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Designs Codes and Cryptography

Generalized oblivious transfer by secret sharing

Tamir Tassa

Published Mar 13, 2010• 74 citations
Publisher
DOI
10.1007/s10623-010-9378-8

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In 1-out-of-qqqOblivious Transfer (OT) protocols, a sender Alice is able to send one ofq≥2q\ge 2q≥2messages to a receiver Bob, all while being oblivious to which message was transferred. Moreover, the receiver learns only one of these messages. Oblivious Transfer combiners takennninstances of OT protocols as input, and produce an OT protocol that is secure if sufficiently many of thennnoriginal OT instances are secure. We present new 1-out-of-qqqOT combiners that are perfectly secure against active adversaries. Our combiners arise from secret sharing techniques. We show that given anFq\mathbb {F}_{q}Fq​-linear secret sharing scheme on a set ofnnnparticipants and adversary structureA\mathcal {A}A, we can construct annnn-server, 1-out-of-qqqOT combiner that is secure against an adversary corrupting either Alice and a set of servers inA\mathcal {A}A, or Bob and a set of serversBBBwithBˉ∉A\bar {B}\notin \mathcal {A}Bˉ∈/A. If the normalized total share size of the scheme is$\ell ,thentheresultingOTcombinerrequires, then the resulting OT combiner requires,thentheresultingOTcombinerrequires\ell callstoOTprotocols,andthetotalamountofbitsexchangedduringtheprotocoliscalls to OT protocols, and the total amount of bits exchanged during the protocol iscallstoOTprotocols,andthetotalamountofbitsexchangedduringtheprotocolis(q^{2}+q+1)\ell \log q.Wealsopresentaconstructionbasedon1−out−of−2OTcombinersthatusestheprotocolofCreˊpeau,BrassardandRobert(FOCS1986).Thisconstructionprovidessmallercommunicationcostsforcertainadversarystructures,suchasthresholdones:Foranyprimepower. We also present a construction based on 1-out-of-2 OT combiners that uses the protocol of Crépeau, Brassard and Robert (FOCS 1986). This construction provides smaller communication costs for certain adversary structures, such as threshold ones: For any prime power.Wealsopresentaconstructionbasedon1−out−of−2OTcombinersthatusestheprotocolofCreˊpeau,BrassardandRobert(FOCS1986).Thisconstructionprovidessmallercommunicationcostsforcertainadversarystructures,suchasthresholdones:Foranyprimepowerq\geq n,thereare, there are,therearen−server,1−out−of−-server, 1-out-of-−server,1−out−of−qOTcombinersthatareperfectlysecureagainstactiveadversariescorruptingeitherAliceorBob,andaminorityoftheOTcandidates,exchangingOT combiners that are perfectly secure against active adversaries corrupting either Alice or Bob, and a minority of the OT candidates, exchangingOTcombinersthatareperfectlysecureagainstactiveadversariescorruptingeitherAliceorBob,andaminorityoftheOTcandidates,exchangingO(qn\log q)$bits in total.