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Аналог системы Эль-Гамаля.




К несекретным данным в данной криптосистеме относятся: конечное поле , определенная на нем эллиптическая кривая и точка-«основания» на ней.

Отправитель общения выбирает целое число , являющееся секретным ключом, и вычисляет открытый ключ Шифруемое сообщение – это точка Для шифровки сообщения необходимо выбрать сеансовый ключ и вычислить координаты двух точек и Координаты точек высылаются получателю сообщении. Для расшифровки сообщения необходимо вычислить:

К недостаткам эллиптического аналога системы Эль-Гамаля следует отнести то, что способ, по которому текстовому сообщению можно поставить в соответствие точки кривой , вообще говоря, неочевиден. Для преодоления этих сложностей был предложен вероятностный алгоритм (более подробно см. в [3]).

В эллиптической криптосистеме Эль-Гамаля приходится передавать четыре числа – координаты точек . Для уменьшения количества передаваемых чисел вместо двух координат точки где – элементы конечного поля , ограничиваются передачей лишь абсциссы этой точки. Это возможно, т.к. связаны соотношением . Поэтому если известно, то вычислить не представляет большого труда. Однако при вычислениях по формуле, приведенной выше, мы получим два значения для Т.к. для расшифровки сообщения в эллиптической криптосистеме Эль-Гамаля требуется знать две координаты для каждой из точек , то в таких случаях используют еще один дополнительный бит – для указания того, какое же из значений является истинным:

Литература:

1. Мао В. Современная криптография: теория и практика / В. Мао. – М.: Издательский дом «Вильямс», 2005. – 768 с.

2. Hoffstein J. An Introduction to Mathematical Cryptography / J. Hoffstein, J. Pipher, J.H. Sinverman. – Brawn University, 2004. – 361 p.

3. Коблиц Н. Курс теории чисел и криптографии / Н. Коблиц. – М.: Научное изд-во ТВП, 2001. – 261 с.

 

 

Задача.Расшифруйте сообщение, зашифрованное с помощью эллиптического аналога криптосистемы Эль-Гамаля. Использованная кривая: Соответствие точек кривой и букв алфавита для разных вариантов дано в Таблице 2.

Таблица 1.

 

  (секретный ключ) (сеансовый ключ) Ответ
1. {5166,3065} {6153,940} {4249,3972} {2832,3491}, {3743,3701}, {575,730}, {648,722}  
2. {2080, 1582} {5078,5099} {4243, 1967} {591, 91}, {4575, 3979}, {1100, 637}, {231, 587}  
3. {4621, 1079} {2586, 963} {5824, 4741} {2109, 2466}, {433, 1213}, {2887, 131}, {3240, 4505}  
4. {917, 2472} {1545, 792} {5581, 4846} {3059, 5547}, {645, 5591}, {5750, 4557}, {5490, 4776}  
5. {5326, 251} {1864, 006} {4232, 6253} {1169, 758}, {2668, 5106}, {3774, 1941}, {5620, 3342}  
6. {1190, 875} {2047, 620} {1522, 184} {1949, 4877}, {3972, 81}, {6387, 4092}, {874, 2502}  
7. {3571, 860}     {5934, 385} {843, 4969} {2826, 1672}, {1949, 1544}, {583, 818}, {657, 1479}  
8. {690, 1952} {1443, 391} {5228, 383} {3265, 6019}, {4813, 6002}, {2670, 1841}, {6241, 1669}  
9. {4638, 661} {5573, 118} {5615, 3989} {4852, 4824}, {3846, 5597}, {5138, 693}, {2773, 1797}  
10. {1898, 2910} {6232, 758} {3959, 2014}   {4100, 6017}, {4245, 6178}, {6032, 1606}, {5521, 1428}  
11. {2891, 2272} {63, 337} {2080, 1582} {2560, 3025}, {4527, 5309}, {4908, 4539}, {2778, 3293}  
12. {997, 402} {2321, 5444} {3662, 893} {3669, 3478}, {253, 5575}, {3351, 6100}, {657, 1479}  
13. {1063, 75}   {5302, 2959} {6130, 4090} {4629, 1081}, {4139, 4318}, {414, 4728}, {757, 4197}  
14. {5297, 3112} {6067, 3181} {2085, 2868} {1567, 3276}, {5055, 5249}, {4235, 4134}, {5913, 3299}  
15. {1276, 1390}   {4288, 2786} {4314, 3675} {978, 1535}, {6119, 972}, {6378, 483}, {3530, 1660}  
16. {1510, 514}   {3673, 5248} {5558, 5177} {4481, 4715}, {183, 2039}, {1872, 3786}, {5439, 4516}  
17. {6305, 1994} {3341, 5451} {4519, 5840} {4515, 5630}, {5249, 2483}, {875, 3977}, {5227, 6190}  

 

Таблица 2.

  Соответствие букв алфавита и точек на кривой
1. "А" -> {5044, 723}, "Б" -> {3075, 1907}, "В" -> {3568, 660}, "Г" -> {2509, 1709}, "Д" -> {1535, 865}, "Е" -> {5301, 1652}, "Ё" -> {3622, 126}, "Ж" -> {3548, 2970}, "З" -> {6052, 1764}, "И" -> {3279, 1630}, "Й" -> {5307, 1707}, "К" -> {3560, 2326}, "Л" -> {3236, 2814}, "М" -> {5216, 2486}, "Н" -> {2819, 355}, "О" -> {3384, 1539}, "П" -> {4481, 1706}, "Р" -> {4385, 2029}, "С" -> {3700, 2573}, "Т" -> {2249, 862}, "У" -> {3988, 1440}, "Ф" -> {2819, 355}, "Х" -> {5231, 134}, "Ц" -> {588, 2346}, "Ч" -> {3927, 1453}, "Ш" -> {2553, 1879}, "Щ" -> {645, 830}, "Ъ" -> {2044, 1122}, "Ы" -> {814, 602}, "Ь" -> {954, 709}, "Э" -> {3412, 87}, "Ю" -> {6144, 332}, "Я" -> {6056, 1970}
2. "А" -> {619, 3065}, "Б" -> {2295, 1551}, "В" -> {4769, 1804}, "Г" -> {5164, 734}, "Д" -> {1796, 2954}, "Е" -> {2138, 895}, "Ё" -> {1425, 2495}, "Ж" -> {5826, 272}, "З" -> {1017, 586}, "И" -> {218, 1704}, "Й" -> {1887, 471}, "К" -> {5287, 1600}, "Л" -> {6240, 2337}, "М" -> {5043, 2564}, "Н" -> {2151, 2579}, "О" -> {2861, 1649}, "П" -> {5423, 3170}, "Р" -> {1253, 248}, "С" -> {5972, 498}, "Т" -> {6230, 1977}, "У" -> {3506, 438}, "Ф" -> {5916, 940}, "Х" -> {3405, 2133}, "Ц" -> {3409, 2401}, "Ч" -> {4753, 874}, "Ш" -> {6144, 332}, "Щ" -> {5295, 61}, "Ъ" -> {1681, 2384}, "Ы" -> {5399, 1567}, "Ь" -> {207, 3069}, "Э" -> {3749, 688}, "Ю" -> {5762, 3096}, "Я" -> {2730, 2268}
3. "А" -> {2732, 335}, "Б" -> {6146, 3108}, "В" -> {3566, 1190}, "Г" -> {5329, 1462}, "Д" -> {5472, 2530}, "Е" -> {5682, 2297}, "Ё" -> {2325, 3050}, "Ж" -> {5737, 1879}, "З" -> {3743, 2720}, "И" -> {1341, 1739}, "Й" -> {3351, 321}, "К" -> {2087, 31}, "Л" -> {6349, 896}, "М" -> {6320, 2940}, "Н" -> {5833, 2525}, "О" -> {4424, 967}, "П" -> {1752, 796}, "Р" -> {4698, 1223}, "С" -> {648, 722}, "Т" -> {3928, 1584}, "У" -> {4927, 1808}, "Ф" -> {4639, 1221}, "Х" -> {1343, 2060}, "Ц" -> {2294, 2809}, "Ч" -> {1937, 2560}, "Ш" -> {1801, 1902}, "Щ" -> {6208, 2725}, "Ъ" -> {2044, 1122}, "Ы" -> {3532, 307}, "Ь" -> {4559, 1225}, "Э" -> {766, 1199}, "Ю" -> {3827, 29}, "Я" -> {1707, 1410}
4. "А" -> {6412, 2895}, "Б" -> {5963, 2430}, "В" -> {1786, 265}, "Г" -> {4013, 464}, "Д" -> {277, 537}, "Е" -> {5179, 2735}, "Ё" -> {4487, 2302}, "Ж" -> {5853, 1219}, "З" -> {4724, 1752}, "И" -> {1744, 1574}, "Й" -> {4927, 1808}, "К" -> {5894, 179}, "Л" -> {5526, 2629}, "М" -> {2669, 2652}, "Н" -> {4496, 353}, "О" -> {2477, 748}, "П" -> {484, 618}, "Р" -> {748, 2043}, "С" -> {2857, 1412}, "Т" -> {4506, 1774}, "У" -> {265, 768}, "Ф" -> {5824, 1680}, "Х" -> {1299, 2539}, "Ц" -> {1159, 2888}, "Ч" -> {2059, 3071}, "Ш" -> {4372, 1850}, "Щ" -> {5450, 998}, "Ъ" -> {6414, 2894}, "Ы" -> {3998, 219}, "Ь" -> {1730, 2994}, "Э" -> {1569, 2385}, "Ю" -> {4715, 2583}, "Я" -> {2623, 2513}
5. "А" -> {5538, 2283}, "Б" -> {3284, 2675}, "В" -> {5115, 1012}, "Г" -> {3973, 538}, "Д" -> {5486, 3077}, "Е" -> {3228, 1999}, "Ё" -> {342, 1923}, "Ж" -> {3856, 995}, "З" -> {5685, 748}, "И" -> {2103, 296}, "Й" -> {2599, 943}, "К" -> {4719, 2912}, "Л" -> {3490, 1198}, "М" -> {3923, 2744}, "Н" -> {484, 618}, "О" -> {1884, 3056}, "П" -> {1624, 432}, "Р" -> {3527, 26}, "С" -> {5740, 2736}, "Т" -> {5716, 3133}, "У" -> {6396, 706}, "Ф" -> {4502, 2218}, "Х" -> {5382, 1079}, "Ц" -> {3694, 809}, "Ч" -> {4583, 1568}, "Ш" -> {4688, 2055}, "Щ" -> {3977, 412}, "Ъ" -> {2924, 2244}, "Ы" -> {4518, 149}, "Ь" -> {3768, 1536}, "Э" -> {390, 91}, "Ю" -> {3801, 1552}, "Я" -> {1015, 2325}
6. "А" -> {3236, 2814}, "Б" -> {4737, 1312}, "В" -> {2325, 3050}, "Г" -> {5096, 1174}, "Д" -> {3983, 1562}, "Е" -> {2474, 1468}, "Ё" -> {1724, 1781}, "Ж" -> {5542, 1723}, "З" -> {1203, 1696}, "И" -> {842, 1238}, "Й" -> {286, 596}, "К" -> {5604, 2774}, "Л" -> {6085, 2532}, "М" -> {3103, 1369}, "Н" -> {3801, 1552}, "О" -> {5651, 1424}, "П" -> {1005, 2142}, "Р" -> {2183, 2297}, "С" -> {1574, 2002}, "Т" -> {5224, 2967}, "У" -> {5470, 188}, "Ф" -> {4736, 496}, "Х" -> {731, 2117}, "Ц" -> {1629, 740}, "Ч" -> {982, 2147}, "Ш" -> {3241, 1431}, "Щ" -> {1264, 1524}, "Ъ" -> {4201, 1670}, "Ы" -> {543, 1591}, "Ь" -> {949, 2205}, "Э" -> {5958, 2051}, "Ю" -> {5294, 2580}, "Я" -> {2573, 627}
7. "А" -> {3741, 1977}, "Б" -> {2070, 958}, "В" -> {5896, 1248}, "Г" -> {4737, 1312}, "Д" -> {6031, 1580}, "Е" -> {3774, 1941}, "Ё" -> {5678, 126}, "Ж" -> {5902, 2095}, "З" -> {905, 2746}, "И" -> {3680, 239}, "Й" -> {4935, 1643}, "К" -> {5460, 579}, "Л" -> {5574, 2149}, "М" -> {480, 2037}, "Н" -> {4603, 558}, "О" -> {4061, 2161}, "П" -> {6261, 2294}, "Р" -> {1175, 2876}, "С" -> {3937, 2033}, "Т" -> {766, 1199}, "У" -> {3184, 2685}, "Ф" -> {3227, 762}, "Х" -> {1575, 2837}, "Ц" -> {1231, 538}, "Ч" -> {5824, 1680}, "Ш" -> {3059, 874}, "Щ" -> {3047, 1380}, "Ъ" -> {2422, 2197}, "Ы" -> {4384, 344}, "Ь" -> {3144, 24}, "Э" -> {2507, 394}, "Ю" -> {5192, 1385}, "Я" -> {4455, 187}
8. "А" -> {5463, 1053}, "Б" -> {1238, 819}, "В" -> {2090, 3091}, "Г" -> {4829, 680}, "Д" -> {1478, 183}, "Е" -> {4080, 1602}, "Ё" -> {5479, 3014}, "Ж" -> {1228, 2992}, "З" -> {5246, 375}, "И" -> {1864, 1006}, "Й" -> {5923, 1377}, "К" -> {2202, 3024}, "Л" -> {1420, 2713}, "М" -> {1985, 1963}, "Н" -> {1025, 2212}, "О" -> {6120, 2273}, "П" -> {353, 1246}, "Р" -> {5201, 577}, "С" -> {4257, 2104}, "Т" -> {2088, 972}, "У" -> {4658, 2294}, "Ф" -> {1871, 881}, "Х" -> {4646, 794}, "Ц" -> {781, 1769}, "Ч" -> {4176, 1639}, "Ш" -> {3255, 863}, "Щ" -> {4271, 2511}, "Ъ" -> {3596, 1316}, "Ы" -> {541, 451}, "Ь" -> {3652, 969}, "Э" -> {3255, 863}, "Ю" -> {857, 767}, "Я" -> {1494, 160}
9. "А" -> {896, 3109}, "Б" -> {2937, 2098}, "В" -> {6323, 2013}, "Г" -> {5959, 2199}, "Д" -> {1741, 549}, "Е" -> {1227, 1011}, "Ё" -> {2873, 2950}, "Ж" -> {3508, 3054}, "З" -> {4796, 2728}, "И" -> {1832, 2494}, "Й" -> {1411, 2279}, "К" -> {5888, 1325}, "Л" -> {1715, 1353}, "М" -> {5544, 2374}, "Н" -> {3472, 2215}, "О" -> {583, 818}, "П" -> {5178, 2560}, "Р" -> {5161, 1570}, "С" -> {4010, 2126}, "Т" -> {2146, 2688}, "У" -> {3864, 2040}, "Ф" -> {503, 213}, "Х" -> {2334, 811}, "Ц" -> {1863, 2799}, "Ч" -> {5893, 1150}, "Ш" -> {2565, 2144}, "Щ" -> {3641, 1665}, "Ъ" -> {5976, 1156}, "Ы" -> {2251, 612}, "Ь" -> {682, 1866}, "Э" -> {657, 1479}, "Ю" -> {5227, 231}, "Я" -> {266, 3049}
10. "А" -> {3761, 1027}, "Б" -> {2704, 2888}, "В" -> {4372, 1850}, "Г" -> {6133, 674}, "Д" -> {3206, 2418}, "Е" -> {1769, 1040}, "Ё" -> {982, 2147}, "Ж" -> {2499, 19}, "З" -> {4832, 2736}, "И" -> {1619, 944}, "Й" -> {297, 2161}, "К" -> {5454, 1449}, "Л" -> {5697, 1611}, "М" -> {1597, 1751}, "Н" -> {1773, 928}, "О" -> {3013, 2805}, "П" -> {2133, 1666}, "Р" -> {4883, 999}, "С" -> {4765, 2396}, "Т" -> {6136, 1867}, "У" -> {3262, 801}, "Ф" -> {2880, 1335}, "Х" -> {2977, 1166}, "Ц" -> {555, 2884}, "Ч" -> {3897, 3162}, "Ш" -> {237, 3022}, "Щ" -> {3509, 3070}, "Ъ" -> {5096, 1174}, "Ы" -> {5896, 1248}, "Ь" -> {5956, 3047}, "Э" -> {5841, 1724}, "Ю" -> {202, 1395}, "Я" -> {2209, 2632}
11. "А" -> {2330, 1420}, "Б" -> {5000, 2908}, "В" -> {6005, 759}, "Г" -> {4535, 2034}, "Д" -> {1945, 1311}, "Е" -> {2825, 1120}, "Ё" -> {2027, 2955}, "Ж" -> {2355, 1968}, "З" -> {595, 1830}, "И" -> {5740, 2736}, "Й" -> {3323, 1117}, "К" -> {982, 2147}, "Л" -> {4413, 427}, "М" -> {4235, 2287}, "Н" -> {4258, 563}, "О" -> {372, 2222}, "П" -> {660, 1740}, "Р" -> {1884, 3056}, "С" -> {3764, 2765}, "Т" -> {3954, 1884}, "У" -> {4450, 1558}, "Ф" -> {5976, 1156}, "Х" -> {4677, 39}, "Ц" -> {2672, 3148}, "Ч" -> {1724, 1781}, "Ш" -> {5903, 2231}, "Щ" -> {2329, 517}, "Ъ" -> {1980, 943}, "Ы" -> {1179, 2577}, "Ь" -> {3487, 2279}, "Э" -> {1863, 2799}, "Ю" -> {2094, 1700}, "Я" -> {3231, 322}
12. "А" -> {3903, 663}, "Б" -> {3376, 1577}, "В" -> {4293, 1814}, "Г" -> {3554, 2334}, "Д" -> {5725, 448}, "Е" -> {5447, 3037}, "Ё" -> {577, 2620}, "Ж" -> {3368, 317}, "З" -> {6153, 940}, "И" -> {3664, 1443}, "Й" -> {3429, 759}, "К" -> {4380, 2341}, "Л" -> {5507, 1339}, "М" -> {3124, 3132}, "Н" -> {895, 1083}, "О" -> {253, 846}, "П" -> {5157, 890}, "Р" -> {3241, 1431}, "С" -> {4506, 1774}, "Т" -> {4061, 2161}, "У" -> {5305, 2054}, "Ф" -> {622, 175}, "Х" -> {4719, 2912}, "Ц" -> {290, 2435}, "Ч" -> {1044, 3048}, "Ш" -> {1715, 1353}, "Щ" -> {2160, 3120}, "Ъ" -> {1520, 19}, "Ы" -> {2646, 2833}, "Ь" -> {3282, 721}, "Э" -> {4903, 2823}, "Ю" -> {3889, 2860}, "Я" -> {2244, 1535}
13. "А" -> {4626, 2395}, "Б" -> {3164, 2121}, "В" -> {2780, 2167}, "Г" -> {5984, 418}, "Д" -> {921, 1940}, "Е" -> {6317, 890}, "Ё" -> {2430, 1400}, "Ж" -> {3411, 1914}, "З" -> {968, 1040}, "И" -> {1840, 2067}, "Й" -> {6213, 3165}, "К" -> {2178, 1962}, "Л" -> {297, 2161}, "М" -> {1936, 295}, "Н" -> {3376, 1577}, "О" -> {5521, 1428}, "П" -> {1524, 770}, "Р" -> {1847, 2739}, "С" -> {565, 1294}, "Т" -> {1584, 3040}, "У" -> {5290, 2532}, "Ф" -> {6163, 3010}, "Х" -> {3580, 2035}, "Ц" -> {3952, 368}, "Ч" -> {1391, 1730}, "Ш" -> {5752, 2819}, "Щ" -> {4365, 555}, "Ъ" -> {1138, 959}, "Ы" -> {3917, 2232}, "Ь" -> {5119, 795}, "Э" -> {2645, 1698}, "Ю" -> {1312, 2548}, "Я" -> {4320, 394}
14. "А" -> {2543, 1179}, "Б" -> {1191, 204}, "В" -> {3610, 121}, "Г" -> {3923, 2744}, "Д" -> {352, 1931}, "Е" -> {6039, 2516}, "Ё" -> {2704, 2888}, "Ж" -> {3972, 81}, "З" -> {3656, 747}, "И" -> {1002, 1772}, "Й" -> {4121, 1489}, "К" -> {5707, 582}, "Л" -> {2013, 2341}, "М" -> {663, 1473}, "Н" -> {829, 97}, "О" -> {6270, 1375}, "П" -> {2206, 2039}, "Р" -> {4710, 1267}, "С" -> {866, 3134}, "Т" -> {1070, 2967}, "У" -> {2819, 355}, "Ф" -> {5737, 1879}, "Х" -> {2757, 2080}, "Ц" -> {1118, 991}, "Ч" -> {2194, 3061}, "Ш" -> {521, 533}, "Щ" -> {3291, 1597}, "Ъ" -> {4325, 2226}, "Ы" -> {1762, 85}, "Ь" -> {1752, 796}, "Э" -> {6166, 709}, "Ю" -> {2592, 3163}, "Я" -> {5088, 2022}
15. "А" -> {268, 1563}, "Б" -> {4422, 1082}, "В" -> {591, 91}, "Г" -> {1888, 985}, "Д" -> {4936, 2571}, "Е" -> {1904, 719}, "Ё" -> {2208, 2841}, "Ж" -> {6043, 459}, "З" -> {2439, 370}, "И" -> {5556, 803}, "Й" -> {2378, 1510}, "К" -> {715, 366}, "Л" -> {2880, 1335}, "М" -> {5033, 119}, "Н" -> {3882, 1088}, "О" -> {1153, 1661}, "П" -> {2577, 600}, "Р" -> {2695, 1713}, "С" -> {2090, 3091}, "Т" -> {4880, 2719}, "У" -> {6389, 3189}, "Ф" -> {5240, 2112}, "Х" -> {2513, 2904}, "Ц" -> {253, 846}, "Ч" -> {3729, 1558}, "Ш" -> {2321, 977}, "Щ" -> {3977, 412}, "Ъ" -> {1715, 1353}, "Ы" -> {2081, 1014}, "Ь" -> {4054, 1481}, "Э" -> {4013, 464}, "Ю" -> {2578, 1438}, "Я" -> {6170, 1081}
16. "А" -> {3825, 2017}, "Б" -> {2880, 1335}, "В" -> {5461, 1986}, "Г" -> {1800, 3203}, "Д" -> {4299, 469}, "Е" -> {2548, 723}, "Ё" -> {4638, 1661}, "Ж" -> {3785, 1662}, "З" -> {650, 72}, "И" -> {1630, 1779}, "Й" -> {1380, 2939}, "К" -> {3345, 2291}, "Л" -> {6182, 3030}, "М" -> {3472, 2215}, "Н" -> {3103, 1369}, "О" -> {4771, 2898}, "П" -> {6192, 1987}, "Р" -> {3680, 239}, "С" -> {3414, 15}, "Т" -> {6160, 2148}, "У" -> {4151, 41}, "Ф" -> {4404, 1342}, "Х" -> {5244, 1222}, "Ц" -> {2346, 3154}, "Ч" -> {4204, 2016}, "Ш" -> {842, 1238}, "Щ" -> {4849, 694}, "Ъ" -> {1100, 637}, "Ы" -> {1387, 3108}, "Ь" -> {1816, 1532}, "Э" -> {482, 732}, "Ю" -> {1676, 3037}, "Я" -> {4152, 2175}
17. "А" -> {1619, 944}, "Б" -> {2833, 887}, "В" -> {3652, 969}, "Г" -> {320, 137}, "Д" -> {5671, 518}, "Е" -> {4959, 3165}, "Ё" -> {3463, 2198}, "Ж" -> {1055, 2150}, "З" -> {231, 587}, "И" -> {1187, 1637}, "Й" -> {207, 3069}, "К" -> {3031, 622}, "Л" -> {298, 2765}, "М" -> {910, 2452}, "Н" -> {5740, 2736}, "О" -> {1821, 2756}, "П" -> {3958, 428}, "Р" -> {2263, 1440}, "С" -> {5507, 1339}, "Т" -> {6335, 1326}, "У" -> {6080, 1311}, "Ф" -> {1707, 1410}, "Х" -> {962, 1520}, "Ц" -> {1910, 864}, "Ч" -> {6241, 1669}, "Ш" -> {4649, 2579}, "Щ" -> {4580, 781}, "Ъ" -> {3224, 1951}, "Ы" -> {4547, 2836}, "Ь" -> {1656, 7}, "Э" -> {1520, 19}, "Ю" -> {6161, 1847}, "Я" -> {1523, 2279}

 


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