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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
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 3/
Mobile Ad hoc Networks• Wireless Networks
• Infrastructure-based Networks• Wireless LANs• Ad hoc Networks
Mahdieh Ahmadi
Useful when • infrastructure not available• Impractical• Expensive
SL based Probabilsitic Key Managment 4/
MANETs :: Complexities
Mahdieh Ahmadi
• Autonomous and infrastructure less
• Multi-hop routing
• Dynamic network topology
• Device heterogeneity
• Bandwidth constrained variable capacity links
• Network ScalabilityA
B AB
SL based Probabilsitic Key Managment 5/
MANETs:: Complexities
• Broadcast nature of the communications
• Lack of mobility awareness by system/applications
• Short battery lifetime
• Limited capacities
• Security
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 6/
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
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 7/
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
Mahdieh Ahmadi
Confidentiality Integrity Authentication Non-repudiation× Availability
Confidentiality× Integrity× Authentication× Non-repudiation× Availability
SL based Probabilsitic Key Managment 8/
Cryptography :: Key Management
Mahdieh Ahmadi
• 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
Mahdieh Ahmadi
𝒂 (𝒍𝒐𝒈𝒌𝒏 )+𝒃
Connectivity Probability : 99.99%Storing only a few keys instead of all keys
SL based Probabilsitic Key Managment 11/
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
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 12/
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
Mahdieh Ahmadi
Standard Logic
Probabilistic Logic
? ?Standard
LogicProbabilistic
LogicBelief Theory ?
SL based Probabilsitic Key Managment 13/
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
Mahdieh Ahmadi
Standard Logic
Probabilistic Logic
Belief Theory Subjective Logic
SL based Probabilsitic Key Managment 14/
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
Mahdieh Ahmadi
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
SL based Probabilsitic Key Managment 16/
Binomial Opinion :: Evidence Notation
• =
• Observation vector of X
Mahdieh Ahmadi
Binomial Opinion r
• Number of observations of x s
• Number of observations of
SL based Probabilsitic Key Managment 17/
Subjective Logic :: Probabilistic Notation
• =
• Expectation value vector of X
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 18/
Subjective Logic :: Operators• Addition• Subtraction• Multiplication• Division• Deduction• Abduction• Discounting• Cumulative fusion• Averaging fusion• Belief Constraining• …
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 19/
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
Mahdieh Ahmadi
Average Fusion Operator:
Case I: For :
Case II: For :
SL based Probabilsitic Key Managment 20/
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
Mahdieh Ahmadi
Uncertainty Favouring DiscountingOpposite Belief Favouring• + • +
SL based Probabilsitic Key Managment 21/
Subjective Logic :: Example
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 22/
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
Mahdieh Ahmadi
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
SL based Probabilsitic Key Managment 23/
SL Based Probabilistic Key Management
• 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
Mahdieh Ahmadi
SL Based Probabilistic Key Management
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)
SL based Probabilsitic Key Managment 25/
SL Based Probabilistic Key Management
• 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
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 26/
Results• Should be tested in ns3
Mahdieh Ahmadi
SL based Probabilsitic Key Managment 27/
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)
Mahdieh Ahmadi
THANK YOU