Secure Location Verification and Stabilization

acceptance zone k r β Secure Location Verification and Stabilization Adnan Vora and Mikhail Nesterenko Kent State University

Location Verification problem • description • have: protected asset • ensure: access to asset only if the principal is in correct location • applications: • wireless network access • keyless car starting • secure gate opening • perimeter protection and friendly force identification, etc. • appeal • immediate practical benefits • non-traditional approach to security • effective algorithmic solutions

Outline • problem statement • basic solution • description and properties • immediate applications: securing arbitrary zones • extensions • improving efficiency • operating with non-circular signal propagation • protecting against directional antennas • using random sensor placement • stabilization and fault-tolerance

protectionzone prover verifiers Problem Statement stated informally in[SSW’03] actors • (potentially malicious) prover(s) • arbitrary protection zone • a set of RF-capable verifiers problem specify: • placement rules for verifiers • prover  verifierscommunication protocol so that the verifiers accept the correctprover only if it is inside the protection zone and reject otherwise protocol is secure if a prover anywhere outside protection zone is rejected • assumptions • prover authentication not required • verifiers are trusted • intra-verifier communication is reliable and secure • signal propagation is perfectly omni-directional (unit-disk)

Previous Approaches • use limited signal propagation speed (e.g. ultrasound) • a verifier radios prover • prover buzzes back • verifier computes roundtrip time and calculates distance • limitations • uncertainties of two mediums: sound and ether (echos, varying propagation speeds) • extra hardware needed: sounders and microphones • requires sequential verification (and time synchronizationbetween verifiers) RF prover .01secs=4ft sound verifiers

Basic Solution idea use broadcast nature of RF signalpropagation specifics • separate roles of verifiers • acceptor – receives signal from prover inside protection zone • rejector – receives signal from outside prover solution • communication protocol: • prover broadcasts signal to distance x, if no decision – increases distance by x • prover is accepted if only acceptors hear from prover, rejected otherwise, informed of decision • placement rules: to come x x acceptedprover x acceptors rejector rejectedprover

Rejection Zone rejection zone – prover (correct ormalicious) is never accepted Lemma 1 [VN04] a pointon a plane is in rejectionzone if it is closer to the nearest rejector than the nearest acceptor Theorem 1 sensor placement is secure iff the rejectors’ Voronoi cells cover the area outside the protection zone rejection zone rejector rejector acceptor Voronoi diagram rejector rejector

why ambiguity zone exists x x x x correct prover rejected malicous prover accepted Acceptance and Ambiguity Zones rejector rejector acceptance zone – correct prover is always accepted ambiguity zone – prover may (not) be accepted acceptor acceptance zone ambiguity zone rejector rejector Lemma 2: a point is in acceptance zoneif it is x closer to the nearest acceptor than to the nearest rejector

Securing Polygons rejection zone protection gap – largest distance frompoint in rejection zone to nearestpoint outside protection zone – measures how far rejection zone encroaches upon protection zone protection is complete if protection gap is zero Lemma 3n-sided convex polygonis completely protected with n+1 verifiers Lemma 4 in this case, if the protection zone contains a circle of radius r, the acceptance zone contains an open disk of radius r-x/2 Theorem 2 An arbitrary n-sided polygonal protection zone can be completely secured with O(n) verifiers ambiguity zone acceptance zone x/2 protection zone boundary

t Securing Arbitrary Protection Zones ambiguity gap – largest distance from a point in ambiguity zoneto nearest point outsideprotection zone Theorem 3 the numberof verifiers required to secure an arbitrary-shaped protection zone of area Sand perimeter P withconstant ambiguity gap is in O(P+S) Proof outline: • divide protection zone in squares with constant side t (number of such squares is in O(P+S)) , • protect each square individually with 5 verifiers acceptance zone x

Protecting against Directional Antennas rejectors assumption: fixed beamwidth β Theorem 5 an arbitrary shapedprotection zone can be securedagainst malicious provers using O(r) verifiers where r is radius of inscribed circle proof outline: idea – place rejectors such that if acceptor isreached so is rejector • inscribe circle with radius r • place rejectors on circumferenceof co-centric circle of radius r-k, where k – constant, space rejectors 2k tan(β/2) apart • place acceptor in the middle,  condition satisfied protectionzone acceptance zone acceptor k r β 2k tan(β/2) malicious prover

Logarithmic Verification Time • basic algorithm: number of verification attempts is d/xwhere d – protection zone diameter • with more acceptors can be made logarithmic • add acceptor placement rule:for every point in the acceptance zone, there exists integer i≥0, such that there are no rejectors closer to this point than x2i+1 and at least one acceptor between x2i and x2i+1 • modify protocol: prover doubles its signal strength every verification attempt Theorem 5 modified protocol is correct and the maximum number of broadcasts is in O(logd)

… x/4 x/4 x/4 x/4 prover rejected no decision accepted a b Shrinking Ambiguity Zone • basic algorithm: ambiguity zone size isproportional to x • can be made arbitrarily small with additional verification attempts • recall: ambiguity zone is dueto discrete signal increments • idea: tune signal strength if rejected • modified protocol:if prover is rejected and the last signal increment is z, the prover decreases the signal strength by z/2and rebroadcasts; if no decision, the prover increases the signal stregth by z/2 and rebroadcasts; process continues until prover accepted Theorem 6 the modified protocol is correct and the number of extra broadcast attempts is proportional log(b-a)

basic model definite r never prover complex model definite possible y never r prover Complex Signal Propagation • basic signal propagation model: unit-disk • complex (more realistic) model: a ring of possible signal reception zone delineation for complex model: Lemma 6: a point is in rejection zone if it is at least y closer to nearest rejector than acceptor Lemma 7: a point is in acceptance zone if it is at least x+y closer to nearest acceptor than rejector results similar to basic model apply signal reception

boundary rejectors acceptance+ ambiguityzones border of protection zone rejectors Random Verifier Placement modified problem • verifiers are not aware of theirlocation • they are informed if they areinside or outside protection zone classification • an outside verifier is rejector • a verifier whose Voronoi neighbor is outside is rejector • rest are acceptors Theorem 7 verification protocol with random placement of verifiers solves location verification problem

outsideverifiers boundary rejectors acceptors insideverifers rejectors Implementation of Random Placement • in practice radio neighborhoodcan be used to approximate Voronoi neighborhood • need to ensure appropriate verifier density on the border of protection zone • placement procedure • verifiers have read-only bit signifying inside/outside placement • classification procedure • if verifier or its neighbors have outside bit set – verifier is rejector, acceptor otherwise

Stabilization of Random Placement • observe: classification decisionis local – depends only on neighborhood topology  very robust • state correction – each verifier periodically checks the inside/outside bits of the neighbors and reevaluates its classification  global state • stabilizes • fault-contains • adaptively • in constant time/space/energy corrupt state

Other Extensions and Further Info • distributed decision making – an acceptor only needs to contact neighboring rejectors • fault-tolerant rejector sets – redundant rejector sets independently covering rejection zone provide extra security and fault-tolerance guarantees • limited power provers – can be serviced with appropriately dense acceptor location • details:A. Vora, M. Nesterenko "Secure Location Verification Using Radio Broadcast”, Techreport TR-KSU-CS-2004-01,http://www.cs.kent.edu/~mikhail/Research/tr-ksu-cs-2004-01.pdf

Secure Location Verification and Stabilization