How to prove security of communication protocols?

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1 1/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees How to prove security of communication protocols? Véronique Cortier, LORIA - CNRS, Nancy Colloquium Morgenstern, June 20th, 2012 Joint work with Hubert Comon-Lundh, Stéphanie Delaune, Steve Kremer, Ben Smyth and Bogdan Warinschi.

2 2/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees Context : cryptographic protocols Cryptographic protocols are widely used in everyday life. They aim at securing communications over public or insecure networks.

3 3/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees Security goals Cryptographic protocols aim at preserving confidentiality of data (e.g. pin code, medical files,...) ensuring authenticity (are you really talking to your bank?) ensuring anonymous communications (for e-voting protocols,...) protecting against repudiation (I never sent this message!)

4 4/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees Difficulty : there are potential powerful attackers! Presence of an attacker may participate to the protocol. may forge and send messages, may read every message sent on the net, may intercept messages,

5 5/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees Attacking Single Sign On Protocol Single Sign On Protocols enables to log in once for several services used e.g. in Google App

6 Attacking Single Sign On Protocol Single Sign On Protocols enables to log in once for several services used e.g. in Google App 5/37 A flaw discovered in 2010, now fixed (Avantssar project) Step 1 An attacker offers an interesting or funny (but malicious) new Google App Step 2 Some clients register to this malicious Application Step 3 The attacker can now access all the other applications of the client, including e.g. Gmail or Google Calendar.

7 /37 Designing protocols is error prone Software testing leaves flaws : Flaw in the authentication protocol used in Google Apps Attack on pay-per-view devices Man-in-the-middle attack These flaws rely on the design of the protocols Not on a bad implementation (bugs) Not on weaknesses of the primitives (e.g. encryption, signatures) Not on generic hacking techniques (e.g. worms, code injection)

8 /37 How to analyse security protocols?? = confidentiality authenticity non-repudiation Methodology 1 Proposing accurate models symbolic models cryptographic/computational models 2 Proving security decision procedures transfer results Running example : electronic voting

9 8/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees Example : Electronic voting Elections are a security-sensitive process which is the cornerstone of modern democracy. Electronic voting promises Convenient, efficient and secure facility for recording and tallying votes for a variety of types of elections : from small committees or on-line communities through to full-scale national elections Already used e.g. in Estonia, Norway, USA, France (from abroad).

10 9/37 Introduction on security protocols Modeling Verification Towards cryptographic guarantees Two main families for e-voting Voting machines Voters have to attend a voting station External authentication system (e.g. ID card) Internet voting Voters vote from home from their own computers Systems in use : Civitas (A. Myers et al), Helios,...

11 10/37 Running example : Helios http ://heliosvoting.org/ Developed by B. Adida et al, already in use : Election at Louvain University Princeton Election of the IACR board (major association in Cryptography)

12 11/37 Helios : a voting protocol for low-coercion environment The security of Helios relies on the assumption that the voter s computer can be trusted. Not suitable for political elections A corrupted machine may : leak the choice of the voter vote for a different candidate e.g. attack by Laurent Grégoire on the Élection législatives pour les francais de l étranger, using code injection.

13 Helios : a voting protocol for low-coercion environment The security of Helios relies on the assumption that the voter s computer can be trusted. 1/37 Not suitable for political elections A corrupted machine may : leak the choice of the voter vote for a different candidate e.g. attack by Laurent Grégoire on the Élection législatives pour les francais de l étranger, using code injection. Suitable for medium issue elections : professional elections scientific councils, students representatives, etc. union representatives Should be compared with previous voting systems in use : e.g. attack (ballot stuffing) on the 2011 paper ballots (with barcodes) CNRS election

14 Behavior of Helios (simplified) Phase 1 : voting 1 Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 Phase 2 : Tallying using homomorphic encryption (El Gamal) n n {v i } pk(s) = { v i } pk(s) i=1 i=1 Only the final result needs to be decrypted. 12/37 pk(s) : public key, the private key being shared among trustees.

15 Behavior of Helios (simplified) Phase 1 : voting 1 {v D } pk(s) Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 Phase 2 : Tallying using homomorphic encryption (El Gamal) n n {v i } pk(s) = { v i } pk(s) i=1 i=1 Only the final result needs to be decrypted. 12/37 pk(s) : public key, the private key being shared among trustees.

16 Behavior of Helios (simplified) Phase 1 : voting 1 Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 David {v D } pk(s) v D = 0 or 1 Phase 2 : Tallying using homomorphic encryption (El Gamal) n n {v i } pk(s) = { v i } pk(s) i=1 i=1 Only the final result needs to be decrypted. 12/37 pk(s) : public key, the private key being shared among trustees.

17 Behavior of Helios (simplified) Phase 1 : voting 1 Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 David {v D } pk(s) v D = 0 or Phase 2 : Tallying using homomorphic encryption (El Gamal) n n {v i } pk(s) = { v i } pk(s) based on g a g b = g a+b i=1 i=1 Only the final result needs to be decrypted! 12/37 pk(s) : public key, the private key being shared among trustees.

18 3/37 This is oversimplified! {v D } pk(s) Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 David {v D } pk(s) Result : {v A +v B +v C +v D + } pk(s)

19 13/37 This is oversimplified! {v D } pk(s) Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 David {v D } pk(s) v D = Result : {v A +v B +v C } pk(s) A malicious voter can cheat!

20 13/37 This is oversimplified! {v D } pk(s) Bulletin Board Alice {v A } pk(s) v A = 0 or 1 Bob {v B } pk(s) v B = 0 or 1 Chris {v C } pk(s) v C = 0 or 1 David {v D } pk(s) v D = Result : {v A +v B +v C +v D + } pk(s) A malicious voter can cheat! In Helios : use of (Signature of) Proof of Knowledge {v D } pk(s),spk{v D = 0 or 1}

21 14/37 How to analyse security protocols? For example, how to prove that Helios is secure?

22 14/37 How to analyse security protocols? For example, how to prove that Helios is secure? Task 1 : Modeling 1 Modeling messages 2 Modeling the behavior of the protocol 3 Modeling security

23 15/37 Modeling messages Idea 1 : keeping only the structure of the messages Messages are abstracted by terms. Example : The message { A,N a } K is represented by : A < > {} N a K

24 15/37 Modeling messages Idea 1 : keeping only the structure of the messages Messages are abstracted by terms. Example : The message { A,N a } K is represented by : A < > {} N a K Idea 2 : Equations for reflecting the properties of the primitives Decryption dec({x} y,y) = x Homomorphic encryption {x 1 } y {x 2 } y = {x 1 +x 2 } y

25 6/37 Modeling protocols Processes of the applied pi-calculus [Abadi & Fournet] Voter id voting v Voter(id,v) = c id ({v} pk(s),spk(v,{v} pk(s) ))

26 16/37 Modeling protocols Processes of the applied pi-calculus [Abadi & Fournet] Voter id voting v Voter(id,v) = c id ({v} pk(s),spk(v,{v} pk(s) )) Bulletin board for n voters BulletinBoard = c id1 (x 1 ). if Valid(x 1 ) then out(x 1 ). c idn (x n ). if Valid(x n ) then out(x n ). c tally (π 1 (x 1 ) π 1 (x n ))

27 16/37 Modeling protocols Processes of the applied pi-calculus [Abadi & Fournet] Voter id voting v Voter(id,v) = c id ({v} pk(s),spk(v,{v} pk(s) )) Bulletin board for n voters BulletinBoard = c id1 (x 1 ). if Valid(x 1 ) then out(x 1 ). c idn (x n ). if Valid(x n ) then out(x n ). c tally (π 1 (x 1 ) π 1 (x n )) Tallying phase Tally = c tally (y).out(dec(y,sk(s)))

28 17/37 Modeling attackers We assume that the network can be controlled by attackers may participate to the protocol. may forge and send messages, may read every message sent on the net, may intercept messages,

29 17/37 Modeling attackers We assume that the network can be controlled by attackers may participate to the protocol. may forge and send messages, may read every message sent on the net, may intercept messages, Attackers in applied pi-calculus A protocol P satisfies some property φ if for all process A A P = φ

30 18/37 Examples of security properties Secrecy of s If A P Q then Q out(s) Q For any attacker A and reachable process Q, the secret s remains unknown. Authentication/Agreement If A P Q then Q = Received(B,N a ) Sent(A,N a ) For any attacker A, if B receives a nonce believing it is from A then A must have sent it. Most properties can be expressed as accessibility properties

31 19/37 What is a secure voting protocol?

32 20/37 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote.

33 20/37 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote. But everyone knows 0 and 1!

34 20/37 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote. Idea 2 : An attacker should not attach my vote to my identity.

35 0/37 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote. Idea 2 : An attacker should not attach my vote to my identity. But everyone can form Alice,0 and Alice,1!

36 20/37 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote. Idea 2 : An attacker should not attach my vote to my identity. Idea 3 : An attacker cannot see the difference when I vote 0 or 1. Voter 1(0) Voter 2(v 2) Voter n(v n) Voter 1(1) Voter 2(v 2) Voter n(v n)

37 20/37 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote. Idea 2 : An attacker should not attach my vote to my identity. Idea 3 : An attacker cannot see the difference when I vote 0 or 1. Voter 1(0) Voter 2(v 2) Voter n(v n) Voter 1(1) Voter 2(v 2) Voter n(v n) The attacker always sees the difference since the tally differs. Unanimity does break privacy.

38 Let s have a closer look to privacy How to state formally : No one should know my vote (0 or 1)? Idea 1 : An attacker should not learn the value of my vote. Idea 2 : An attacker should not attach my vote to my identity. Idea 3 : An attacker cannot see the difference when I vote 0 or 1. Voter 1(0) Voter 2(v 2) Voter n(v n) Voter 1(1) Voter 2(v 2) Voter n(v n) Idea 4 : An attacker cannot see when votes are swapped. Voter 1 (0) Voter 2 (1) Voter 1 (1) Voter 2 (0) 20/37 S. Kremer & M. Ryan

39 21/37 How to analyse security protocols?? = confidentiality authenticity non-repudiation Methodology 1 Proposing accurate models symbolic models cryptographic/computational models 2 Proving security decision procedures transfer results

40 22/37 How to analyse security protocols? First, we consider accessibility properties, e.g. confidentiality, authentication, etc. Automatic verification Unfortunately, security (e.g. confidentiality) is undecidable. No generic algorithm can work. Identification of decidable fragments analysis of a finite number of sessions restriction on the class of protocols Semi-decision procedure : ProVerif

41 23/37 How does ProVerif work? Developed by Bruno Blanchet, ENS Paris, France. Implements a sound semi-decision procedure (that may not terminate). The applied pi-calculus is translated into first-order logic, more precisely into Horn clauses. Based on a resolution strategy well adapted to protocols.

42 24/37 Horn clauses for the intruder Horn clauses perfectly reflect the attacker symbolic manipulations on terms. x y I(x),I(y) I(< x,y >) pairing x y I(x),I(y) I({x} y ) encryption x y I({x} y ),I(y) I(x) decryption x y I(< x,y >) I(x) projection x y I(< x,y >) I(y) projection

43 25/37 Horn clauses for the protocol Protocol WMF : A S : {n a,b,k} ka S B : {n s,a,k} kb B A : {m ab } k Horn clauses : I({n a,b,k} ka ) I({x,b,y} ka ) I({n s (x,y),a,y} kb ) I({x,a,y} kb ) I({m ab } y )

44 25/37 Horn clauses for the protocol Protocol WMF : A S : {n a,b,k} ka S B : {n s,a,k} kb B A : {m ab } k Horn clauses : I({n a,b,k} ka ) I({x,b,y} ka ) I({n s (x,y),a,y} kb ) I({x,a,y} kb ) I({m ab } y ) Secrecy property is a reachability (accessibility) property I(m ab ) Checking security reduces to checking satisfiability There exists an attack iff the set of formulas corresponding to Intruder manipulations + protocol + property is NOT satisfiable.

45 26/37 How to decide satisfiability? Resolution techniques : Binary resolution D 1 D k B A 1 A n C A1 θ = Bθ (D 1 D k A 2 A n C)θ

46 6/37 How to decide satisfiability? Resolution techniques : Binary resolution D 1 D k B A 1 A n C A1 θ = Bθ (D 1 D k A 2 A n C)θ It does not terminate. Example : I(s) I(x),I(y) I( x,y )

47 6/37 How to decide satisfiability? Resolution techniques : Binary resolution D 1 D k B A 1 A n C A1 θ = Bθ (D 1 D k A 2 A n C)θ It does not terminate. Example : I(s) I(x),I(y) I( x,y ) I(y) I( s,y )

48 6/37 How to decide satisfiability? Resolution techniques : Binary resolution D 1 D k B A 1 A n C A1 θ = Bθ (D 1 D k A 2 A n C)θ It does not terminate. Example : I(s) I(x),I(y) I( x,y ) I(y) I( s,y ) I( s,s ) I( s, s,s ) I( s, s, s,s ) I( s, s, s, s,s ) I( s, s, s, s, s,s )

49 27/37 Efficient and sound resolution strategy Idea : Resolution is only applied on selected literals A 1,B that do not belong to a forbidden set S. Typically S = {I(x)}. Theorem Resolution based on selection, avoiding S, is complete w.r.t. satisfiability. If the fixed point does not contain the empty clause, then the corresponding protocol is secure. ProVerif may not terminate.

50 Efficient and sound resolution strategy Idea : Resolution is only applied on selected literals A 1,B that do not belong to a forbidden set S. Typically S = {I(x)}. Theorem Resolution based on selection, avoiding S, is complete w.r.t. satisfiability. 27/37 If the fixed point does not contain the empty clause, then the corresponding protocol is secure. ProVerif may not terminate. Performs very well in practice! Works on most of existing protocols in the literature Is also used on industrial protocols (e.g. certified protocol, JFK, Plutus filesystem) Can handle various cryptographic primitives (various encryption, signatures, blind signatures, hash, etc.)

51 28/37 Security of Helios Privacy Voter 1 (0) Voter 2 (1) Voter 1 (1) Voter 2 (0) Not an accessibility property ProVerif cannot be applied (yet).

52 28/37 Security of Helios Privacy Voter 1 (0) Voter 2 (1) Voter 1 (1) Voter 2 (0) Not an accessibility property ProVerif cannot be applied (yet). Helios is actually subject to replay attack, which breaks privacy! The fixed version (weeding duplicated ballots) provably ensures privacy

53 Security of Helios Privacy Voter 1 (0) Voter 2 (1) Voter 1 (1) Voter 2 (0) Not an accessibility property ProVerif cannot be applied (yet). Helios is actually subject to replay attack, which breaks privacy! The fixed version (weeding duplicated ballots) provably ensures privacy 28/37 Verifiability Individual verifiability : voter can check that her own ballot is included in the election s bulletin board. Universal verifiability : anyone can check that the election outcome corresponds to the ballots published on the bulletin board. Helios provably satisfies both verifiability properties.

54 29/37 Limitations of this approach? Are you ready to use any protocol verified with this technique?

55 29/37 Limitations of this approach? Are you ready to use any protocol verified with this technique? Side channel attacks Representing messages by a term algebra abstracts away many mathematical properties.

56 30/37 Setting for cryptographic/computational models Messages : (Bitstrings) Protocol : Message exchange program Use cryptographic algorithms

57 30/37 Setting for cryptographic/computational models Messages : (Bitstrings) Protocol : Message exchange program Use cryptographic algorithms Adversary A : any probabilistic polynomial Turing machine, i.e. any probabilistic polynomial program. polynomial : captures what is feasible probabilistic : the adversary may try to guess some information

58 31/37 Computational secrecy One-Wayness : The probability for an adversary A to compute the secret s against a protocol P is negligible (smaller than any inverse of polynomial).

59 1/37 Computational secrecy One-Wayness : The probability for an adversary A to compute the secret s against a protocol P is negligible (smaller than any inverse of polynomial). Not strong enough! The adversary may be able to compute half of the secret message. There is no guarantee in case that some partial information on the secret is known.

60 31/37 Computational secrecy One-Wayness : The probability for an adversary A to compute the secret s against a protocol P is negligible (smaller than any inverse of polynomial). Not strong enough! The adversary may be able to compute half of the secret message. There is no guarantee in case that some partial information on the secret is known. Indistinguishability : the adversary should not learn a bit of the secret. Pr[A P(n 0 ) 1] Pr[A P(n 1 ) 1] is negligible

61 32/37 Formal and Cryptographic approaches Formal approach Cryptographic approach Messages terms bitstrings Encryption idealized algorithm Adversary idealized any polynomial algorithm Guarantees unclear strong Proof automatic by hand, tedious and error-prone Link between the two approaches?

62 33/37 Proving cryptographic security through symbolic models Symbolic models Computational models < > {} K A N a

63 33/37 Proving cryptographic security through symbolic models Symbolic models Computational models < > {} K A N a Idea : soundness result Show that security in symbolic models implies security in computational ones. [Abadi Rogaway 00]

64 34/37 Soundness of equivalences in the applied pi-calculus Result : Assuming a strong encryption scheme (IND-CCA2 hypothesis) P 1 P 2 [[P 1 ]] [[P 2 ]] Symbolic equivalence of processes P 1 and P 2 Indistinguishability of the implementation of P 1 and P 2

65 4/37 Soundness of equivalences in the applied pi-calculus Result : Assuming a strong encryption scheme (IND-CCA2 hypothesis) P 1 P 2 [[P 1 ]] [[P 2 ]] Symbolic equivalence of processes P 1 and P 2 Indistinguishability of the implementation of P 1 and P 2 Key technique Any attack trace from the concrete adversary is an attack against the symbolic protocol, or the adversary breaks encryption. Consequence : Security in symbolic models directly implies security in cryptographic models, against arbitrary attackers.

66 35/37 Benefit : modularity Cryptographic security guarantees can be obtained at the symbolic level Ideal protocol Formal approach: verification of idealized protocols Implemented protocol signature algorithm encryption algorithm Cryptographers: verification of the cryptographic primitives

67 6/37 Conclusion Formal methods form a powerful approach for analyzing security protocols Use of existing techniques : term algebra, equational theories, clauses and resolution techniques, tree automata, etc. Many decision procedures Several successful automatic tools e.g. ProVerif, Avispa/Avantssar, Scyther, NRL Protocol Analyzer Detect attacks (e.g. flaw in Gmail) Prove security of standard protocols (e.g. IKE, JFK, Certified , Helios,...) Provides cryptographic guarantees under classical assumptions on the implementation of the primitives

68 37/37 The end Special thanks to : Hubert Comon-Lundh Ben Smyth Stéphanie Delaune Bogdan Warinschi Steve Kremer

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