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text , λ(y) ≤ − ℓ\n

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text md̃\n

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text ln(1− y).\n

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text 10 Matthieu Finiasz and Kannan Ramchandran\n

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text Fig. 4. Column weight distribution for the harmonic distribution of order 7.\n

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text λ (x) D =\n

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text 1\n

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text H(D)\n

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text D ∑\n

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text ϵ will asymptotically allow to recover most documents.\n

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text The Enhanced Harmonic Distribution. One problem with the harmonic distribution is the presence\n

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text of columns of weight 2 (see Fig. 4). Any distribution containing columns of weight 2 will have\n

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text a non-zero asymptotic probability of having identical 6 columns . Indeed, the collision probability\n

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text , which is not negligible for the harmonic distribution.\n

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text This means that, using the harmonic distribution, decoding will often end with a few (O(1)\n

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text asymptotically) unrecovered documents. In order to deal with this issue, one solution is to combine\n

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text this distribution with a constant weight distribution to obtain what we call the enhanced harmonic\n

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text distribution. Each column of H is the concatenation of a column of length ℓ − ℓ3 following the\n

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text √\n

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text ℓ) would be a reasonable choice. Asymptotically, these ℓ3 additional rows in H do not\n

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text increase the expansion factor. Using this enhanced harmonic distribution with an order D harmonic\n

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text distribution, the probability of recovering all m documents tends to 1 when ℓ\n

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text m > 1 + 1\n

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text D and m\n

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text tends to infinity.\n

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text Communication Cost. We compared the different distributions through simulations and the\n

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text results we obtained are shown in Fig. 5. One can clearly see the limitation of the weight 3 distri-\n

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text bution: for m > .8ℓ ≃ ℓ\n

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text 1.22 the probability of full recovery drops. However, the enhanced harmonic\n

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text 6 If two documents fi generate the exact same column H , i our decoding algorithm will be unable to recover any of\n

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text them as no singleton can ever be obtained.\n

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text Communication Optimal Private Stream Search 11\n

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text average ratio of recovered documents\n

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text average ratio of recovered documents\n

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text full recovery probability\n

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In (a) and (d) ℓ = 100 and ℓ3 = 10,\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": 10, "line_id": 110170, "_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": 6, "name": "size", "value": "10", "is_mergeable": true }, { "start": 0, "end": 6, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 7, "end": 9, "name": "size", "value": "10", "is_mergeable": true }, { "start": 7, "end": 9, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 10, "end": 18, "name": "size", "value": "10", "is_mergeable": true }, { "start": 10, "end": 18, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 19, "end": 28, "name": "size", "value": "10", "is_mergeable": true }, { "start": 19, "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": "WLPKII+CMMI10", "is_mergeable": true }, { "start": 29, "end": 30, "name": "italic", "value": "True", "is_mergeable": true }, { "start": 31, "end": 34, "name": "size", "value": "10", "is_mergeable": true }, { "start": 31, "end": 34, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 35, "end": 43, "name": "size", "value": "10", "is_mergeable": true }, { "start": 35, "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": 55, "name": "size", "value": "10", "is_mergeable": true }, { "start": 50, "end": 55, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 56, "end": 58, "name": "size", "value": "10", "is_mergeable": true }, { "start": 56, "end": 58, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 59, "end": 61, "name": "size", "value": "10", "is_mergeable": true }, { "start": 59, "end": 61, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 62, "end": 65, "name": "size", "value": "10", "is_mergeable": true }, { "start": 62, "end": 65, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 66, "end": 69, "name": "size", "value": "10", "is_mergeable": true }, { "start": 66, "end": 69, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 70, "end": 73, "name": "size", "value": "10", "is_mergeable": true }, { "start": 70, "end": 73, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 74, "end": 75, "name": "size", "value": "10", "is_mergeable": true }, { "start": 74, "end": 75, "name": "style", "value": "WLPKII+CMMI10", "is_mergeable": true }, { "start": 74, "end": 75, "name": "italic", "value": "True", "is_mergeable": true }, { "start": 76, "end": 77, "name": "size", "value": "10", "is_mergeable": true }, { "start": 76, "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": 88, "name": "size", "value": "7", "is_mergeable": true }, { "start": 86, "end": 88, "name": "style", "value": "BOTFMA+CMR8", "is_mergeable": true }, { "start": 89, "end": 90, "name": "size", "value": "10", "is_mergeable": true }, { "start": 89, "end": 90, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 91, "end": 94, "name": "size", "value": "10", "is_mergeable": true }, { "start": 91, "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\": 0.3712121212121212, \"width\": 0.7630718954248366, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_110170", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "

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text number of matching documents m for different buffer sizes ℓ. In (a) and (d) ℓ = 100 and ℓ3 = 10,\n

", "default_label": "not_paragraph" }, "168": { "id": 168, "task_path": "images/000044_000168_img_bbox_txt_a07cd9230110538d6746589cbdc45042_110193.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 10, "bbox": { "x_top_left": 72, "y_top_left": 308, "width": 379, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 110193, "_line": "in (b) and (e) ℓ = 1000 and ℓ3 = 35, and in (c) and (f) ℓ = 10000 and ℓ3 = 100.\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": 10, "line_id": 110193, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 80, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 80, "name": "spacing", "value": "8", "is_mergeable": true }, { "start": 0, "end": 2, "name": "size", "value": "10", 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text in (b) and (e) ℓ = 1000 and ℓ3 = 35, and in (c) and (f) ℓ = 10000 and ℓ3 = 100.\n

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page_id 10

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text distribution holds to our expectations, it behaves as expected with a probability of recovering all\n

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page_id 10

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text m matching documents close to 1 even for expansion rates smaller than 5%.\n

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text With this algorithm and the enhanced harmonic distribution, the asymptotic communication\n

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text cost of private stream search with m matching documents of S bits is thus 2m(S+64+O(log |Ω|))\n

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text using Paillier’s cryptosystem, or m(S + 1088 + O(log |Ω|)) using the Damg̊ard-Jurik extension,\n

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text which corresponds to an expansion rate of 1 compared to a non-private search.\n

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text 4 Further Improvements\n

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text An Offline-Online Construction. Using any of our two new schemes, it is possible to reduce\n

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text the size of the reply from the server almost to the size of a non-private search result. However, the\n

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text size of the query the user sends remains large. The size of a private query is linear in |Ω| whereas it\n

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text is logarithmic for a non-private query. To improve this, we propose an offline-online scheme, where\n

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text the linear query is sent offline and a logarithmic query is sent online.\n

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text A Single Keyword Scheme. We first focus on queries containing a single keyword. In this case, any\n

", "default_label": "paragraph" }, "182": { "id": 182, "task_path": "images/000044_000182_img_bbox_txt_a07cd9230110538d6746589cbdc45042_110395.jpg", "labeled": [ "not_paragraph" ], "data": { "location": { "page_number": 10, "bbox": { "x_top_left": 72, "y_top_left": 558, "width": 420, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 110395, "_line": "query can be obtained as a cyclic shift of any other query. The scheme works 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": 10, "line_id": 110395, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 89, "name": "indentation", "value": "72", "is_mergeable": true }, { "start": 0, "end": 89, "name": "spacing", "value": "8", "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": 9, "name": "size", "value": "10", "is_mergeable": true }, { "start": 6, "end": 9, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 10, "end": 12, "name": "size", "value": "10", "is_mergeable": true }, { "start": 10, "end": 12, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 13, "end": 21, "name": "size", "value": "10", "is_mergeable": true }, { "start": 13, "end": 21, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 22, "end": 24, "name": "size", "value": "10", "is_mergeable": true }, { "start": 22, "end": 24, "name": "style", "value": "JCQZQQ+CMR10", "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": 33, "name": "size", "value": "10", "is_mergeable": true }, { "start": 27, "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": 42, "name": "size", "value": "10", "is_mergeable": true }, { "start": 40, "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": 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": 70, "name": "size", "value": "10", "is_mergeable": true }, { "start": 64, "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": 79, "name": "size", "value": "10", "is_mergeable": true }, { "start": 77, "end": 79, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 80, "end": 88, "name": "size", "value": "10", "is_mergeable": true }, { "start": 80, "end": 88, "name": "style", "value": "JCQZQQ+CMR10", "is_mergeable": true }, { "start": 0, "end": 89, "name": "bounding box", "value": "{\"x_top_left\": 0.11764705882352941, \"y_top_left\": 0.7045454545454546, \"width\": 0.6862745098039216, \"height\": 0.006313131313131313, \"page_width\": 612, \"page_height\": 792}", "is_mergeable": false } ], "_uid": "txt_a07cd9230110538d6746589cbdc45042_110395", "original_document": "1709714755_101.pdf", "color": "#ffa500" }, "additional_info": "

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text query can be obtained as a cyclic shift of any other query. The scheme works as follows:\n

", "default_label": "not_paragraph" }, "183": { "id": 183, "task_path": "images/000044_000183_img_bbox_txt_a07cd9230110538d6746589cbdc45042_110419.jpg", "labeled": [ "paragraph" ], "data": { "location": { "page_number": 10, "bbox": { "x_top_left": 77, "y_top_left": 578, "width": 462, "height": 5 }, "name": "", "rotated_angle": 0.0 }, "order": 110419, "_line": "– offline, the user generates a query Qj = {q , 1 . . . , q } |Ω| where qj = E(1) and qi = E(0) otherwise\n", "_metadata": { "tag_hierarchy_level": { "level_1": null, "level_2": null, "can_be_multiline": false, "line_type": "unknown" }, "hierarchy_level": { "level_1": 5, "level_2": 1, "can_be_multiline": false, "line_type": "list_item" }, "page_id": 10, "line_id": 110419, "_LineMetadata__other_fields": {} }, "_annotations": [ { "start": 0, "end": 104, "name": "indentation", "value": "77", "is_mergeable": true }, { "start": 0, "end": 104, "name": "spacing", "value": "14", "is_mergeable": true }, { "start": 0, "end": 1, "name": 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text – offline, the user generates a query Qj = {q , 1 . . . , q } |Ω| where qj = E(1) and qi = E(0) otherwise\n

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text (with j picked uniformly at random), and sends it to the server,\n

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text – offline, the server computes all possible cyclic shifts of Qj by i ∈ [0, [Ω| − 1] positions and\n

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text executes the corresponding queries. It stores each result in a separate buffer B . i\n

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text – online, the user wants to query the server for the ′ j -th keywords and sends ′ j − j mod |Ω| to\n

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text of online communication is very close to the non-private case: the query is only log |Ω| bits long,\n

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text and with the PSS schemes we have presented ′ Bj −j can also be small.\n

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text 12 Matthieu Finiasz and Kannan Ramchandran\n

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text The offline amount of communication is still the same as for the standard scheme, but the\n

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text amount of computation on the server side is multiplied by |Ω|. However, as this is offline work,\n

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text it can easily be outsourced to distant server farm and does not have to be run on the “online”\n

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text the shifted query online as in the standard scheme: the amount of work the server has to do will\n

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text then be the same as in the normal scheme, but most of the communication will be done offline.\n

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text Dealing with Multiple Keywords. To maintain privacy, the offline query the user sends has to be\n

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text completely random and, at the same time, it should be possible to modify it into any other query the\n

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text user might later want to ask. For a single keyword, cyclic shifts work well whatever the dictionary\n

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text When k = 2, a solution is to transform each index j into Aj+B mod |Ω|, where A ∈ [1, |Ω|−1]\n

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text and B ∈ [0, |Ω| − 1] are the “permutation index” that the user will send to the server in the online\n

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text phase. If |Ω| is prime, then any pair j , 1 j2 can be transformed into any pairs ′ j , 1 j\n

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text ′\n

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text 2 by choosing\n

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text Building such families of permutations for larger values of k is not always simple, and the\n

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text number of permutations in the family will always have to be at least\n

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text (|Ω|\n

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text k\n

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text )\n

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text . This means that the\n

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text presented work fine.\n

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text Using Fully Homomorphic Encryption. The PSS schemes we presented do not need fully\n

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text homomorphic encryption (FHE) to work: computing a syndrome only requires addition and multi-\n

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text plication by a scalar. However, having an efficient FHE scheme would allow AND queries, whereas\n

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text the current schemes are limited to OR queries. Also, the query size could be reduced to O(log |Ω|)\n

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text encrypted bits without having to use an offline-online 7 scheme . For instance, when querying for\n

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text keyword w , i the user could decompose the index i in log |Ω| bits and send two ciphertexts E(0)\n

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text and E(1) (or E(1) and E(0)) for each of these bits. The server can then regenerate the whole query\n

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text Q by homomorphically multiplying the encrypted bits corresponding to each index. Continuing on\n

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text this idea, it would even be possible to get rid of the dictionary by simply querying a bit string: a\n

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text query of 2n encrypted bits would be enough to search any pattern of n bits.\n

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text One interesting aspect of this application of FHE, is that the number of homomorphic multi-\n

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text plications that have to be done is independent of the database size: multiplications are required to\n

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text reconstruct the query, but not for the stream search itself. Even if homomorphic multiplication is\n

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text very expensive, or if the chosen scheme only allows for a few multiplications, it is still possible use\n

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text it for very large databases: only the dictionary size is limited.\n

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text be a public key scheme: only the user encrypts and decrypts elements (the server simply has to be\n

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text able to initialize the buffer to E(0), but the user could send him this initial value). A symmetric\n

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text Application to Set Reconciliation. An interesting aspect of the LDPC scheme we presented is\n

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text that, even though it behaves like a randomized scheme, it is fully deterministic. What this means\n

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text 7 Not that a query of O(log |Ω|) bits is probably not achievable in practice as current FHE schemes all require some\n

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text important message expansion, but a poly-logarithmic query is possible.\n

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