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text each coefficient in the matrix, that is mt exponentiations instead of t in the other algorithms.\n
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text However, on the user side, the computational cost remains similar and will be dominated by the\n
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text can be upper bounded by 2 2 O(m (logN) ). Decryption costs 3 O(m(logN) ), which will dominate as\n
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text long as m is smaller than a few thousands. The document extraction process is thus no longer linear\n
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text in m, but the quadratic component is negligible in practice.\n
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text the Paillier encryption scheme: using a different homomorphic encryption scheme could improve\n
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text this. The remaining 2 log t bits are however necessary to get a deterministic zero-error algorithm:\n
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text the user has to solve an error correction problem, meaning he will have to learn both the value\n
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text way of looking at the zero-error property: the user knows when he misses something, whereas in\n
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text but has no information wether he got all the documents or not. With our Reed-Solomon scheme,\n
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text it is possible to imagine an interactive 4 protocol where the user can query for additional syndrome\n
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text 3 In practice, this can be reduced to 3 logm+3 log |K| bits, but having a dependency on m is not really convenient.\n
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text offer much better performance than Reed-Solomon codes. However, the error correction problem\n
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text also has to be transformed into an erasure correction problem. This is possible by combining the\n
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text following ideas (which is similar to what Danezis and Dı́az proposed):\n
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text – instead of using a fix LDPC matrix, generate it from the documents fi themselves,\n
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text – use a decoding algorithm similar to the erasure correction algorithm proposed by Luby and\n
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text Mitzenmacher for verification codes [8].\n
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text Query Construction. This step is the same as for the Ostrovsky-Skeith construction. Instead of m\n
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text and γ, the user sends the desired buffer length ℓ to the server.\n
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Then,\n", + "_metadata": { + "tag_hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "unknown" + }, + "hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "raw_text" + }, + "page_id": 6, + "line_id": 70248, + "_LineMetadata__other_fields": {} + }, + "_annotations": [ + { + "start": 0, + "end": 98, + "name": "indentation", + "value": "72", + "is_mergeable": true + }, + { + "start": 0, + "end": 98, + "name": "spacing", + "value": "19", + "is_mergeable": true + }, + { + "start": 0, + "end": 5, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 0, + "end": 5, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 6, + "end": 16, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 6, + "end": 16, + "name": "style", + "value": "JVOVRL+CMTI10", + "is_mergeable": true + }, + { + "start": 6, + "end": 16, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 17, + "end": 20, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 17, + "end": 20, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 21, + "end": 27, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 21, + "end": 27, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 28, + "end": 32, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 28, + "end": 32, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 33, + "end": 44, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 33, + "end": 44, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 45, + "end": 46, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 45, + "end": 46, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 47, + "end": 52, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 47, + "end": 52, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 53, + "end": 54, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 53, + "end": 54, + "name": "style", + "value": "WLPKII+CMMI10", + "is_mergeable": true + }, + { + "start": 53, + "end": 54, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 55, + "end": 57, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 55, + "end": 57, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 58, + "end": 62, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 58, + "end": 62, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 63, + "end": 64, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 63, + "end": 64, + "name": "style", + "value": "WLPKII+CMMI10", + "is_mergeable": true + }, + { + "start": 63, + "end": 64, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 65, + "end": 67, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 65, + "end": 67, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 68, + "end": 72, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 68, + "end": 72, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 73, + "end": 75, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 73, + "end": 75, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 76, + "end": 81, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 76, + "end": 81, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 82, + "end": 91, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 82, + "end": 91, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 92, + "end": 97, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 92, + "end": 97, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 0, + "end": 98, + "name": "bounding box", + "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.553030303030303, \"width\": 0.7630718954248366, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_70248", + "original_document": "1709714755_101.pdf", + "color": "#f00" + }, + "additional_info": "page_id 6
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text Query Execution. The server first initializes a buffer B of size ℓ to E(0) in every position. Then,\n
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text for each document fi it proceeds as follows:\n
", + "default_label": "not_paragraph" + }, + "95": { + "id": 95, + "task_path": "images/000043_000095_img_bbox_txt_a07cd9230110538d6746589cbdc45042_70260.jpg", + "labeled": [ + "paragraph" + ], + "data": { + "location": { + "page_number": 6, + "bbox": { + "x_top_left": 77, + "y_top_left": 466, + "width": 64, + "height": 1 + }, + "name": "", + "rotated_angle": 0.0 + }, + "order": 70260, + "_line": "– compute f ′\n", + "_metadata": { + "tag_hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": true, + "line_type": "unknown" + }, + "hierarchy_level": { + "level_1": 5, + "level_2": 1, + "can_be_multiline": false, + "line_type": "list_item" + }, + "page_id": 6, + "line_id": 70260, + "_LineMetadata__other_fields": {} + }, + "_annotations": [ + { + "start": 0, + "end": 14, + "name": "indentation", + "value": "77", + "is_mergeable": true + }, + { + "start": 0, + "end": 14, + "name": "spacing", + "value": "9", + "is_mergeable": true + }, + { + "start": 0, + "end": 1, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 0, + "end": 1, + "name": "style", + "value": "WLPKII+CMMI10", + "is_mergeable": true + }, + { + "start": 0, + "end": 1, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 2, + "end": 9, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 2, + "end": 9, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 10, + "end": 11, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 10, + "end": 11, + "name": "style", + "value": "WLPKII+CMMI10", + "is_mergeable": true + }, + { + "start": 10, + "end": 11, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 12, + "end": 13, + "name": "size", + "value": "7", + "is_mergeable": true + }, + { + "start": 12, + "end": 13, + "name": "style", + "value": "XYBRXS+CMSY8", + "is_mergeable": true + }, + { + "start": 12, + "end": 13, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 0, + "end": 14, + "name": "bounding box", + "value": "{\"x_top_left\": 0.12581699346405228, \"y_top_left\": 0.5883838383838383, \"width\": 0.10457516339869281, \"height\": 0.0012626262626262627, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_70260", + "original_document": "1709714755_101.pdf", + "color": "#ffa500" + }, + "additional_info": "page_id 6
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text – compute f ′\n
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text 5 an integer d following a given\n
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text distribution (the best choice for this distribution is discussed at the end of this section) and\n
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text generate a uniformly random binary column vector Hi of length ℓ and Hamming weight d,\n
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+ "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 0, + "end": 99, + "name": "bounding box", + "value": "{\"x_top_left\": 0.12581699346405228, \"y_top_left\": 0.6906565656565656, \"width\": 0.7320261437908496, \"height\": 0.0012626262626262627, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_70371", + "original_document": "1709714755_101.pdf", + "color": "#ffa500" + }, + "additional_info": "page_id 6
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text – for every non-zero position in H , i multiply the corresponding position in buffer B by E(c f i ′\n
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text singletons. The detection of singletons is made easy by the structure of f ′\n
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text Fig. 3. Local view of the bipartite graph as a tree. The dashed lines correspond to nodes/edges\n
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text Choosing an Optimal Column Weight Distribution. In order to analyze the decoding al-\n
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text parity node. Then the decoding algorithm consists in repeating the following steps:\n
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text – select all edges with right degree 1 (edges connected to singletons),\n
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text – remove these edges from the graph as well as the associated left and right nodes,\n
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text – remove all other edges that were connected to the left nodes (no other edges were connected to\n
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text the right nodes).\n
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text Decoding is successful if, at the end, all the edges have been removed from the graph.\n
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text Studying the probability of success of this algorithm for given parameters m and ℓ is difficult,\n
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text and ℓ tend to infinity, the asymptotic proportion of edges removed at each step can be computed.\n
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text Let λ(x) =\n
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text ∑\n
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text edge of the graph has left (resp. right) degree i. Also, let bj denote the proportion of edges of the\n
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text graph that are still present after step j of the algorithm. Then b0 = 1 (all the edges are present\n
", + "default_label": "not_paragraph" + }, + "160": { + "id": 160, + "task_path": "images/000043_000160_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80401.jpg", + "labeled": [ + "not_paragraph" + ], + "data": { + "location": { + "page_number": 7, + "bbox": { + "x_top_left": 71, + "y_top_left": 563, + "width": 156, + "height": 5 + }, + "name": "", + "rotated_angle": 0.0 + }, + "order": 80401, + "_line": "before the algorithm starts) and:\n", + "_metadata": { + "tag_hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": true, + "line_type": "unknown" + }, + "hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": true, + "line_type": "raw_text" + }, + "page_id": 7, + "line_id": 80401, + "_LineMetadata__other_fields": {} + }, + "_annotations": [ + { + "start": 0, + "end": 34, + "name": "indentation", + "value": "71", + "is_mergeable": true + }, + { + "start": 0, + "end": 34, + "name": "spacing", + "value": "8", + "is_mergeable": true + }, + { + "start": 0, + "end": 6, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 0, + "end": 6, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 7, + "end": 10, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 7, + "end": 10, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 11, + "end": 20, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 11, + "end": 20, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 21, + "end": 28, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 21, + "end": 28, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 29, + "end": 33, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 29, + "end": 33, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 0, + "end": 34, + "name": "bounding box", + "value": "{\"x_top_left\": 0.11601307189542484, \"y_top_left\": 0.7108585858585859, \"width\": 0.2549019607843137, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_80401", + "original_document": "1709714755_101.pdf", + "color": "#ffa500" + }, + "additional_info": "page_id 7
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text before the algorithm starts) and:\n
", + "default_label": "not_paragraph" + }, + "161": { + "id": 161, + "task_path": "images/000043_000161_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80407.jpg", + "labeled": [ + "Other" + ], + "data": { + "location": { + "page_number": 7, + "bbox": { + "x_top_left": 249, + "y_top_left": 590, + "width": 290, + "height": 5 + }, + "name": "", + "rotated_angle": 0.0 + }, + "order": 80407, + "_line": "bj+1 = λ(1− ρ(1− b )). j (1)\n", + "_metadata": { + "tag_hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "unknown" + }, + "hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "raw_text" + }, + "page_id": 7, + "line_id": 80407, + "_LineMetadata__other_fields": {} + }, + "_annotations": [ + { + "start": 0, + "end": 29, + "name": "indentation", + "value": "249", + "is_mergeable": true + }, + { + "start": 0, + "end": 29, + "name": "spacing", + "value": "21", + "is_mergeable": true + }, + { + "start": 0, + "end": 4, + "name": "size", + "value": "7", + "is_mergeable": true + }, + { + "start": 0, + "end": 4, + "name": "style", + "value": "AXUDEI+CMSY10", + "is_mergeable": true + }, + { + "start": 0, + "end": 4, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 5, + "end": 6, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 5, + "end": 6, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 7, + "end": 11, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 7, + "end": 11, + "name": "style", + "value": "AXUDEI+CMSY10", + "is_mergeable": true + }, + { + "start": 7, + "end": 11, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 12, + "end": 16, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 12, + "end": 16, + "name": "style", + "value": "AXUDEI+CMSY10", + "is_mergeable": true + }, + { + "start": 12, + "end": 16, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 17, + "end": 18, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 17, + "end": 18, + "name": "style", + "value": "MTKKCR+CMMI8", + "is_mergeable": true + }, + { + "start": 17, + "end": 18, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 19, + "end": 22, + "name": "size", + "value": "7", + "is_mergeable": true + }, + { + "start": 19, + "end": 22, + "name": "style", + "value": "MTKKCR+CMMI8", + "is_mergeable": true + }, + { + "start": 19, + "end": 22, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 23, + "end": 24, + "name": "size", + "value": "7", + "is_mergeable": true + }, + { + "start": 23, + "end": 24, + "name": "style", + "value": "MTKKCR+CMMI8", + "is_mergeable": true + }, + { + "start": 23, + "end": 24, + "name": "italic", + "value": "True", + "is_mergeable": true + }, + { + "start": 25, + "end": 28, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 25, + "end": 28, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 0, + "end": 29, + "name": "bounding box", + "value": "{\"x_top_left\": 0.4068627450980392, \"y_top_left\": 0.7449494949494949, \"width\": 0.4738562091503268, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_80407", + "original_document": "1709714755_101.pdf", + "color": "#f00" + }, + "additional_info": "page_id 7
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text bj+1 = λ(1− ρ(1− b )). j (1)\n
", + "default_label": "paragraph" + }, + "162": { + "id": 162, + "task_path": "images/000043_000162_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80424.jpg", + "labeled": [ + "paragraph" + ], + "data": { + "location": { + "page_number": 7, + "bbox": { + "x_top_left": 88, + "y_top_left": 611, + "width": 451, + "height": 5 + }, + "name": "", + "rotated_angle": 0.0 + }, + "order": 80424, + "_line": "We will not prove this formula here (please refer to [9,11] for the complete proof), but the\n", + "_metadata": { + "tag_hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "unknown" + }, + "hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "raw_text" + }, + "page_id": 7, + "line_id": 80424, + "_LineMetadata__other_fields": {} + }, + "_annotations": [ + { + "start": 0, + "end": 93, + "name": "indentation", + "value": "88", + "is_mergeable": true + }, + { + "start": 0, + "end": 93, + "name": "spacing", + "value": "14", + "is_mergeable": true + }, + { + "start": 0, + "end": 2, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 0, + "end": 2, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 3, + "end": 7, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 3, + "end": 7, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 8, + "end": 11, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 8, + "end": 11, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 12, + "end": 17, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 12, + "end": 17, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 18, + "end": 22, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 18, + "end": 22, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 23, + "end": 30, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 23, + "end": 30, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 31, + "end": 35, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 31, + "end": 35, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 36, + "end": 43, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 36, + "end": 43, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 44, + "end": 49, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 44, + "end": 49, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 50, + "end": 52, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 50, + "end": 52, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 53, + "end": 59, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 53, + "end": 59, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 60, + "end": 63, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 60, + "end": 63, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 64, + "end": 67, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 64, + "end": 67, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 68, + "end": 76, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 68, + "end": 76, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 77, + "end": 84, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 77, + "end": 84, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 85, + "end": 88, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 85, + "end": 88, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 89, + "end": 92, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 89, + "end": 92, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 0, + "end": 93, + "name": "bounding box", + "value": "{\"x_top_left\": 0.1437908496732026, \"y_top_left\": 0.7714646464646465, \"width\": 0.7369281045751634, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_80424", + "original_document": "1709714755_101.pdf", + "color": "#f00" + }, + "additional_info": "page_id 7
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text We will not prove this formula here (please refer to [9,11] for the complete proof), but the\n
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text intuition is the following. During the decoding process, the neighborhood around the edge (v, c)\n
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text between a message node v and a parity node c, can be seen as a tree (see Fig. 3). The decoding\n
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text process then consists in letting information flow from the leaves of the tree to its root: the (v, c)\n
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text of the sons of v is a leaf. The probability this happens can easily be computed if the distributions\n
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text λ and ρ are known. The parity node ′ c is connected to k edges with probability ρk and it is a\n
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text Communication Optimal Private Stream Search 9\n
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text Table 1. Minimal expansion rates asymptotically allowing recovery of all the documents, for\n
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text (1− k P ) singleton\n
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text ′ k −1 = 1− λ(1− P ). singleton The edge\n
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text (v, c) is thus removed at step j+1 with probability 1−λ(1−ρ(1− b )), j which leads to formula (1).\n
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text Asymptotically, the decoding algorithm is successful if bj\n
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text j→∞ −→ 0, which will be the case if:\n
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text Application to Danezis and Dı́az’s Algorithm. In [4], Danezis and Dı́az propose to use a constant\n
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text weight d for the columns of H. This means that λ(x) = d−1 x . The distribution ρ is a little more\n
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text complicated as it is induced by λ. A row of H will have a weight following a binomial distribution: it\n
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text is the sum ofm random coins equal to 1 with probability d\n
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text ℓ . We thus have ρi =\n
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text iℓ\n
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text md\n
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text (\n
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text m\n
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text )(\n
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text The best asymptotic choice is d = 3, which does not mean that for a given m and ℓ the best\n
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text probability of success is always achieved for d = 3. This however proves that Danezis and Dı́az’s\n
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text algorithm with constant column weight d can indeed achieve communications linear in m. However,\n
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text it can never have a good probability of success for expansion rates smaller than 1.22.\n
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text The Harmonic Distribution. Using a constant column weight makes the description of the algorithm\n
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text easy and gives very good results for some parameters (see Fig. 5), but it is not optimal.\n
", + "default_label": "not_paragraph" + }, + "231": { + "id": 231, + "task_path": "images/000043_000231_img_bbox_txt_a07cd9230110538d6746589cbdc45042_90469.jpg", + "labeled": [ + "paragraph" + ], + "data": { + "location": { + "page_number": 8, + "bbox": { + "x_top_left": 88, + "y_top_left": 553, + "width": 451, + "height": 5 + }, + "name": "", + "rotated_angle": 0.0 + }, + "order": 90469, + "_line": "The decoding problem we are facing is very close to the problem of decoding LT-codes [7] and,\n", + "_metadata": { + "tag_hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "unknown" + }, + "hierarchy_level": { + "level_1": null, + "level_2": null, + "can_be_multiline": false, + "line_type": "raw_text" + }, + "page_id": 8, + "line_id": 90469, + "_LineMetadata__other_fields": {} + }, + "_annotations": [ + { + "start": 0, + "end": 94, + "name": "indentation", + "value": "88", + "is_mergeable": true + }, + { + "start": 0, + "end": 94, + "name": "spacing", + "value": "8", + "is_mergeable": true + }, + { + "start": 0, + "end": 3, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 0, + "end": 3, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 4, + "end": 12, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 4, + "end": 12, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 13, + "end": 20, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 13, + "end": 20, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 21, + "end": 23, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 21, + "end": 23, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 24, + "end": 27, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 24, + "end": 27, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 28, + "end": 34, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 28, + "end": 34, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 35, + "end": 37, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 35, + "end": 37, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 38, + "end": 42, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 38, + "end": 42, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 43, + "end": 48, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 43, + "end": 48, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 49, + "end": 51, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 49, + "end": 51, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 52, + "end": 55, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 52, + "end": 55, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 56, + "end": 63, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 56, + "end": 63, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 64, + "end": 66, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 64, + "end": 66, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 67, + "end": 75, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 67, + "end": 75, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 76, + "end": 84, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 76, + "end": 84, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 85, + "end": 88, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 85, + "end": 88, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 89, + "end": 93, + "name": "size", + "value": "10", + "is_mergeable": true + }, + { + "start": 89, + "end": 93, + "name": "style", + "value": "JCQZQQ+CMR10", + "is_mergeable": true + }, + { + "start": 0, + "end": 94, + "name": "bounding box", + "value": "{\"x_top_left\": 0.1437908496732026, \"y_top_left\": 0.6982323232323232, \"width\": 0.7369281045751634, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", + "is_mergeable": false + } + ], + "_uid": "txt_a07cd9230110538d6746589cbdc45042_90469", + "original_document": "1709714755_101.pdf", + "color": "#f00" + }, + "additional_info": "page_id 8
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text The decoding problem we are facing is very close to the problem of decoding LT-codes [7] and,\n
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text for LT-codes, it was proven that the optimal distribution choice is the so-called Robust Soliton\n
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text distribution. However, in our context, using this distribution would correspond to fixing ρ(x), that\n
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