x^3 - x^5 - x^7 + x^10 + x^11 - x^12 - x^13 + x^14 + x^15 + x^16 - x^17 + x^18 + 2x^19 - x^20 + x^23 - x^24 + 2x^25 - x^26 - x^27 - 2x^28 + x^30 - x^31 - x^32 + x^34 + 2x^35 + x^36 - x^39 + x^42 - x^43 - x^45 + x^46 - 2x^47 - 2x^50 - x^51 + x^52 + x^53 + 2x^55 + x^56 + x^58 + x^59 + x^61 - 2x^62 - 2x^65 + x^66 + x^67 - x^68 + 2x^69 - x^70 + 2x^72 - x^74 + x^75 + x^77 + x^78 - x^81 - x^82 - x^84 - x^90 - x^91 + 2x^92 + x^93 - 2x^96 + x^100 + x^102 + x^105 + x^106 + x^108 - x^109 + 2x^110 - x^112 + x^113 - 2x^114 - 2x^115 - x^117 + x^118 + x^119 - x^120 + x^121 + x^122 + 2x^123 + x^126 + x^130 - x^132 - x^133 - x^134 - x^137 - x^139 + x^141 - x^143 + x^145 + x^148 - x^150 + x^152 - x^153 + 2x^155 + x^157 - x^158 + x^160 - x^164 + 2x^165 - x^167 - x^169 - x^170 + x^171 - 2x^172 + x^173 + x^177 + x^178 - x^179 + x^181 - 2x^182 - x^183 - x^184 + x^186 - 2x^188 + x^190 + x^191 + x^194 + 2x^195 - x^196 + x^197 + x^199 + 2x^200 - x^201 - 2x^202 + x^203 - x^204 + x^214 + 2x^215 - x^217 + x^218 - 2x^219 + x^223 - x^224 - x^225 - 2x^226 + x^227 - x^228 + x^229 - 2x^230 + x^231 - x^232 - 2x^233 + x^234 - 2x^235 - x^236 - x^241 - x^243 - x^245 + x^247 + x^249 - 2x^250 - x^251 - x^252 + x^253 + x^254 - x^255$

مقدار نهایی $v$ برای این بلوک:

$v_{ 1 } = 843 + 215x - 849x^2 - 1025x^3 - 1093x^4 + 1620x^5 + 102x^6 - 1497x^7 - 280x^8 - 631x^9 + 983x^10 - 839x^11 - 1462x^12 - 909x^13 + 1039x^14 + 64x^15 - 1204x^16 + 1645x^17 - 895x^18 + 1568x^19 + 710x^20 - 806x^21 - 723x^22 + 451x^23 + 1317x^24 + 95x^25 - 1638x^26 + 51x^27 + 1535x^28 + 1292x^29 - 1257x^30 - 948x^31 - 228x^32 + 1280x^33 - 724x^34 + 189x^35 + 694x^36 - 757x^37 + 1485x^38 + 177x^39 + 1364x^40 - 1326x^41 + 1172x^42 + 1082x^43 + 1241x^44 - 597x^45 + 396x^46 + 636x^47 + 788x^48 + 513x^49 - 1051x^50 + 208x^51 - 650x^52 - 989x^53 + 916x^54 + 876x^55 - 35x^56 - 693x^57 + 920x^58 + 396x^59 - 331x^60 - 909x^61 - 153x^62 + 1638x^63 + 1160x^64 + 1292x^65 - 1016x^66 - 1570x^67 + 463x^68 - 61x^69 + 700x^70 - 303x^71 - 1080x^72 - 185x^73 - 199x^74 - 1406x^75 + 432x^76 + 945x^77 - 604x^78 + 668x^79 + 1653x^80 + 1090x^81 + 866x^82 + 328x^83 - 482x^84 - 1375x^85 - 878x^86 - 148x^87 - 1464x^88 - 1559x^89 + 586x^90 + 464x^91 - 548x^92 - 569x^93 - 1638x^94 + 1319x^95 + 412x^96 - 147x^97 - 459x^98 + 1173x^99 + 1009x^100 + 880x^101 + 128x^102 - 1541x^103 + 334x^104 + 127x^105 - 525x^106 + 161x^107 - 976x^108 + 1514x^109 - 427x^110 + 1452x^111 + 729x^112 + 507x^113 + 639x^114 - 101x^115 - 1660x^116 + 526x^117 + 1439x^118 + 1471x^119 - 759x^120 - 353x^121 + 56x^122 + 1499x^123 + 410x^124 - 282x^125 + 92x^126 - 137x^127 + 1403x^128 + 1333x^129 - 1159x^130 - 231x^131 + 145x^132 + 157x^133 + 1598x^134 + 920x^135 + 1198x^136 - 1327x^137 + 648x^138 + 846x^139 + 315x^140 - 933x^141 + 1450x^142 + 518x^143 - 865x^144 + 1518x^145 + 1375x^146 - 1011x^147 - 1237x^148 - 281x^149 + 1624x^150 - 525x^151 + 964x^152 - 506x^153 - 774x^154 - 981x^155 - 1122x^156 - 804x^157 - 791x^158 - 256x^159 - 912x^160 - 425x^161 - 270x^162 + 1001x^163 + 1459x^164 - 84x^165 + 1622x^166 - 203x^167 - 1374x^168 + 755x^169 - 1136x^170 + 226x^171 + 671x^172 + 1490x^173 - 1234x^174 - 270x^175 + 1636x^176 + 1223x^177 + 237x^178 + 710x^179 + 1414x^180 + 1220x^181 - 1515x^182 + 943x^183 + 714x^184 - 79x^185 - 825x^186 - 593x^187 - 1259x^188 - 77x^189 - 599x^190 - 500x^191 + 105x^192 + 467x^193 + 522x^194 - 1110x^195 + 1206x^196 + 1030x^197 + 1023x^198 + 247x^199 - 1563x^200 - 1292x^201 + 470x^202 + 1479x^203 - 1531x^204 + 4x^205 - 6x^206 - 637x^207 + 1504x^208 + 1265x^209 - 969x^210 - 589x^211 + 321x^212 + 1402x^213 - 436x^214 - 1515x^215 - 1329x^216 + 43x^217 + 1280x^218 + 657x^219 - 1331x^220 + 1592x^221 + 686x^222 + 1391x^223 + 110x^224 - 1527x^225 - 456x^226 - 167x^227 - 291x^228 - 1228x^229 - 941x^230 + 442x^231 - 1163x^232 + 165x^233 + 71x^234 + 26x^235 + 406x^236 - 854x^237 + 326x^238 + 1653x^239 + 590x^240 - 1092x^241 - 272x^242 - 1359x^243 + 566x^244 - 1145x^245 + 328x^246 + 1068x^247 - 359x^248 + 818x^249 + 741x^250 + 1423x^251 + 1577x^252 - 1004x^253 + 250x^254 + 1259x^255$

بلوک 2 پیام (تا ۲۵۶ بیت):

چندجمله‌ای پیام $m(x)$:

$m(x) = x^2 + x^9 + x^12 + x^13 + x^14 + x^17 + x^18 + x^23 + x^25 + x^26 + x^28 + x^30 + x^33 + x^34 + x^39 + x^41 + x^42 + x^45 + x^46 + x^49 + x^50 + x^52 + x^55$

پیام مقیاس شده $1665 \cdot m(x)$:

$\bar{m}(x) = 1665x^2 + 1665x^9 + 1665x^12 + 1665x^13 + 1665x^14 + 1665x^17 + 1665x^18 + 1665x^23 + 1665x^25 + 1665x^26 + 1665x^28 + 1665x^30 + 1665x^33 + 1665x^34 + 1665x^39 + 1665x^41 + 1665x^42 + 1665x^45 + 1665x^46 + 1665x^49 + 1665x^50 + 1665x^52 + 1665x^55$

خطای تصادفی $e_2 \in CBD_2$:

$e_2 = 1 + x^2 + x^4 + x^5 + 2x^6 + x^7 - x^8 + x^13 - x^14 + x^15 + x^16 + x^18 + x^20 - x^21 + x^26 + x^29 + x^30 - x^31 - 2x^32 - x^33 - x^34 + x^35 - 2x^36 + x^37 + 2x^38 - x^42 + x^43 + x^44 - 2x^45 + x^46 - 2x^47 + 2x^49 - x^50 + 2x^51 + x^52 + x^53 - x^55 + x^56 - x^57 + 2x^60 + x^62 + 2x^63 - x^64 - x^65 - x^68 + x^69 + x^70 + x^72 - x^74 - x^75 - 2x^76 - x^80 + x^81 - x^82 - x^83 - x^84 - x^85 + x^86 - x^87 - x^88 - 2x^90 + 2x^91 - x^93 + x^95 - x^96 + x^97 + 2x^98 + x^99 + x^100 - 2x^103 - x^107 - x^108 + x^110 + x^112 - x^113 - x^115 - x^119 + 2x^120 - x^121 + 2x^122 + x^126 + x^127 - x^128 - x^129 + x^131 + x^132 + 2x^136 - x^139 + 2x^141 + x^142 - 2x^144 + x^146 - x^150 + 2x^151 + 2x^152 + 2x^155 + x^156 - x^157 - x^158 + 2x^159 + x^160 + x^162 + x^165 + x^166 - 2x^168 + x^169 - x^170 - x^171 - x^175 - x^176 + x^177 - 2x^179 + 2x^180 + x^181 - x^182 + x^184 + x^185 + x^186 + x^187 + x^188 - x^190 + x^191 - x^194 + x^195 + x^197 - x^198 + x^199 - x^200 - x^202 - x^203 + x^205 - 2x^206 - x^208 + 2x^210 - x^211 + 2x^212 - x^215 + x^216 - x^218 - 2x^221 - 2x^222 + x^223 + x^226 + 2x^227 - x^228 + x^229 - 2x^230 - x^231 - x^232 + x^234 - x^237 + x^239 - x^240 + x^242 - x^243 + x^245 + x^246 + x^252 - x^253 - x^254$

مقدار نهایی $v$ برای این بلوک:

$v_{ 2 } = 844 - 1450x + 819x^2 - 1026x^3 - 1092x^4 + 1622x^5 + 104x^6 + 169x^7 - 281x^8 - 631x^9 - 683x^10 - 840x^11 + 204x^12 - 907x^13 - 627x^14 + 64x^15 - 1204x^16 + 1646x^17 - 895x^18 - 99x^19 + 712x^20 + 857x^21 + 941x^22 - 1214x^23 + 1318x^24 + 93x^25 - 1636x^26 + 52x^27 - 127x^28 + 1293x^29 + 408x^30 + 716x^31 - 229x^32 + 1279x^33 - 726x^34 + 188x^35 - 974x^36 + 908x^37 - 178x^38 - 1486x^39 + 1364x^40 - 1326x^41 + 1170x^42 + 1084x^43 + 1242x^44 + 1067x^45 + 396x^46 - 1029x^47 + 788x^48 + 515x^49 - 1050x^50 + 211x^51 + 1015x^52 + 675x^53 + 916x^54 + 873x^55 - 35x^56 + 970x^57 - 746x^58 + 395x^59 - 329x^60 + 754x^61 - 150x^62 + 1640x^63 + 1159x^64 + 1293x^65 + 647x^66 - 1571x^67 + 463x^68 - 62x^69 + 702x^70 - 303x^71 - 1081x^72 + 1479x^73 - 199x^74 - 1408x^75 + 430x^76 + 944x^77 + 1059x^78 - 997x^79 + 1652x^80 - 573x^81 - 799x^82 - 1338x^83 - 482x^84 - 1376x^85 + 787x^86 - 149x^87 - 1465x^88 + 105x^89 - 1080x^90 - 1198x^91 + 1114x^92 - 571x^93 - 1638x^94 - 345x^95 + 413x^96 + 1518x^97 + 1207x^98 - 491x^99 + 1009x^100 + 880x^101 + 127x^102 - 1543x^103 + 334x^104 - 1539x^105 + 1138x^106 - 1505x^107 - 978x^108 - 150x^109 - 428x^110 + 1452x^111 + 731x^112 - 1160x^113 - 1024x^114 - 100x^115 + 4x^116 - 1138x^117 - 227x^118 - 196x^119 - 756x^120 + 1309x^121 - 1608x^122 + 1497x^123 + 410x^124 + 1382x^125 - 1573x^126 + 1528x^127 + 1402x^128 - 333x^129 + 504x^130 + 1434x^131 + 147x^132 + 158x^133 - 66x^134 + 920x^135 + 1200x^136 + 338x^137 - 1017x^138 + 846x^139 + 315x^140 - 932x^141 + 1451x^142 - 1146x^143 - 867x^144 - 148x^145 - 289x^146 + 653x^147 - 1238x^148 - 281x^149 + 1624x^150 - 523x^151 + 965x^152 + 1159x^153 + 890x^154 - 981x^155 + 543x^156 - 806x^157 - 791x^158 - 254x^159 - 912x^160 + 1239x^161 + 1395x^162 - 664x^163 - 205x^164 - 85x^165 + 1623x^166 + 1462x^167 - 1376x^168 + 757x^169 + 528x^170 + 224x^171 + 673x^172 + 1489x^173 - 1234x^174 - 271x^175 + 1635x^176 - 442x^177 + 236x^178 + 709x^179 + 1416x^180 + 1220x^181 + 150x^182 - 721x^183 + 716x^184 + 1586x^185 + 839x^186 - 592x^187 + 408x^188 + 1587x^189 - 601x^190 - 500x^191 + 105x^192 - 1198x^193 - 1145x^194 - 1111x^195 + 1207x^196 + 1030x^197 + 1022x^198 - 1418x^199 - 1566x^200 + 373x^201 - 1194x^202 - 188x^203 - 1530x^204 + 5x^205 + 1656x^206 + 1027x^207 + 1503x^208 - 400x^209 + 697x^210 + 1074x^211 + 323x^212 + 1402x^213 + 1227x^214 + 146x^215 - 1328x^216 + 44x^217 - 387x^218 + 659x^219 + 333x^220 - 75x^221 + 684x^222 + 1391x^223 + 111x^224 - 1526x^225 + 1211x^226 - 166x^227 - 291x^228 - 1228x^229 - 941x^230 + 440x^231 - 1163x^232 - 1498x^233 + 71x^234 + 28x^235 + 407x^236 + 809x^237 + 326x^238 + 1654x^239 + 589x^240 + 573x^241 + 1393x^242 + 305x^243 + 566x^244 - 1143x^245 - 1336x^246 + 1067x^247 - 359x^248 + 817x^249 - 922x^250 + 1424x^251 - 86x^252 + 658x^253 - 1417x^254 + 1260x^255$

خروجی کامل متن رمزنگاری شده Kyber-768 ($c = (u, v)$)

جفت خروجی متن رمزنگاری شده $$c = (u, v)$$ شامل بردار چندجمله‌ای $u \in R_q^3$ و چندجمله‌ای‌های $v \in R_q$ به صورت خوانا آماده ارسال برای آلیس می‌باشد.

$$ Ciphertext \quad c = \left( u_{3 \times 1}, v \right) $$

مرحله سوم - رمزگشایی (Decryption)

$m' = \text{Round}_q(v - s^T u)$

در این مرحله، آلیس متن رمز شده $$c=(u, v)$$ را دریافت نموده و با استفاده از کلید خصوصی خود ( $s \in R_q^3$ ) چندجمله‌ای پیام اولیه را بازیابی می‌کند.

یادآوری کلید خصوصی آلیس (

s \in R_q^3

$$ m = \text{Round}_q \left( v - s^T \cdot u \right) $$

مراحل محاسباتی بازیابی پیام برای هر بلوک:

بلوک 1: تطابق دقیق با پیام اولیه

محاسبه ضرب داخلی $s^T u \pmod q$:

$s^T u = 885 - 1468x - 851x^2 - 1032x^3 - 1004x^4 + 1638x^5 + 172x^6 + 147x^7 - 322x^8 + 1048x^9 - 729x^10 - 921x^11 - 1446x^12 + 727x^13 + 1050x^14 + 41x^15 - 1199x^16 - 46x^17 + 828x^18 - 90x^19 + 663x^20 + 812x^21 + 990x^22 + 441x^23 + 1315x^24 - 1586x^25 + 72x^26 + 82x^27 + 1595x^28 + 1307x^29 - 1320x^30 + 714x^31 - 260x^32 - 391x^33 + 956x^34 + 233x^35 - 930x^36 + 977x^37 - 187x^38 + 177x^39 + 1341x^40 + 365x^41 - 511x^42 + 1116x^43 + 1288x^44 - 658x^45 - 1297x^46 - 955x^47 + 765x^48 - 1070x^49 + 627x^50 + 206x^51 - 696x^52 + 724x^53 + 857x^54 - 738x^55 - 77x^56 + 958x^57 - 702x^58 + 372x^59 - 296x^60 + 705x^61 - 37x^62 - 1658x^63 + 1141x^64 + 1331x^65 + 704x^66 - 1584x^67 + 507x^68 - 70x^69 + 761x^70 - 275x^71 - 1169x^72 + 1459x^73 - 234x^74 - 1391x^75 + 372x^76 + 948x^77 + 1049x^78 - 1006x^79 + 1575x^80 - 587x^81 - 817x^82 - 1326x^83 - 427x^84 - 1377x^85 + 813x^86 - 139x^87 - 1481x^88 + 125x^89 - 1064x^90 - 1238x^91 + 1068x^92 - 556x^93 + 1642x^94 - 332x^95 + 385x^96 + 1482x^97 + 1098x^98 - 520x^99 + 986x^100 + 840x^101 + 75x^102 - 1526x^103 + 374x^104 - 1558x^105 + 1177x^106 - 1537x^107 - 932x^108 - 172x^109 - 384x^110 + 1402x^111 + 699x^112 - 1099x^113 - 983x^114 - 158x^115 + 16x^116 - 1125x^117 - 235x^118 - 217x^119 - 828x^120 + 1335x^121 - 1557x^122 + 1516x^123 + 400x^124 + 1374x^125 - 1529x^126 + 1484x^127 + 1484x^128 - 347x^129 + 488x^130 + 1390x^131 + 132x^132 + 196x^133 - 81x^134 + 962x^135 + 1152x^136 + 346x^137 - 994x^138 + 868x^139 + 335x^140 - 933x^141 + 1361x^142 - 1156x^143 - 883x^144 - 133x^145 - 263x^146 + 676x^147 - 1185x^148 - 283x^149 - 1660x^150 - 535x^151 + 1007x^152 + 1177x^153 + 844x^154 - 899x^155 + 602x^156 - 783x^157 - 816x^158 - 276x^159 - 795x^160 + 1243x^161 + 1408x^162 - 622x^163 - 210x^164 - 69x^165 + 1632x^166 + 1401x^167 - 1443x^168 + 753x^169 + 449x^170 + 280x^171 + 684x^172 + 1538x^173 - 1286x^174 - 279x^175 + 1608x^176 - 458x^177 + 222x^178 + 743x^179 + 1505x^180 + 1169x^181 + 182x^182 - 736x^183 + 691x^184 + 1593x^185 + 810x^186 - 645x^187 + 373x^188 + 1595x^189 - 550x^190 - 481x^191 + 107x^192 - 1220x^193 - 1098x^194 - 1099x^195 + 1220x^196 + 1016x^197 + 1079x^198 - 1432x^199 - 1525x^200 + 388x^201 - 1136x^202 - 182x^203 - 1523x^204 - 99x^205 + 1660x^206 + 1013x^207 + 1504x^208 - 448x^209 + 636x^210 + 1059x^211 + 358x^212 + 1326x^213 + 1256x^214 + 177x^215 - 1328x^216 + 38x^217 - 400x^218 + 696x^219 + 322x^220 - 68x^221 + 757x^222 + 1405x^223 + 117x^224 - 1550x^225 + 1142x^226 - 164x^227 - 287x^228 - 1178x^229 - 935x^230 + 478x^231 - 1144x^232 - 1479x^233 + 111x^234 + 35x^235 + 432x^236 + 814x^237 + 387x^238 + 1595x^239 + 632x^240 + 598x^241 + 1375x^242 + 247x^243 + 574x^244 - 1188x^245 - 1350x^246 + 1091x^247 - 434x^248 + 842x^249 - 950x^250 + 1450x^251 - 89x^252 + 596x^253 - 1357x^254 + 1322x^255$

تفریق $v - s^T u \pmod q$:

$v - s^T u = -42 - 1646x + 2x^2 + 7x^3 - 89x^4 - 18x^5 - 70x^6 - 1644x^7 + 42x^8 + 1650x^9 - 1617x^10 + 82x^11 - 16x^12 - 1636x^13 - 11x^14 + 23x^15 - 5x^16 - 1638x^17 + 1606x^18 + 1658x^19 + 47x^20 - 1618x^21 + 1616x^22 + 10x^23 + 2x^24 - 1648x^25 + 1619x^26 - 31x^27 - 60x^28 - 15x^29 + 63x^30 - 1662x^31 + 32x^32 - 1658x^33 + 1649x^34 - 44x^35 + 1624x^36 + 1595x^37 - 1657x^38 + 23x^40 + 1638x^41 - 1646x^42 - 34x^43 - 47x^44 + 61x^45 - 1636x^46 + 1591x^47 + 23x^48 + 1583x^49 + 1651x^50 + 2x^51 + 46x^52 + 1616x^53 + 59x^54 + 1614x^55 + 42x^56 - 1651x^57 + 1622x^58 + 24x^59 - 35x^60 - 1614x^61 - 116x^62 - 33x^63 + 19x^64 - 39x^65 + 1609x^66 + 14x^67 - 44x^68 + 9x^69 - 61x^70 - 28x^71 + 89x^72 - 1644x^73 + 35x^74 - 15x^75 + 60x^76 - 3x^77 - 1653x^78 - 1655x^79 + 78x^80 - 1652x^81 - 1646x^82 + 1654x^83 - 55x^84 + 2x^85 + 1638x^86 - 9x^87 + 17x^88 + 1645x^89 + 1650x^90 - 1627x^91 - 1616x^92 - 13x^93 + 49x^94 + 1651x^95 + 27x^96 - 1629x^97 - 1557x^98 - 1636x^99 + 23x^100 + 40x^101 + 53x^102 - 15x^103 - 40x^104 - 1644x^105 + 1627x^106 - 1631x^107 - 44x^108 - 1643x^109 - 43x^110 + 50x^111 + 30x^112 + 1606x^113 + 1622x^114 + 57x^115 + 1653x^116 + 1651x^117 - 1655x^118 - 1641x^119 + 69x^120 + 1641x^121 + 1613x^122 - 17x^123 + 10x^124 - 1656x^125 + 1621x^126 - 1621x^127 - 81x^128 - 1649x^129 - 1647x^130 - 1621x^131 + 13x^132 - 39x^133 - 1650x^134 - 42x^135 + 46x^136 + 1656x^137 + 1642x^138 - 22x^139 - 20x^140 + 89x^142 - 1655x^143 + 18x^144 + 1651x^145 + 1638x^146 + 1642x^147 - 52x^148 + 2x^149 - 45x^150 + 10x^151 - 43x^152 + 1646x^153 - 1618x^154 - 82x^155 + 1605x^156 - 21x^157 + 25x^158 + 20x^159 - 117x^160 + 1661x^161 + 1651x^162 + 1623x^163 - 1660x^164 - 15x^165 - 10x^166 - 1604x^167 + 69x^168 + 2x^169 - 1585x^170 - 54x^171 - 13x^172 - 48x^173 + 52x^174 + 9x^175 + 28x^176 - 1648x^177 + 15x^178 - 33x^179 - 91x^180 + 51x^181 + 1632x^182 - 1650x^183 + 23x^184 + 1657x^185 - 1635x^186 + 52x^187 - 1632x^188 + 1657x^189 - 49x^190 - 19x^191 - 2x^192 - 1642x^193 + 1620x^194 - 11x^195 - 14x^196 + 14x^197 - 56x^198 - 1650x^199 - 38x^200 + 1649x^201 + 1606x^202 + 1661x^203 - 8x^204 + 103x^205 + 1663x^206 - 1650x^207 - 1616x^209 - 1605x^210 - 1648x^211 - 37x^212 + 76x^213 + 1637x^214 + 1637x^215 - x^216 + 5x^217 - 1649x^218 - 39x^219 - 1653x^220 + 1660x^221 - 71x^222 - 14x^223 - 7x^224 + 23x^225 - 1598x^226 - 3x^227 - 4x^228 - 50x^229 - 6x^230 - 36x^231 - 19x^232 + 1644x^233 - 40x^234 - 9x^235 - 26x^236 + 1661x^237 - 61x^238 + 58x^239 - 42x^240 + 1639x^241 - 1647x^242 - 1606x^243 - 8x^244 + 43x^245 - 1651x^246 - 23x^247 + 75x^248 - 24x^249 - 1638x^250 - 27x^251 - 1663x^252 - 1600x^253 + 1607x^254 - 63x^255$

چندجمله‌ای بازگشتی $m'(x) = \text{Round}_q(\cdot)$:

$m'(x) = x + x^7 + x^9 + x^10 + x^13 + x^17 + x^18 + x^19 + x^21 + x^22 + x^25 + x^26 + x^31 + x^33 + x^34 + x^36 + x^37 + x^38 + x^41 + x^42 + x^46 + x^47 + x^49 + x^50 + x^53 + x^55 + x^57 + x^58 + x^61 + x^66 + x^73 + x^78 + x^79 + x^81 + x^82 + x^83 + x^86 + x^89 + x^90 + x^91 + x^92 + x^95 + x^97 + x^98 + x^99 + x^105 + x^106 + x^107 + x^109 + x^113 + x^114 + x^116 + x^117 + x^118 + x^119 + x^121 + x^122 + x^125 + x^126 + x^127 + x^129 + x^130 + x^131 + x^134 + x^137 + x^138 + x^143 + x^145 + x^146 + x^147 + x^153 + x^154 + x^156 + x^161 + x^162 + x^163 + x^164 + x^167 + x^170 + x^177 + x^182 + x^183 + x^185 + x^186 + x^188 + x^189 + x^193 + x^194 + x^199 + x^201 + x^202 + x^203 + x^206 + x^207 + x^209 + x^210 + x^211 + x^214 + x^215 + x^218 + x^220 + x^221 + x^226 + x^233 + x^237 + x^241 + x^242 + x^243 + x^246 + x^250 + x^252 + x^253 + x^254$

بلوک 2: تطابق دقیق با پیام اولیه

محاسبه ضرب داخلی $s^T u \pmod q$:

تفریق $v - s^T u \pmod q$:

$v - s^T u = -41 + 18x - 1659x^2 + 6x^3 - 88x^4 - 16x^5 - 68x^6 + 22x^7 + 41x^8 + 1650x^9 + 46x^10 + 81x^11 + 1650x^12 - 1634x^13 + 1652x^14 + 23x^15 - 5x^16 - 1637x^17 + 1606x^18 - 9x^19 + 49x^20 + 45x^21 - 49x^22 - 1655x^23 + 3x^24 - 1650x^25 + 1621x^26 - 30x^27 + 1607x^28 - 14x^29 - 1601x^30 + 2x^31 + 31x^32 - 1659x^33 + 1647x^34 - 45x^35 - 44x^36 - 69x^37 + 9x^38 - 1663x^39 + 23x^40 + 1638x^41 - 1648x^42 - 32x^43 - 46x^44 - 1604x^45 - 1636x^46 - 74x^47 + 23x^48 + 1585x^49 + 1652x^50 + 5x^51 - 1618x^52 - 49x^53 + 59x^54 + 1611x^55 + 42x^56 + 12x^57 - 44x^58 + 23x^59 - 33x^60 + 49x^61 - 113x^62 - 31x^63 + 18x^64 - 38x^65 - 57x^66 + 13x^67 - 44x^68 + 8x^69 - 59x^70 - 28x^71 + 88x^72 + 20x^73 + 35x^74 - 17x^75 + 58x^76 - 4x^77 + 10x^78 + 9x^79 + 77x^80 + 14x^81 + 18x^82 - 12x^83 - 55x^84 + x^85 - 26x^86 - 10x^87 + 16x^88 - 20x^89 - 16x^90 + 40x^91 + 46x^92 - 15x^93 + 49x^94 - 13x^95 + 28x^96 + 36x^97 + 109x^98 + 29x^99 + 23x^100 + 40x^101 + 52x^102 - 17x^103 - 40x^104 + 19x^105 - 39x^106 + 32x^107 - 46x^108 + 22x^109 - 44x^110 + 50x^111 + 32x^112 - 61x^113 - 41x^114 + 58x^115 - 12x^116 - 13x^117 + 8x^118 + 21x^119 + 72x^120 - 26x^121 - 51x^122 - 19x^123 + 10x^124 + 8x^125 - 44x^126 + 44x^127 - 82x^128 + 14x^129 + 16x^130 + 44x^131 + 15x^132 - 38x^133 + 15x^134 - 42x^135 + 48x^136 - 8x^137 - 23x^138 - 22x^139 - 20x^140 + x^141 + 90x^142 + 10x^143 + 16x^144 - 15x^145 - 26x^146 - 23x^147 - 53x^148 + 2x^149 - 45x^150 + 12x^151 - 42x^152 - 18x^153 + 46x^154 - 82x^155 - 59x^156 - 23x^157 + 25x^158 + 22x^159 - 117x^160 - 4x^161 - 13x^162 - 42x^163 + 5x^164 - 16x^165 - 9x^166 + 61x^167 + 67x^168 + 4x^169 + 79x^170 - 56x^171 - 11x^172 - 49x^173 + 52x^174 + 8x^175 + 27x^176 + 16x^177 + 14x^178 - 34x^179 - 89x^180 + 51x^181 - 32x^182 + 15x^183 + 25x^184 - 7x^185 + 29x^186 + 53x^187 + 35x^188 - 8x^189 - 51x^190 - 19x^191 - 2x^192 + 22x^193 - 47x^194 - 12x^195 - 13x^196 + 14x^197 - 57x^198 + 14x^199 - 41x^200 - 15x^201 - 58x^202 - 6x^203 - 7x^204 + 104x^205 - 4x^206 + 14x^207 - x^208 + 48x^209 + 61x^210 + 15x^211 - 35x^212 + 76x^213 - 29x^214 - 31x^215 + 6x^217 + 13x^218 - 37x^219 + 11x^220 - 7x^221 - 73x^222 - 14x^223 - 6x^224 + 24x^225 + 69x^226 - 2x^227 - 4x^228 - 50x^229 - 6x^230 - 38x^231 - 19x^232 - 19x^233 - 40x^234 - 7x^235 - 25x^236 - 5x^237 - 61x^238 + 59x^239 - 43x^240 - 25x^241 + 18x^242 + 58x^243 - 8x^244 + 45x^245 + 14x^246 - 24x^247 + 75x^248 - 25x^249 + 28x^250 - 26x^251 + 3x^252 + 62x^253 - 60x^254 - 62x^255$

چندجمله‌ای بازگشتی $m'(x) = \text{Round}_q(\cdot)$:

$m'(x) = x^2 + x^9 + x^12 + x^13 + x^14 + x^17 + x^18 + x^23 + x^25 + x^26 + x^28 + x^30 + x^33 + x^34 + x^39 + x^41 + x^42 + x^45 + x^46 + x^49 + x^50 + x^52 + x^55$

متن بازگشایی شده نهایی (Decrypted Plaintext Output):

                "Advanced Cryptography Class, Dr. Najafi"
            

پیام رمزگشایی شده کاملاً یکسان و برابر با متن اولیه انتخاب شده است!

بخش تعاملی (Interactive Playground) - سیستم Kyber-768

Live Execution Engine

راهنمای عملکرد:

سیستم با مشخصات فیکس‌شده استاد ($q=3329, n=256, k=3, \eta_1=2, \eta_2=2$) اجرا می‌گردد. با ارسال فرم فوق، کلیدهای جدید و بردارهای تصادفی تولید شده و خروجی تمام مراحل ریاضی در زبانه‌های مربوطه بروزرسانی خواهد شد.