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NP-hardness of the shortest vector problem
and its relation to the closest vector problem
Jeroen van Wier
Interdisciplinary Centre for Security, Reliability and Trust, University of Luxembourg
June 12, 2019
Jeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 1 / 17
Overview
1Problem denitions
2Reducing GapCVP' to GapSVP
3Reducing SetCover to GapCVP'
4Super Awesome Bonus
Jeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 2 / 17
SVP/CVP denition
Denition
TheoptimisationproblemSVPis dened on instancesB2Znk, with the objective to nd a non-zero vector ~u2 L(B) such that k ~uk=(B):Denition TheoptimisationproblemCVPis dened on instances (B;~v), where B2Znkand~v2Zn, with the objective to nd a vector~u2 L(B) such that k ~u~vk= min~w2L(B)k~w~vk:Jeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 3 / 17
SVP/CVP denition
Denition
TheoptimisationproblemSVPis dened on instancesB2Znk, with the objective to nd a non-zero vector ~u2 L(B) such that k ~uk=(B):Denition TheoptimisationproblemCVPis dened on instances (B;~v), where B2Znkand~v2Zn, with the objective to nd a vector~u2 L(B) such that k ~u~vk= min~w2L(B)k~w~vk:Jeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 3 / 17
SVP/CVP denition
Denition
TheoptimisationproblemSVP
is dened on instancesB2Znk, with the objective to nd a non-zero vector ~u2 L(B) such that k ~uk (B):Denition
TheoptimisationproblemCVP
is dened on instances (B;~v), where B2Znkand~v2Zn, with the objective to nd a vector~u2 L(B) such that k ~u~vk min~w2L(B)k~w~vk:Jeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 3 / 17
GapSVP denition
Denition
TheoptimisationproblemSVP
is dened on instancesB2Znk, with the objective to nd a non-zero vector ~u2 L(B) such that k ~uk (B):Denition
ThepromiseproblemGapSVP
is dened on instances (B;d), where B2Znkandd2R+, with the objective to distinguish between the cases: yes:(B)d no:(B)> dJeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 4 / 17
GapSVP denition
Denition
TheoptimisationproblemSVP
is dened on instancesB2Znk, with the objective to nd a non-zero vector ~u2 L(B) such that k ~uk (B):Denition
ThepromiseproblemGapSVP
is dened on instances (B;d), where B2Znkandd2R+, with the objective to distinguish between the cases: yes:(B)d no:(B)> dJeroen van Wier (Uni.lu - SnT)SVP is NP-hardJune 12, 2019 4 / 17
GapCVP denition
Denition
TheoptimisationproblemCVP
is dened on instances (B;~v), where B2Znkand~v2Zn, with the objective to nd a vector~u2 L(B) such thatquotesdbs_dbs2.pdfusesText_4
Sexually Violent Predator Assessment Process - SOAB