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text fi E(c ) i , the server will embed several values in an integer f ′\n
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text j,i ∈ ZN as shown in Fig. 1. This requires\n
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text two different Reed-Solomon codes RS and ′ RS . The code RS will be used to recover the ci and is\n
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text defined on Zpt where pt is the smallest prime greater than the database size t (it should also be\n
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text greater than the dictionary size |Ω|) and the coefficients of its parity check matrix are thus defined\n
", "default_label": "not_paragraph" }, "6": { "id": 6, "task_path": "images/000043_000006_img_bbox_txt_a07cd9230110538d6746589cbdc45042_50493.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 4, "bbox": { "x_top_left": 72, "y_top_left": 677, "width": 468, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 50493, "_line": "as RSj,i = j i mod p . t Similarly, ′ RS will be used to recover the values of fi and is defined over\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": 4, "line_id": 50493, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 101, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 101, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 2, "name": "size", 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text as RSj,i = j i mod p . t Similarly, ′ RS will be used to recover the values of fi and is defined over\n
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text Zpf where pf is the largest prime that can fit in the remaining bits of one plaintext (Paillier or\n
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Thus, for each of the m positions of buffer B, the\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": 4, "line_id": 50532, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 98, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 98, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 16, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 16, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 17, "end": 19, "name": "size", "value": "10", "is_mergeable": true }, { "start": 17, "end": 19, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 20, "end": 24, "name": "size", "value": "10", "is_mergeable": true }, { "start": 20, "end": 24, 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text (see Fig. 1) to avoid overflows. The user thus starts by reducing the last two elements modulo pt\n
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text for each j ∈ [1,m], and gets 2m syndrome positions in RS of the sparse vector (c ). i This is enough\n
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text to recover the values and positions of the m non-zero ci elements using the Reed-Solomon error\n
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text correcting algorithm.\n
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text Then, the user reduces the first part of each plaintext modulo pf to obtainm syndrome positions\n
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text in ′ RS of the sparse vector (c f ). i i As the non-zero ci elements are known, the positions of the non-\n
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text zero c f i i are also known, and the user has to solve an erasure problem. The m syndrome positions\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 buffer decryption step. Reed-Solomon decoding costs 2 O(m ) multiplications in Zpt and Zpf , which\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 Communication Cost. With this scheme, a buffer of size m is enough to recover m matching\n
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text documents, which is optimal. However, part of each buffer position is reserved for the recovery of\n
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For each document, an overhead of 5 log t + 3 log |Ω| bits has to be\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": 5, "line_id": 60386, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 98, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 98, "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": 6, "name": "size", "value": "7", "is_mergeable": true }, { "start": 4, "end": 6, "name": "style", "value": "MTKKCR+CMMI8", "is_mergeable": true }, { "start": 4, "end": 6, "name": "italic", "value": "True", "is_mergeable": true }, { "start": 7, 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text the ci and for zero-padding. For each document, an overhead of 5 log t + 3 log |Ω| bits has to be\n
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The 3 log t + 3 log |Ω| padding 3 bits are wasted bits that are due to the structure of\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": 5, "line_id": 60407, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 101, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 101, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 12, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 12, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 13, "end": 16, "name": "size", "value": "10", "is_mergeable": true }, { "start": 13, "end": 16, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 17, "end": 18, "name": "size", "value": "10", "is_mergeable": true }, { "start": 17, "end": 18, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 19, "end": 22, "name": "size", "value": "10", "is_mergeable": true }, { "start": 19, "end": 22, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 23, "end": 24, "name": "size", "value": "10", "is_mergeable": true }, { "start": 23, "end": 24, "name": "style", "value": "WLPKII+CMMI10", "is_mergeable": true }, { "start": 23, "end": 24, "name": "italic", "value": "True", "is_mergeable": true }, { "start": 25, "end": 26, "name": "size", "value": "10", "is_mergeable": true }, { "start": 25, "end": 26, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 27, "end": 28, "name": "size", "value": "10", "is_mergeable": true }, { "start": 27, "end": 28, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 29, "end": 32, "name": "size", "value": "10", "is_mergeable": true }, { "start": 29, "end": 32, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 33, "end": 36, "name": "size", "value": "10", "is_mergeable": true }, { "start": 33, "end": 36, "name": "style", "value": "AXUDEI+CMSY10", "is_mergeable": true }, { "start": 33, "end": 36, "name": "italic", "value": "True", "is_mergeable": true }, { "start": 37, "end": 44, "name": "size", "value": "10", "is_mergeable": true }, { "start": 37, "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": 51, "name": "size", "value": "10", "is_mergeable": true }, { "start": 47, "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": 62, "name": "size", "value": "10", "is_mergeable": true }, { "start": 56, "end": 62, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 63, "end": 67, "name": "size", "value": "10", "is_mergeable": true }, { "start": 63, "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": 76, "name": "size", "value": "10", "is_mergeable": true }, { "start": 73, "end": 76, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 77, "end": 80, "name": "size", "value": "10", "is_mergeable": true }, { "start": 77, "end": 80, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 81, "end": 83, "name": "size", "value": "10", "is_mergeable": true }, { "start": 81, "end": 83, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 84, "end": 87, "name": "size", "value": "10", "is_mergeable": true }, { "start": 84, "end": 87, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 88, "end": 97, "name": "size", "value": "10", "is_mergeable": true }, { "start": 88, "end": 97, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 98, "end": 100, "name": "size", "value": "10", "is_mergeable": true }, { "start": 98, "end": 100, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 101, "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_60407", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 5
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text transmitted. The 3 log t + 3 log |Ω| padding 3 bits are wasted bits that are due to the structure of\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 and the position of the errors, leading to an overhead of O(log t) bits per document. Overall, for m\n
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text matching documents of S bits, this scheme requires 2m(S+O(log t)) bits using the original Paillier\n
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text cryptosystem or m(S+1024+O(log t)) using the Damg̊ard-Jurik extension. Asymptotically, when\n
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text m and S tend to infinity, this corresponds to an expansion ratio (compared to a non-private search)\n
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text of 2 using Paillier and 1 using Damg̊ard-Jurik.\n
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text Note that these ratios are valid only if the user knows in advance the number m of matches to\n
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text expect. This can be the case in some scenarios, but not in a typical keyword search. In this case, the\n
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text user will query form positions and will receive a syndrome of sizem: if less thanm documents match\n
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text the query he will be able to extract all of them with probability 1, but if more than m documents\n
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text match the query he will not be able to decode and will not get any document. This gives another\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 the Ostrovsky-Skeith scheme, the user will usually be able to extract at least a few documents,\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 It can however be easily reduced to 3 log t+ 3c for any c if queries are limited c 2 keywords at a time.\n
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text 4 Special care has to be taken when designing such an interactive protocol, so as not lose too much privacy by\n
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text Communication Optimal Private Stream Search 7\n
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text 3.2 An Asymptotically Optimal Construction Using Irregular LDPC Codes\n
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text Description of the Construction. In order to improve the asymptotic communication cost and\n
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text remove any dependency on the database size t, it is necessary to use a randomized scheme (thus\n
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text with a non-zero probability of failure): in that case, it is well known that (irregular) LDPC codes can\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
", "default_label": "not_paragraph" }, "92": { "id": 92, "task_path": "images/000043_000092_img_bbox_txt_a07cd9230110538d6746589cbdc45042_70230.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 6, "bbox": { "x_top_left": 72, "y_top_left": 413, "width": 297, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 70230, "_line": "and γ, the user sends the desired buffer length ℓ to the server.\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": 6, "line_id": 70230, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 64, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 64, "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": 6, "name": "size", "value": "10", "is_mergeable": true }, { "start": 4, "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": 15, "name": "size", "value": "10", "is_mergeable": true }, { "start": 11, "end": 15, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 16, "end": 21, "name": "size", "value": "10", "is_mergeable": true }, { "start": 16, "end": 21, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 22, "end": 25, "name": "size", "value": "10", "is_mergeable": true }, { "start": 22, "end": 25, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 26, "end": 33, "name": "size", "value": "10", "is_mergeable": true }, { "start": 26, "end": 33, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 34, "end": 39, "name": "size", "value": "10", "is_mergeable": true }, { "start": 34, "end": 39, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 40, "end": 46, "name": "size", "value": "10", "is_mergeable": true }, { "start": 40, "end": 46, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 47, "end": 48, "name": "size", "value": "10", "is_mergeable": true }, { "start": 47, "end": 48, "name": "style", "value": "WLPKII+CMMI10", "is_mergeable": true }, { "start": 47, "end": 48, "name": "italic", "value": "True", "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": 0, "end": 64, "name": "bounding box", "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.5214646464646465, \"width\": 0.4852941176470588, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_70230", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 6
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text and γ, the user sends the desired buffer length ℓ to the server.\n
", "default_label": "not_paragraph" }, "93": { "id": 93, "task_path": "images/000043_000093_img_bbox_txt_a07cd9230110538d6746589cbdc45042_70248.jpg", "labeled": [ "paragraph" ], "data": { "location": { "page_number": 6, "bbox": { "x_top_left": 72, "y_top_left": 438, "width": 467, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 70248, "_line": "Query Execution. The server first initializes a buffer B of size ℓ to E(0) in every position. 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
", "default_label": "paragraph" }, "94": { "id": 94, "task_path": "images/000043_000094_img_bbox_txt_a07cd9230110538d6746589cbdc45042_70256.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 6, "bbox": { "x_top_left": 72, "y_top_left": 451, "width": 208, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 70256, "_line": "for each document fi it proceeds as follows:\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": 6, "line_id": 70256, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 45, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 45, "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": 8, "name": "size", "value": "10", "is_mergeable": true }, { "start": 4, "end": 8, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 9, "end": 17, "name": "size", "value": "10", "is_mergeable": true }, { "start": 9, "end": 17, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 18, "end": 20, "name": "size", "value": "7", "is_mergeable": true }, { "start": 18, "end": 20, "name": "style", "value": "MTKKCR+CMMI8", "is_mergeable": true }, { "start": 18, "end": 20, "name": "italic", "value": "True", "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": 32, "name": "size", "value": "10", "is_mergeable": true }, { "start": 24, "end": 32, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 33, "end": 35, "name": "size", "value": "10", "is_mergeable": true }, { "start": 33, "end": 35, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 36, "end": 44, "name": "size", "value": "10", "is_mergeable": true }, { "start": 36, "end": 44, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 45, "name": "bounding box", "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.5694444444444444, \"width\": 0.33986928104575165, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_70256", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 6
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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 – compute E(c f i ′\n
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text – using a pseudo-random number generator seeded by f , i generate\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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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 ). i\n
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text ′\n
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text )) i contains the encrypted syndrome of the sparse (c f i\n
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text ′\n
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text ) i vector with\n
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text respect to the parity check matrix H of an irregular LDPC code.\n
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text Document Extraction. When receiving buffer B, the user starts by decrypting it and looks for\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 i (see Fig. 2). The isolated\n
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text 1 allows to read the value of ci directly and divide c f i\n
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text ′\n
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text i by it: if it is indeed a singleton, the checksum\n
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text the same PRNG as the server) the corresponding column H . i Knowing Hi (and c ) i it is possible to\n
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text 5 Note that Danezis and Dı́az use the document index i to generate the column H , i which requires to number\n
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text documents. Using the document itself as the seed makes application to streaming data more natural.\n
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text 8 Matthieu Finiasz and Kannan Ramchandran\n
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text c\n
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text v\n
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text c’\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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The edge between v and c will be removed at step j +1 as one son ′ c\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": 80049, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 99, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 99, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 7, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 7, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 8, "end": 10, "name": "size", "value": "10", "is_mergeable": true }, { "start": 8, "end": 10, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 11, "end": 14, "name": "size", "value": "10", "is_mergeable": true }, { "start": 11, 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text removes at the end of step j. The edge between v and c will be removed at step j +1 as one son ′ c\n
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text of v is a singleton (it has no sons remaining at the end of step j).\n
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text remove document fi from the other syndrome positions where it has been added, thus uncovering\n
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text new singletons, which in turn can reveal new documents f . i This gives an iterative algorithm which\n
", "default_label": "not_paragraph" }, "133": { "id": 133, "task_path": "images/000043_000133_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80100.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 7, "bbox": { "x_top_left": 72, "y_top_left": 276, "width": 151, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 80100, "_line": "we analyse asymptotically here.\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": 80100, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 32, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 32, "name": "spacing", "value": "8", "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": 10, "name": "size", "value": "10", "is_mergeable": true }, { "start": 3, "end": 10, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 11, "end": 25, "name": "size", "value": "10", "is_mergeable": true }, { "start": 11, "end": 25, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 26, "end": 31, "name": "size", "value": "10", "is_mergeable": true }, { "start": 26, "end": 31, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 32, "name": "bounding box", "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.3484848484848485, \"width\": 0.24673202614379086, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_80100", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 7
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text we analyse asymptotically here.\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 gorithm, the matrix H can be transformed into a bipartite graph. On the left of the graph are m\n
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text ′\n
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text ) i elements) and on the right are ℓ parity nodes (the decrypted\n
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text edges connected to its information node and its right degree the number of edges connected to its\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
", "default_label": "not_paragraph" }, "144": { "id": 144, "task_path": "images/000043_000144_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80245.jpg", "labeled": [ "paragraph" ], "data": { "location": { "page_number": 7, "bbox": { "x_top_left": 77, "y_top_left": 419, "width": 398, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 80245, "_line": "– remove these edges from the graph as well as the associated left and right nodes,\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": 7, "line_id": 80245, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 84, "name": "indentation", "value": "77", "is_mergeable": true }, { "start": 0, "end": 84, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 1, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 1, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 2, "end": 8, "name": "size", "value": "10", "is_mergeable": true }, { "start": 2, "end": 8, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 9, "end": 14, "name": "size", "value": "10", "is_mergeable": true }, { "start": 9, "end": 14, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 15, "end": 20, "name": "size", "value": "10", "is_mergeable": true }, { "start": 15, "end": 20, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 21, "end": 25, "name": "size", "value": "10", "is_mergeable": true }, { "start": 21, "end": 25, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 26, "end": 29, "name": "size", "value": "10", "is_mergeable": true }, { "start": 26, "end": 29, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 30, "end": 35, "name": "size", "value": "10", "is_mergeable": true }, { "start": 30, "end": 35, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 36, "end": 38, "name": "size", "value": "10", "is_mergeable": true }, { "start": 36, "end": 38, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 39, "end": 43, "name": "size", "value": "10", "is_mergeable": true }, { "start": 39, "end": 43, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 44, "end": 46, "name": "size", "value": "10", "is_mergeable": true }, { "start": 44, "end": 46, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 47, "end": 50, "name": "size", "value": "10", "is_mergeable": true }, { "start": 47, "end": 50, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 51, "end": 61, "name": "size", "value": "10", "is_mergeable": true }, { "start": 51, "end": 61, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 62, "end": 66, "name": "size", "value": "10", "is_mergeable": true }, { "start": 62, "end": 66, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 67, "end": 70, "name": "size", "value": "10", "is_mergeable": true }, { "start": 67, "end": 70, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 71, "end": 76, "name": "size", "value": "10", "is_mergeable": true }, { "start": 71, "end": 76, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 77, "end": 83, "name": "size", "value": "10", "is_mergeable": true }, { "start": 77, "end": 83, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 84, "name": "bounding box", "value": "{\"x_top_left\": 0.12581699346405228, \"y_top_left\": 0.5290404040404041, \"width\": 0.6503267973856209, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_80245", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 7
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text – remove these edges from the graph as well as the associated left and right nodes,\n
", "default_label": "not_paragraph" }, "145": { "id": 145, "task_path": "images/000043_000145_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80263.jpg", "labeled": [ "paragraph" ], "data": { "location": { "page_number": 7, "bbox": { "x_top_left": 77, "y_top_left": 433, "width": 462, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 80263, "_line": "– remove all other edges that were connected to the left nodes (no other edges were connected to\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": 7, "line_id": 80263, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 97, "name": "indentation", "value": "77", "is_mergeable": true }, { "start": 0, "end": 97, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 1, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 1, "name": "style", "value": "QSTXPN+CMBX10", "is_mergeable": true }, { "start": 0, "end": 1, "name": "bold", "value": "True", "is_mergeable": true }, { "start": 2, "end": 8, "name": "size", "value": "10", "is_mergeable": true }, { "start": 2, "end": 8, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 9, "end": 12, "name": "size", "value": "10", "is_mergeable": true }, { "start": 9, "end": 12, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 13, "end": 18, "name": "size", "value": "10", "is_mergeable": true }, { "start": 13, "end": 18, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 19, "end": 24, "name": "size", "value": "10", "is_mergeable": true }, { "start": 19, "end": 24, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 25, "end": 29, "name": "size", "value": "10", "is_mergeable": true }, { "start": 25, "end": 29, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 30, "end": 34, "name": "size", "value": "10", "is_mergeable": true }, { "start": 30, "end": 34, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 35, "end": 44, "name": "size", "value": "10", "is_mergeable": true }, { "start": 35, "end": 44, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 45, "end": 47, "name": "size", "value": "10", "is_mergeable": true }, { "start": 45, "end": 47, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 48, "end": 51, "name": "size", "value": "10", "is_mergeable": true }, { "start": 48, "end": 51, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 52, "end": 56, "name": "size", "value": "10", "is_mergeable": true }, { "start": 52, "end": 56, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 57, "end": 62, "name": "size", "value": "10", "is_mergeable": true }, { "start": 57, "end": 62, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 63, "end": 66, "name": "size", "value": "10", "is_mergeable": true }, { "start": 63, "end": 66, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 67, "end": 72, "name": "size", "value": "10", "is_mergeable": true }, { "start": 67, "end": 72, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 73, "end": 78, "name": "size", "value": "10", "is_mergeable": true }, { "start": 73, "end": 78, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 79, "end": 83, "name": "size", "value": "10", "is_mergeable": true }, { "start": 79, "end": 83, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 84, "end": 93, "name": "size", "value": "10", "is_mergeable": true }, { "start": 84, "end": 93, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 94, "end": 96, "name": "size", "value": "10", "is_mergeable": true }, { "start": 94, "end": 96, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 97, "name": "bounding box", "value": "{\"x_top_left\": 0.12581699346405228, \"y_top_left\": 0.5467171717171717, \"width\": 0.7549019607843137, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_80263", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 7
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text – remove all other edges that were connected to the left nodes (no other edges were connected to\n
", "default_label": "not_paragraph" }, "146": { "id": 146, "task_path": "images/000043_000146_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80266.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 7, "bbox": { "x_top_left": 88, "y_top_left": 446, "width": 79, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 80266, "_line": "the right nodes).\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": 80266, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 18, "name": "indentation", "value": "88", "is_mergeable": true }, { "start": 0, "end": 18, "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": 9, "name": "size", "value": "10", "is_mergeable": true }, { "start": 4, "end": 9, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 10, "end": 17, "name": "size", "value": "10", "is_mergeable": true }, { "start": 10, "end": 17, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 18, "name": "bounding box", "value": "{\"x_top_left\": 0.1437908496732026, \"y_top_left\": 0.5631313131313131, \"width\": 0.12908496732026145, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_80266", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 7
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text the right nodes).\n
", "default_label": "not_paragraph" }, "147": { "id": 147, "task_path": "images/000043_000147_img_bbox_txt_a07cd9230110538d6746589cbdc45042_80282.jpg", "labeled": [ "paragraph" ], "data": { "location": { "page_number": 7, "bbox": { "x_top_left": 72, "y_top_left": 468, "width": 409, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 80282, "_line": "Decoding is successful if, at the end, all the edges have been removed from the graph.\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": 80282, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 87, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 87, "name": "spacing", "value": "16", "is_mergeable": true }, { "start": 0, "end": 8, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 8, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 9, "end": 11, "name": "size", "value": "10", "is_mergeable": true }, { "start": 9, "end": 11, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 12, "end": 22, "name": "size", "value": "10", "is_mergeable": true }, { "start": 12, "end": 22, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 23, "end": 26, "name": "size", "value": "10", "is_mergeable": true }, { "start": 23, "end": 26, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 27, "end": 29, "name": "size", "value": "10", "is_mergeable": true }, { "start": 27, "end": 29, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 30, "end": 33, "name": "size", "value": "10", "is_mergeable": true }, { "start": 30, "end": 33, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 34, "end": 38, "name": "size", "value": "10", "is_mergeable": true }, { "start": 34, "end": 38, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 39, "end": 42, "name": "size", "value": "10", "is_mergeable": true }, { "start": 39, "end": 42, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 43, "end": 46, "name": "size", "value": "10", "is_mergeable": true }, { "start": 43, "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": 57, "name": "size", "value": "10", "is_mergeable": true }, { "start": 53, "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": 70, "name": "size", "value": "10", "is_mergeable": true }, { "start": 63, "end": 70, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 71, "end": 75, "name": "size", "value": "10", "is_mergeable": true }, { "start": 71, "end": 75, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 76, "end": 79, "name": "size", "value": "10", "is_mergeable": true }, { "start": 76, "end": 79, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 80, "end": 86, "name": "size", "value": "10", "is_mergeable": true }, { "start": 80, "end": 86, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 87, "name": "bounding box", "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.5909090909090909, \"width\": 0.6683006535947712, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_80282", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 7
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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 however, as proven in [9], if the left and right degree distribution of edges remains constant and m\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 i λ x i\n
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text i−1, where λi (resp. ρ ) i denote the probability that an\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
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text before the algorithm starts) and:\n
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text bj+1 = λ(1− ρ(1− b )). j (1)\n
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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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The decoding\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": 80460, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 95, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 95, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 7, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 7, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 8, "end": 9, "name": "size", "value": "10", "is_mergeable": true }, { "start": 8, "end": 9, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 10, "end": 17, "name": "size", "value": "10", "is_mergeable": true }, { "start": 10, "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": 24, "name": "size", "value": "10", "is_mergeable": true }, { "start": 23, "end": 24, "name": "style", "value": "WLPKII+CMMI10", "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": 29, "end": 30, "name": "size", "value": "10", "is_mergeable": true }, { "start": 29, "end": 30, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 31, "end": 37, "name": "size", "value": "10", "is_mergeable": true }, { "start": 31, "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": 45, "name": "size", "value": "10", "is_mergeable": true }, { "start": 43, "end": 45, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 46, "end": 49, "name": "size", "value": "10", "is_mergeable": true }, { "start": 46, "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": 57, "name": "size", "value": "10", "is_mergeable": true }, { "start": 53, "end": 57, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 58, "end": 60, "name": "size", "value": "10", "is_mergeable": true }, { "start": 58, "end": 60, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 61, "end": 62, "name": "size", "value": "10", "is_mergeable": true }, { "start": 61, "end": 62, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 63, "end": 67, "name": "size", "value": "10", "is_mergeable": true }, { "start": 63, "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": 77, "name": "size", "value": "10", "is_mergeable": true }, { "start": 73, "end": 77, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 78, "end": 81, "name": "size", "value": "10", "is_mergeable": true }, { "start": 78, "end": 81, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 82, "end": 85, "name": "size", "value": "10", "is_mergeable": true }, { "start": 82, "end": 85, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 86, "end": 94, "name": "size", "value": "10", "is_mergeable": true }, { "start": 86, "end": 94, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 95, "name": "bounding box", "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.8055555555555556, \"width\": 0.7630718954248366, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_80460", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "page_id 7
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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 edge will be removed from the graph at step j +1 if the value of v can be determined at step j +1\n
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text and it can thus send its value to c. As can be seen on Fig. 3, the value of the message v can be\n
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text determined if it has a son ′ c that is a singleton at the end of step j of the algorithm, meaning one\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 specific values of d, in the algorithm of Danezis and Dı́az.\n
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text singleton at the end of step j if its k − 1 sons in the Fig. 3 tree have been removed: this happens\n
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text with probability Psingleton =\n
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text ∑\n
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text k−1 = ρ(1− b ). j\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 ∀x ∈ [0, 1], λ(1− ρ(1− x)) ≤ x.\n
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text Of course, for finite values of m and ℓ, the algorithm can fail even if this condition is verified.\n
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text The sequence of bj represents the expected evolution of the number of edges in the graph, but the\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 (\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
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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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text is, choosing a row weight distribution. This is not possible as each instance of our decoding problem\n
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text corresponds to a set of m random columns of H, meaning we only have full control on the column\n
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text weight distribution λ(x). The row distribution will always be given by ρ(x) = exp\n
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text (\n
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text − md̃\n
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text i is the average column weight of H. Thus, the constraint on λ(x) is:\n
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text ∀x ∈ [0, 1], λ\n
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