Vote-Independence: A Powerful Privacy Notion for Voting Protocols - - PowerPoint PPT Presentation

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Vote-Independence: A Powerful Privacy Notion for Voting Protocols - - PowerPoint PPT Presentation

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Vote-Independence: A Powerful Privacy Notion for Voting Protocols Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Universit Grenoble 1, CNRS,

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Vote-Independence: A Powerful Privacy Notion for Voting Protocols

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech

Université Grenoble 1, CNRS, Verimag

FPS 2011: May 13, 2011

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 2

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 3

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 4

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Voting machines are not a recent technology

They have been in use in the US for over 100 years!

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Electronic voting machines. . .

. . . are used all over the world

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Internet voting

Available in Estonia France Switzerland . . .

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Security Requirements

Security Requirements

Eligibility Fairness Robustness Individual Verifiability Universal Verifiability Privacy Receipt-Freeness Coercion-Resistance

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 9

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Security Requirements

Security Requirements

Eligibility Fairness Robustness Individual Verifiability Universal Verifiability Privacy Receipt-Freeness Coercion-Resistance

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Helios [?] is a web based open-source voting system based on homomorphic encryption. Alice Bob A B Server =A,B

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Helios [?] is a web based open-source voting system based on homomorphic encryption. Alice Bob A B Server A =A,B

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Helios [?] is a web based open-source voting system based on homomorphic encryption. Alice Bob A B Server A B =A,B

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Helios [?] is a web based open-source voting system based on homomorphic encryption. Alice Bob A B Server A B + =A,B

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Eve can attack Alice’s privacy by copying her vote: Alice Bob A B Server Eve =A,A,B To prevent this attack, we have to enforce Vote-Independence.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Eve can attack Alice’s privacy by copying her vote: Alice Bob A B Server A Eve =A,A,B To prevent this attack, we have to enforce Vote-Independence.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Eve can attack Alice’s privacy by copying her vote: Alice Bob A B Server A B Eve =A,A,B To prevent this attack, we have to enforce Vote-Independence.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 18

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Eve can attack Alice’s privacy by copying her vote: Alice Bob A B Server A B Eve A =A,A,B To prevent this attack, we have to enforce Vote-Independence.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 19

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Eve can attack Alice’s privacy by copying her vote: Alice Bob A B Server A B Eve A A =A,A,B To prevent this attack, we have to enforce Vote-Independence.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion What is electronic voting? An Attack on Privacy in Helios

Attack on Privacy in Helios [?]

Eve can attack Alice’s privacy by copying her vote: Alice Bob A B Server A B Eve A A + + =A,A,B To prevent this attack, we have to enforce Vote-Independence.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Vote-Privacy [?]

Main idea: Observational equivalence between two situations. Alice Bob Vote A B

≈l

Vote B A

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Receipt-Freeness [?]

Again: Observational equivalence between two situations, but Alice tries to create a receipt or a fake. Alice Bob Mallory A B

≈l

B A Secret Data Fake Data

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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SLIDE 25

Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Receipt-Freeness [?]

Again: Observational equivalence between two situations, but Alice tries to create a receipt or a fake. Alice Bob Mallory A B

≈l

B A Secret Data Fake Data

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Coercion-Resistance [?]

Observational equivalence between two situations, but Alice is under control by Mallory or only pretends to be so. Alice Bob Mallory A B

≈l

B A Secret Data Fake Data

Orders Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Coercion-Resistance [?]

Observational equivalence between two situations, but Alice is under control by Mallory or only pretends to be so. Alice Bob Mallory A B

≈l

B A Secret Data Fake Data

Orders Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Vote-Independence

Main idea: Privacy, but with a voter under control of the attacker. If he can relate his vote to e.g. Alice’s vote, Mallory can distinguish both sides. Alice Bob Mallory Vote A B

≈l

Vote B A Chuck ? ?

O r d e r s Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Defining Vote-Independence

Main idea: Privacy, but with a voter under control of the attacker. If he can relate his vote to e.g. Alice’s vote, Mallory can distinguish both sides. Alice Bob Mallory Vote A B

≈l

Vote B A Chuck ? ?

O r d e r s Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Vote-Independence with Passive Collaboration

“Receipt-Freeness with Chuck”: Alice Bob Mallory A B

≈l

B A Secret Data Fake Data Chuck ? ?

O r d e r s Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Vote-Independence with Passive Collaboration

“Receipt-Freeness with Chuck”: Alice Bob Mallory A B

≈l

B A Secret Data Fake Data Chuck ? ?

O r d e r s Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Vote-Independence with active Collaboration

“Coercion-Resistance with Chuck”: Alice Bob Mallory A B

≈l

B A Secret Data Fake Data

Orders

Chuck ? ?

O r d e r s Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion Privacy Vote-Independence

Vote-Independence with active Collaboration

“Coercion-Resistance with Chuck”: Alice Bob Mallory A B

≈l

B A Secret Data Fake Data

Orders

Chuck ? ?

O r d e r s Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

The Applied Pi Calculus [?]

Syntax P, Q, R := processes null process P|Q parallel composition !P replication νn.P name restriction (“new”) if M = N then P else Q conditional in(u, x) message input

  • ut(u, x)

message output {M/x} substitution

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Modeling a voting protocol

Definition (Voting Process [?]) A voting process is a closed plain process VP ≡ ν ˜ n.(V σ1| . . . |V σn|A1| . . . |Am). We define an evaluation context S which is like VP, but has a hole instead of three V σi, and an evaluation context S′ which is like VP, but has a hole instead of two V σi.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Vote-Privacy: The formal definition

Definition (Vote-Privacy [?]) A voting process respects Vote-Privacy (P) if for all votes a and b we have S′ [VA {a/v} |VB {b/v}] ≈l S′ [VA {b/v} |VB {a/v}]

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Vote-Independence (without Collaboration): The formal definition

Definition (Vote-Independence) A voting process respects Vote-Independence (VI) if for all votes a and b we have S

  • VA {a/v} |VB {b/v} |V c1,c2

C

  • ≈l S
  • VA {b/v} |VB {a/v} |V c1,c2

C

  • Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech

Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Receipt-Freeness: The formal definition

Definition (Receipt-Freeness [?]) A voting process respects Receipt-Freeness (RF) if there exists a closed plain process V ′ such that for all votes a and c we have V ′\out(chc,·) ≈l VA {a/v} and S′ VA {b/v}chc |VB {a/v}

  • ≈l S′

V ′|VB {b/v}

  • Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech

Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Vote-Independence with Passive Collaboration: The formal definition

Definition (Vote-Independence with Passive Collaboration) A voting process respects Vote-Independence with Passive Collaboration (VI-PC) if there exists a closed plain process V ′ such that for all votes a and c we have V ′\out(chc,·) ≈l VA {a/v} and S

  • VA {b/v}chc |VB {a/v} |V c1,c2

C

  • ≈l S
  • V ′|VB {b/v} |V c1,c2

C

  • Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech

Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Relations among the notions

CR RF VP VI-AC VI-PC VI

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Examples

CR [?] RF VP VI-AC [?] VI-PC [?] VI [?]

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Plan

1 Introduction

What is electronic voting? An Attack on Privacy in Helios

2 Intuitive Definitions

Privacy Vote-Independence

3 Formal Definitions 4 Analysis and Case Studies 5 Conclusion

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Conclusion

Attack on Helios Extended threat model Formal definition of “Vote-Independence” Strictly stronger than standard Vote-Privacy Generalized to passive and active collaboration Case studies: even Coercion-Resistant protocols may not ensure Vote-Independence

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Future Work

Generalized definition of voting protocols Tools to automate and/or verify the proofs (at least partly) Computational definition

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Thank you for your attention!

Questions?

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Ben Adida, Olivier De Marneffe, Olivier Pereira, and Jean-Jacques Quisquater. Electing a university president using open-audit voting: analysis

  • f real-world use of helios.

In Proceedings of the 2009 conference on Electronic voting technology/workshop on trustworthy elections, EVT/WOTE’09, pages 10–10, Berkeley, CA, USA, 2009. USENIX Association. Martín Abadi and Cédric Fournet. Mobile values, new names, and secure communication. In Proceedings of the 28th ACM SIGPLAN-SIGACT symposium on Principles of programming languages, POPL ’01, pages 104–115, New York, 2001. ACM. Jens-Matthias Bohli, Jörn Müller-Quade, and Stefan Röhrich.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Introduction Intuitive Definitions Formal Definitions Analysis and Case Studies Conclusion

Bingo voting: Secure and coercion-free voting using a trusted random number generator. In Ammar Alkassar and Melanie Volkamer, editors, E-Voting and Identity, volume 4896 of Lecture Notes in Computer Science, pages 111–124. Springer Berlin / Heidelberg, 2007. Stéphanie Delaune, Steve Kremer, and Mark Ryan. Verifying privacy-type properties of electronic voting protocols. Journal of Computer Security, 17:435–487, December 2009. Atsushi Fujioka, Tatsuaki Okamoto, and Kazuo Ohta. A practical secret voting scheme for large scale elections. In Jennifer Seberry and Yuliang Zheng, editors, Advances in Cryptology – AUSCRYPT ’92, volume 718 of Lecture Notes in Computer Science, pages 244–251. Springer Berlin / Heidelberg, 1992.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Byoungcheon Lee, Colin Boyd, Ed Dawson, Kwangjo Kim, Jeongmo Yang, and Seungjae Yoo. Providing receipt-freeness in mixnet-based voting protocols. In Jong In Lim and Dong Hoon Lee, editors, Information Security and Cryptology - ICISC 2003, volume 2971 of Lecture Notes in Computer Science, pages 245–258. Springer Berlin / Heidelberg, 2004. Tatsuaki Okamoto. An electronic voting scheme. In Proceedings of the IFIP World Conference on IT Tools, pages 21–30, 1996. Ben Smyth and Veronique Cortier. Attacking and fixing helios: An analysis of ballot secrecy. Cryptology ePrint Archive, Report 2010/625, 2010. http://eprint.iacr.org/.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Coercion-Resistance: The formal definition

Definition (Coercion-Resistance [?]) A voting process respects Coercion-Resistance (CR) if there exists a closed plain process V ′ such that for any C = νc1.νc2.(_|P) satisfying ˜ n ∩ fn(C) = ∅ and S′ [C [VA {?/v}c1,c2] |VB {a/v}] ≈l S′ VA {b/v}chc |VB {a/v}

  • and for all votes a and c we have

C

  • V ′\out(chc,·) ≈l VA {a/v}

and S′ [C [VA {?/v}c1,c2] |VB {a/v}] ≈l S′ C

  • V ′

|VB {b/v}

  • Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech

Vote-Independence

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Vote-Independence with Active Collaboration: The formal definition

Definition (Vote-Independence with Active Collaboration) A voting process respects Vote-Independence with Active Collaboration (VI-AC) if there exists a closed plain process V ′ such that for any C = νc1.νc2.(_|P) satisfying ˜ n ∩ fn(C) = ∅ and S

  • C [VA {?/v}c1,c2] |VB {a/v} |V c3,c4

C

  • ≈l S
  • VA {b/v}chc |VB {a/v} |V c3,c4

C

  • and for all votes a and c we have

C [V ′]\out(chc,·) ≈l VA {a/v} S

  • C [VA {?/v}c1,c2] |VB {a/v} |V c3,c4

C

  • ≈l S
  • C [V ′] |VB {b/v} |V c3,c4

C

  • Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech

Vote-Independence

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Definition (Process Pch [?]) Let P be a process and ch be a channel. We define Pch as follows: 0ch ˆ = 0, (P|Q)ch ˆ = Pch|Qch, (νn.P)ch ˆ = νn.out(ch, n).Pch when n is a name of base type, (νn.P)ch ˆ = νn.Pch otherwise, (in(u, x).P)ch ˆ = in(u, x).out(ch, x).Pch when x is a variable

  • f base type,

(in(u, x).P)ch ˆ = in(u, x).Pch otherwise, (out(u, M).P)ch ˆ = out(u, M).Pch, (!P)ch ˆ = !Pch, (if M = N then P else Q)ch ˆ = if M = N then Pch else Qch.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Definition (Process Pc1,c2 [?]) Let P be a process, c1, c2 channels. We define Pc1,c2 as follows: 0c1,c2 ˆ = 0, (P|Q)c1,c2 ˆ = Pc1,c2|Qc1,c2, (νn.P)c1,c2 ˆ = νn.out(c1, n).Pc1,c2 if n is a name of base type, (νn.P)c1,c2 ˆ = νn.Pc1,c2 otherwise, (in(u, x).P)c1,c2 ˆ = in(u, x).out(c1, x).Pc1,c2 if x is a variable

  • f base type & x is a fresh variable,

(in(u, x).P)c1,c2 ˆ = in(u, x).Pc1,c2 otherwise, (out(u, M).P)c1,c2 ˆ = in(c2, x).out(u, x).Pc1,c2, (!P)c1,c2 ˆ = !Pc1,c2, (if M = N then P else Q)c1,c2 ˆ = in(c2, x).if x = true then Pc1,c2 else Qc1,c2 where x is a fresh variable and true is a constant.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Definition (Process A\out(ch,·) [?]) Let A be an extended process. We define the process A\out(ch,·) as νch.(A|!in(ch, x)).

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Definition (Equivalence in a Frame) Two terms M and N are equal in the frame φ, written (M = N)φ, if and only if φ ≡ ν ˜ n.σ, Mσ = Nσ, and {˜ n} ∩ (fn(M) ∪ fn(N)) = ∅ for some names ˜ n and some substitution σ. Definition (Static Equivalence (≈s)) Two closed frames φ and ψ are statically equivalent, written φ ≈s ψ, when dom(φ) =dom(ψ) and when for all terms M and N (M = N)φ if and only if (M = N)ψ. Two extended processes A and B are statically equivalent (A ≈s B) if their frames are statically equivalent.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence

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Definition (Labelled Bisimilarity (≈l)) Labelled bisimilarity is the largest symmetric relation R on closed extended processes, such that A R B implies

1 A ≈s B, 2 if A → A′, then B → B′ and A′ R B′ for some B′, 3 if A α

− → A′ and fv(α) ⊆ dom(A) and bn(α) ∩ fn(B) = ∅, then B →∗ α − →→∗ B′ and A′ R B′ for some B′.

Jannik Dreier, Pascal Lafourcade, Yassine Lakhnech Vote-Independence