,

,

109 6.2.1 MapReduce Grouping and Aggregation with COUNT Operation . . 109 6.2.2 MapReduce Grouping and Aggregation with SUM Operation, p.111 ,

3.2 Secure MapReduce Grouping and Aggregation with COUNT Operation ,

, Secure MapReduce Grouping and Aggregation with SUM Operation 114

, Secure MapReduce Grouping and Aggregation with AVG Operation 114

, Secure MapReduce Grouping and Aggregation with MIN Operation 116

,

, , p.117

,

,

,

, Secure Joins with MapReduce, vol.131

, 3.2 Secure n-ary Joins with MapReduce and Cascade Protocol

,

,

,

We focus on a scenario where the data owner outsources her relation on an semi-honest public cloud. More specifically, we evaluate grouping and aggregation with COUNT, We propose secure multiparty protocols computing grouping and aggregation operations on a relation using the MapReduce paradigm, 2018. ,

, 6.1.1 Grouping and Aggregation with MapReduce, p.107

109 6.2.1 MapReduce Grouping and Aggregation with COUNT Operation 109 6.2.2 MapReduce Grouping and Aggregation with SUM Operation . . 110 6.2.3 MapReduce Grouping and Aggregation with AVG Operation . . 110 6.2.4 MapReduce Grouping and Aggregation with MIN Operation ,

112 6.3.2 Secure MapReduce Grouping and Aggregation with COUNT Operation ,

, Secure MapReduce Grouping and Aggregation with SUM Operation114

, Secure MapReduce Grouping and Aggregation with AVG Operation114

, Secure MapReduce Grouping and Aggregation with MIN Operation116

,

, , p.117

,

,

,

, Contents

3.2 Secure n-ary Joins with MapReduce and Cascade Protocol ,

,

,

,

, Relation Links(From, To) consists of the set of pairs of URLs, such that the first has one or more links to the second

, Schematic of a MapReduce computation

, Running time vs the number of matrix multiplications using Paillier's cryptosystem [Pai99] and the BFV [FV12a] scheme

,

Paillier interactive multiplicative homomorphic protocol, p.24 ,

,

,

Map and Reduce functions for MM-2R protocol, p.31 ,

Map and Reduce functions for the MM-1R protocol, p.32 ,

Preprocessing algorithms for secure matrix multiplication, vol.34 ,

Map and Reduce functions for the SP-2R protocol, p.34 ,

, Map and Reduce functions for SP-1R protocol

Map and Reduce functions for the CRSP-2R protocol, p.36 ,

37 3.10 CPU time vs the matrices' dimension for the two state-of-the-art protocols [LRU14] computing the matrix multiplication ,

,

88 5.2 Example of intersection with MapReduce between three relations. First, data owners outsource their respective relation on the public cloud. The public cloud runs the Map function, then the Reduce function verifies if a key is associated to a list of three values, vol.89 ,

,

, MapReduce protocol to compute the intersection of n relations, p.91

Map and Reduce functions of our secure approach SI, p.93 ,

and Mossad * after the preprocessing phase of our secure protocol SI ,

, Example of intersection with MapReduce between three relations using our secure protocol SI. First, data owners outsource their respective encrypted relation on the public cloud. The public cloud runs the Map function, then the Reduce function verifies if keys are associated to a list of three values

96 5.10 CPU time vs the number of intersected relations for the standard MapReduce protocol [LRU14] and our secure approach SI computing the intersection between two relations ,

,

, , p.99

,

U for the proof of Lemma 27 ,

, Adversary B for the proof of Lemma 27

,

, , p.107

,

Map and Reduce functions for the COUNT protocol, p.110 ,

, Map and Reduce functions for the SUM protocol

, Map and Reduce functions for the AVG protocol

, Map and Reduce functions for the MIN protocol

, , p.113

Map and Reduce functions for the SGA COUNT protocol, p.114 ,

Map and Reduce functions for the SGA SUM protocol, p.115 ,

Map and Reduce functions for the SGA AVG protocol, p.115 ,

116 6.13 CPU time vs the number of tuples for no-secure and secure protocols which perform grouping and aggregation with the COUNT, SUM, AVG, and ,

,

SUM C for the proof of Lemma 30 ,

, Adversary A for the proof of Lemma 30

for the proof of Lemma 30 ,

, Adversary B for the proof of Lemma 30

, , p.125

, Adversary A for the proof of Lemma 33

, , p.127

, Adversary B for the proof of Lemma 33

, Adversary D for the proof of Lemma 33

, Joins between relations R 1, vol.132

Cascade of joins with MapReduce between n relations, p.133 ,

, Underlined tuples correspond to tuples that participate to the final join result

, Map and Reduce functions for CAS protocol

, Map and Reduce functions for HYP protocol

,

, We denote ciphertexts of an IND-CPA encryption scheme by {·}, and pseudo-random evaluations by integers

, Map and Reduce functions for SCAS protocol

140 7.11 CPU time vs the number of tuples per relation for CAS and HYP protocols, and their respective secure approach ,

,

, , p.144

, Adversary A for the proof of Lemma 36

, Adversary B for the proof of Lemma 36

, , p.146

, , p.148

, Adversary A for the proof of Lemma 39

$ ) be the cost of multiplication (resp. addition, exponentiation, encryption, decryption, inversion, sampling) ,

|R n |) be the biggest cardinal of relations R i with i ? 1, n . Let C ? be the computation cost of a bitwise exclusive OR operation) ,

, Complexity of original MapReduce grouping and aggregation protocols and of our SGA protocols

be the cost of addition (resp. modular multiplication, asymmetric encryption, division, comparison), p.117 ,

Wall clock times of the secure protocols preprocessing, p.119 ,

Wall clock times of the secure protocols preprocessing, p.141 ,

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