A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT OF C RYPTOGRAPHIC - S CHEMES K EY A SSIGNMENT S CHEMES Speaker: Ridha Khedri Khair Eddin Sabri and Ridha Khedri Intro. Motivation and Foundations & Practice of Security Symposium (Oct. 2012) Contribution The main idea Mathematical Background Key Structure Key Assignment Schemes Specifying the Akl-Taylor technique Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Presentation Outline A G ENERIC A LGEBRAIC M ODEL Introduction 1 FOR THE A NALYSIS OF Motivation and Contribution 2 C RYPTOGRAPHIC - K EY A SSIGNMENT The main idea 3 S CHEMES Mathematical Background Speaker: Ridha 4 Khedri Order Intro. Semiring Motivation and keyStructure 5 Contribution Key Assignment Schemes 6 The main idea Specifying the Akl-Taylor technique 7 Mathematical Background Specifying the Chinese Remainder Technique 8 Key Structure 9 Verification of secrecy properties Key Assignment 10 Conclusion and Future Work Schemes Specifying the Akl-Taylor technique Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Introduction A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC - K EY A SSIGNMENT S CHEMES Speaker: Ridha Khedri ¡ Intro. Motivation and Contribution Agent 1 Agent 1 The main idea Data Store Data Store Mathematical Background Data Server Agent 2 Encrypted Key Structure Agent 2 Data Key Assignment Schemes Specifying the Agent 3 Akl-Taylor Agent 3 technique Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Introduction A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS Encrypted-data stores require OF C RYPTOGRAPHIC - K EY A SSIGNMENT S CHEMES Encryption of information Speaker: Ridha Khedri Intro. Distribution of keys to users Motivation and Contribution The main idea Cipher? Mathematical Either, a common cipher is used by all agents Background Key Structure Or, each agent uses in a quasi-permanent way a set Key Assignment Schemes of already agreed-on ciphers Specifying the Akl-Taylor technique Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Introduction What governs key-assignments? A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS Schemes for key assignments are adopted OF C RYPTOGRAPHIC - K EY A SSIGNMENT S CHEMES Object-based scheme : focuses on objects and the Speaker: Ridha Khedri required conditions to decrypt each one of them Intro. Motivation and Key-based scheme : Ð Ý Our focus Contribution Objects are partially ordered ( i.e., ď is transitive, The main idea reflexive, and antisymmetric) Mathematical Background Key Structure c i ď c j : security level c j is more sensitive than the Key Assignment security level c i Schemes Specifying the ù ñ User at c j can also have an access to an Akl-Taylor technique information classified c i Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Introduction Key-based scheme: A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS K1 OF C RYPTOGRAPHIC - K EY A SSIGNMENT Dean S CHEMES Speaker: Ridha K2 Khedri K3 Prof. Chair Intro. K4 Motivation and Contribution Student The main idea Mathematical Background Key Structure Key k 1 can be used to derive the keys k 2 , k 3 and k 4 Key Assignment Schemes However, no practical way to derive a key associated Specifying the to a node n from those associated to its Akl-Taylor technique descendants Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Motivation and Contribution A G ENERIC Several techniques exist in the literature to handle A LGEBRAIC M ODEL FOR THE A NALYSIS key assignment: OF C RYPTOGRAPHIC - r AklTaylor1983, AtallahBlantonFazio2009, K EY A SSIGNMENT S CHEMES KuoShenChenLai1999, Sandhu1987 s Speaker: Ridha Khedri Problem: Lack of formal means to proof their Intro. correctness / secrecy Motivation and Contribution The main idea Several of them have been found to be flawed or Mathematical very weak in preserving secrecy Background Key Structure Key Assignment Crampton et al. advocate the adoption of a generic Schemes model for key assignment schemes Specifying the For evaluating proposals for key assignment Akl-Taylor technique schemes Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Motivation and Contribution What do we propose? A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS A generic model for the specification and analysis of OF cryptographic-key assignment schemes C RYPTOGRAPHIC - K EY A SSIGNMENT S CHEMES Speaker: Ridha An analysis of two representative schemes: Khedri Akl-Taylor key assignment r AklTaylor1983 r scheme Intro. Motivation and A scheme based on the Chinese remainder Contribution theorem r ChenChung2002 s The main idea Mathematical Background A generalized and extended scheme to assign more Key Structure than one key to a security class Key Assignment Schemes Specifying the The automation of the analysis of systems that use Akl-Taylor technique key assignment schemes (Prover9) Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES The main idea A G ENERIC The key-structure within a set of structures: A LGEBRAIC M ODEL FOR THE A NALYSIS OF Message Structure C RYPTOGRAPHIC - K EY A SSIGNMENT S CHEMES Speaker: Ridha Khedri Envelope Structure Intro. Motivation and Contribution Key Structure Cipher Structure Secret Structure The main idea Mathematical Background Key Structure Key Assignment Structure B is a building block A B Schemes of structure A Specifying the Akl-Taylor Fundamenta Informaticae , 112(4):305–335, 2011. technique Specifying the Chinese Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Remainder
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Mathematical Background Order Let C be a set. A G ENERIC A LGEBRAIC M ODEL A partial order (or order) on C is a binary relation ă FOR THE A NALYSIS OF on C such that, for all x , y , z P C , C RYPTOGRAPHIC - K EY A SSIGNMENT x ă x , Reflexive 1 S CHEMES x ă y ^ y ă x ù ñ x “ y , Antisym. 2 Speaker: Ridha Khedri x ă y ^ y ă z ù ñ x ă z Trans. 3 Intro. A set equipped with a partial order is called an Motivation and Contribution ordered set, partially ordered set, or poset The main idea Mathematical Background A pre-ordered set (or quasi-ordered set): satisfies Order Semiring only (1) and (3), but not (2) Key Structure Key Assignment For a pre-ordered set p P , ă q , its dual p P , ăq is Schemes Specifying the def defined as for all x , y , we have x ă y ð ñ y ă x Akl-Taylor technique Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Specifying the
A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTOGRAPHIC -K EY A SSIGNMENT S CHEMES Mathematical Background Semiring A G ENERIC Definition (Semiring) A LGEBRAIC M ODEL FOR THE A NALYSIS OF Let S ‰ H be a set and ` and ¨ binary operations on S , C RYPTOGRAPHIC - K EY A SSIGNMENT ` ˘ S , ` , ¨ named addition and multiplication. Then is S CHEMES ` ˘ called a semiring if S , ` is a commutative semigroup, Speaker: Ridha Khedri ` ˘ S , ¨ is a semigroup, and ¨ distributes over ` on both the Intro. left and right. Motivation and Contribution ` ˘ S , ` is an idempotent semigroup The main idea ` ˘ S , ` , ¨ an additively idempotent ❀ Mathematical Background semiring Order Semiring ` ˘ S , ¨ is a commutative semigroup Key Structure ` ˘ S , ` , ¨ a commutative semiring ❀ Key Assignment ` ˘ S , ` , ¨ Schemes is an additively idempotent semiring Specifying the there exists a natural ordering relation ❀ Akl-Taylor technique Speaker: Ridha Khedri A G ENERIC A LGEBRAIC M ODEL FOR THE A NALYSIS OF C RYPTO Specifying the
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