SUBJECTIVE LOGIC BASED PROBABILISTIC KEY MANAGEMENT FOR MANETS Mahdieh Ahmadi Performance and...

SUBJECTIVE LOGIC BASED PROBABILISTIC KEY MANAGEMENT FOR

MANETSMahdieh Ahmadi

Performance and Dependability Laboratory

Sharif University of Technology

Spring 2014

SL based Probabilsitic Key Managment 2/

Outline

• Mobile Ad hoc networks(MANETs)

• Probabilistic Key Management

• Subjective Logic

• Proposed Algorithm

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Mobile Ad hoc Networks• Wireless Networks

• Infrastructure-based Networks• Wireless LANs• Ad hoc Networks

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Useful when • infrastructure not available• Impractical• Expensive


MANETs :: Complexities

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• Autonomous and infrastructure less

• Multi-hop routing

• Dynamic network topology

• Device heterogeneity

• Bandwidth constrained variable capacity links

• Network ScalabilityA

B AB


MANETs:: Complexities

• Broadcast nature of the communications

• Lack of mobility awareness by system/applications

• Short battery lifetime

• Limited capacities

• Security

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MANETs:: Security

• Nodes rely on other nodes for communication

• No centralized trusted authorities

• Intermediate nodes are able to Read, Drop or Change

messages before resending them

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Cryptography• Symmetric key cryptography

• Define a shared key between every two node• Shared or pairwise

• Pairwise : every node should store n-1 keys• Shared : compromising just one node destroys the security of the whole

network

• Asymmetric cryptography without third party• Stores all the public keys in every node• Provides authentication, integrity and non-repudiation

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Confidentiality Integrity Authentication Non-repudiation× Availability

Confidentiality× Integrity× Authentication× Non-repudiation× Availability


Cryptography :: Key Management

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• Provide secure procedures for handling cryptographic keying materials

Key Management :: Probabilistic Key Management

Mahdieh Ahmadi SL based Probabilsitic Key Managment 9

j

. …

Destination

Source

i

j

k

i

. … k

. …

j

a

b

a

b

Confidentiality Integrity Authentication Non-repudiation× Availability Need limited capacity

Introduced by Gharib et al., 2013.

SL based Probabilsitic Key Managment 10

Probabilistic Key Management :: Features

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𝒂 (𝒍𝒐𝒈𝒌𝒏 )+𝒃

Connectivity Probability : 99.99%Storing only a few keys instead of all keys


Probabilistic Key Management :: Concerns

• Intermediate decryption-encryption processes• The intermediate node that decrypts and encrypts the message can read or

change it.

• Manifolded traffic• The overall path length is manifold by increasing the average cryptographic

path length.

• Solution• Minimizing • Using the shortest and the most trusted route• Using subjective logic to model the problem

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Subjective Logic

• Extend probabilistic logic by expressing uncertainty

• Reason in presence of uncertain or incomplete evidence• Prepositions such as “I don’t know” cannot be expressed

ignorance or uncertainty

• Additivity Principle• Sum of mutually disjoint elements in a state space add up to 1• Probabilistic Logic YES• Belief Theory NO : main reason for creating belief theory• Reality closer to belief theory

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Standard Logic

Probabilistic Logic

? ?Standard

LogicProbabilistic

LogicBelief Theory ?


Subjective Logic :: Definitions• Frame

• state space(X) with cardinality k

• Base rates • A priori probability in the absence of committed belief mass

• Belief masses can be distributed over the X or over the reduced power set of X

• Uncertainty mass u x • Uncertainty about the probability expectation value

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Standard Logic

Probabilistic Logic

Belief Theory Subjective Logic


Subjective Logic :: Opinion • Opinion

• Applies to a frame(X)• Has an attribute that identifies the belief owner (A)• Function of belief masses, uncertainty mass and base rate

• According to uncertainty• uncertain opinion

• U x > 0

• Dogmatic opinion• U x = 0

• According to type of frame• Binomial Opinions

• Binary frame

• Multinomial Opinions• Frames larger than binary but singletons are focal elements

• Hyper Opinions• Frames larger than binary but there are focal elements of any class

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15

Opinion:: Binomial Opinion• Frame

• Binary frame or binary partition of n-array frame

• Binomial opinion about the truth of state x • w x = (b, d, u, a)

• b + d + u = 1• E x = b + au

• b = 1 TRUE • d = 1 FALSE• b + d =1 Probabilistic Logic• b + d <1 Degrees of uncertainty • b + d = 0 total uncertainty

Mahdieh Ahmadi SL based Probabilsitic Key Managment


Binomial Opinion :: Evidence Notation

• =

• Observation vector of X

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Binomial Opinion r

• Number of observations of x s

• Number of observations of


Subjective Logic :: Probabilistic Notation

• =

• Expectation value vector of X

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Subjective Logic :: Operators• Addition• Subtraction• Multiplication• Division• Deduction• Abduction• Discounting• Cumulative fusion• Averaging fusion• Belief Constraining• …

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Subjective Logic :: Fusion• Combines evidence from multiple sources• Two observers’ respective evidence opinions

• If observations are independent• Cumulative Fusion Operator

• If observations are dependent• Average Fusion Operator

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Average Fusion Operator:

Case I: For :

Case II: For :


Subjective Logic :: Trust Transitivity• A trusts B• B believes that proposition x is true• Agent A will also believe that proposition x is true

• What is the effect of A disbelieving that B will give a good advice?• A thinks that B ignores the truth value of x• A thinks that B consistently recommends the opposite of his real

opinion about the truth value of x

• Base Rate Sensitive Discounting

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Uncertainty Favouring DiscountingOpposite Belief Favouring• + • +


Subjective Logic :: Example

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SL Based Probabilistic Key Management

• K = 2• : x’s opinion that the shortest path

from itself to ‘dest’ is via ‘i’• Every node stores binomial

opinions for each destination node i.e. opinions

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kDestination

a

b

c

d

ef

g

(0, 0, 1, 0.5)

(0, 0, 1, 0.5)

(0, 0, 1, 0.5)

(0, 0, 1, 0.5)

(0, 0, 1, 0.5)

(0, 0, 1, 0.5)

(1, 0, 0, 0.5)

(0, 0, 1, 0.5)

(1, 0, 0, 0.5)

(0, 0, 1, 0.5)

(1, 0, 0, 0.5)

(0, 0, 1, 0.5)(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

• C(x, y): node ‘x’ opinion toward node ‘y’• Initial value

• C(x, y) = (Threshold, 0, (1-Threshold))• Optimistic

• Threshold > 0.5

• Pessimistic• Threshold < 0.5



• Definitions

• x’s opinion that the best path from itself to ‘y’ is via ‘i’

• Node ‘x’ opinion toward node ‘y’

• • x’s opinion that the best path from itself to ‘y’ is via whom• Where max is taken over all cryptographic neighbors of x.

• If then• ‘x’ asks its cryptographic neighbors’ opinions about the best path to node ‘y’• When node ‘x’ receives answer from its neighbors, it updates its own opinion using

equation below.• = • For all cryptographic neighbors of x.• Again

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Mahdieh Ahmadi SL based Probabilsitic Key Managment 24

Destination

Source

i

j

d

s

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0, 0, 1, 0.5)

(0, 0, 1, 0.5)

(1, 0, 0, 0.5)

…

(0.7, 0, 0.3, 0.5)

…

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0.7, 0, 0.3, 0.5)

(0, 0, 1, 0.5)

(0.7, 0, 0.3, 0.5)

(0.49, 0, 0.51, 0.5)



• Characteristics• Proactive Routing • Trusts update when time passes

• Using nodes’ behavior

• Opinions fade(decrease) when times passes• Using exponential relation

• Loop Prevention• Using TTL = • Pass Path

• Features• Does not suffer from

• Honest Elicitation• Free Riding

• Decreases the number of untrusted nodes who decrypt the message

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Results• Should be tested in ns3

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References• Mohammed Gharib, Ehsan Emamjomeh-Zadeh, Ashkan Norouzi-Fard, and Ali

Movaghar. A novel probabilistic key management algorithm for largescale manets. In Proceedings of the 2013 27th International Conference on Advanced Information Networking and Applications, AINA '13, pages 349-356. IEEE Computer Society, 2013.

• Anurag Kumar, D. Manjunath, and Joy Kuri. 2008. Wireless Networking. Morgan Kaufmann Publishers Inc., San Francisco, CA, USA.

• Audun Jøsang. Subjective Logic. Draft book, February 2013. (http://folk.uio.no/josang/papers/subjective_logic.pdf, February 18 2013)

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http://folk.uio.no/josang/papers/subjective_logic.pdf

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