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Robust Sender Anonymity Tamara Rezk FMCrypto (work in progress) G.Barthe, A.Hevia, Z.Luo, T.Rezk, B.Warinschi April, 28 th – Campinas, Brazil

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Anonymity Protocols Hide the identity associated to a message The message may be public. Example:voting Different kind of anonymity properties

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Anonymity Properties Receiver anonymity Sender Unlinkability (SUL) Receiver Unlinkability (RUL) Sender-Receiver Unlinkability (UL) Sender Anonymity (SA) Strong Sender Anonymity (SA*) Receiver Anonymity (RA) Strong Receiver Anonymity (RA*) Sender-Receiver Anonymity (SRA) Unobservability (UO) Sender Unlinkability (SUL) Receiver Unlinkability (RUL) Sender-Receiver Unlinkability (UL) Sender Anonymity (SA) Strong Sender Anonymity (SA*) Receiver Anonymity (RA) Strong Receiver Anonymity (RA*) Sender-Receiver Anonymity (SRA) Unobservability (UO)

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Anonymity Properties Characterizations [Micciancio&Hevia06] c a b a 4 3 2 1 8 7 6 5 c a b a 4 3 2 1 8 7 6 5 43218765 d d m ij = sets of messages from party i to party j M = 7 (Thanks Alejandro for this slide)

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Capturing information leaks By restricting the matrix pair M 0,M 1 – Let f(M) be the information leaked – Requirement: f(M 0 ) = f(M 1 ) M0M0 c d d c = multiset for each row i M1M1 Example of leaked information: (Thanks Alejandro for this slide)

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The anonymity property for protocol P Hypothesis: f( M0 ) = f( M1 ) CA:=b := {0,1}; if (b = 0) then {m := M0} else {m := M1}; S P( m ) g A( S,f(m) ) | Pr[CA; g = b] - ½ | is negligible on the security parameter

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Motivation Anonymity in the case of active adversaries Case study: DC-Nets

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Motivation Anonymity in the case of active adversaries Case study: DC-Nets Robustness was not what we expected it to be Work: definition of robustness

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Robust anonymous protocol 1)A protocol that is anonymous (it does not leak the identity of the participants)

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Robust anonymous protocol 1)A protocol that is anonymous even if some of the participants are corrupt

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Robust anonymous protocol 1)A protocol that is anonymous even if some of the participants are corrupt 2)Honest messages can be delivered even if dishonest participants do not follow the protocol

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Robust anonymous protocol 1)Anonymity property for active adversaries 2)Robustness property

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The anonymity property for protocol P for active adversaries Hypothesis: f(M0) = f(M1) CRA:=b := {0,1}; if (b = 0) then {m := M0} else {m := M1}; g A[P( m )] ( f(m) ) | Pr[CRA; g = b] - ½ | is negligible on the security parameter

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Dinning Cryptographers: all started in a restaurant …

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Dinning Cryptographers Protocol (DC-nets) Bitwise XOR [Chaum88] – Not robust Bilinear Maps [GolleJuels04] – Robust What does exactly the word robust assure?

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The robust DC-nets protocol 1/4 inizialization In this phase: a non-degenerate pairing e : G1 x G1 G2 generators g, h of a cyclic group G1 a hash function H: {0,1}* G1 a private key xi and public key yi = g^xi (secret xi is (t,n)- shared ) a common reference string

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The robust DC-nets protocol 2/4 inizialization In this phase: each participant computes a vector that contains a padding and a unique message that cannot be distinguished from the padding. transmission

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In this phase: each participant computes a vector that contains a padding and a unique message that cannot be distinguished from the padding. transmission n i 2 1 1/3

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In this phase: each participant computes a vector that contains a padding and a unique message that cannot be distinguished from the padding. transmission n i 2 1 2/3 e(H(s||2), yj)^xi*c j i

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In this phase: each participant computes a vector that contains a padding and a unique message that cannot be distinguished from the padding. transmission n i 2 1 3/3 e(H(s||2), yj)^xi*c j i Padding participant i. Coefficient c is 1 if i<j or -1 otherwise.

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In this phase: each participant computes a vector that contains a padding and a unique message that cannot be distinguished from the padding. transmission n i 2 1 3/3 e(H(s||2), yj)^xi*c j i * m Message m transmission

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If each participant transmits exactly one message without collisions then multiplication of vectors yields the messages. transmission n 2 1 n 2 1 n 2 1 ** … Vector Party 1 Vector Party n = n 2 1 m1 m2 … mn

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Example for 2 paticipants: n=2 1/9 transmission

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Example for 2 paticipants: n=2 2/9 transmission Vector Party 1 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2

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Example for 2 paticipants: n=2 3/9 transmission Vector Party 1Vector Party 2 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2

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Example for 2 paticipants: n=2 4/9 transmission * Vector Party 1Vector Party 2 = 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1 m2 transmission result

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Example for 2 paticipants: n=2 5/9 e(H(s||1), y2)^x1 * e(H(s||1), y1)^-x2 * m1 transmission * Vector Party 1Vector Party 2 = 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1 m2 transmission result

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Example for 2 paticipants: n=2 6/9 e(H(s||1), y2)^x1 * e(H(s||1), y1)^-x2 * m1 = {public key inlining} e(H(s||1), x2g)^x1 * e(H(s||1), x1g)^-x2 * m1 transmission * Vector Party 1Vector Party 2 = 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1 m2 transmission result

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Example for 2 paticipants: n=2 7/9 e(H(s||1), y2)^x1 * e(H(s||1), y1)^-x2 * m1 = {public key inlining} e(H(s||1), x2g)^x1 * e(H(s||1), x1g)^-x2 * m1 = {bilinearity} e(H(s||1), x1x2g) * e(H(s||1), x2x1g)^-1 * m1 transmission * Vector Party 1Vector Party 2 = 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1 m2 transmission result

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Example for 2 paticipants: n=2 8/9 e(H(s||1), y2)^x1 * e(H(s||1), y1)^-x2 * m1 = {public key inlining} e(H(s||1), x2g)^x1 * e(H(s||1), x1g)^-x2 * m1 = {bilinearity} e(H(s||1), x1x2g) * e(H(s||1), x2x1g)^-1 * m1 = {conmutativity} e(H(s||1), x1x2g) * e(H(s||1), x1x2g)^-1 * m1 transmission * Vector Party 1Vector Party 2 = 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1 m2 transmission result

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Example for 2 paticipants: n=2 9/9 e(H(s||1), y2)^x1 * e(H(s||1), y1)^-x2 * m1 = {public key inlining} e(H(s||1), x2g)^x1 * e(H(s||1), x1g)^-x2 * m1 = {bilinearity} e(H(s||1), x1x2g) * e(H(s||1), x2x1g)^-1 * m1 = {conmutativity} e(H(s||1), x1x2g) * e(H(s||1), x1x2g)^-1 * m1 = {inverse *} m1 transmission * Vector Party 1Vector Party 2 = 2 1 e(H(s||1), y2)^x1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1 m2 transmission result

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If there is a collision, or the padding is incorrect, or there is more than one message in the vector, recuperation of messages fail! transmission n 2 1 n 2 1 n 2 1 ** … Vector Party 1Vector Party n = n 2 1 m1 m2 … mn

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Vectors are transmitted with a proof of knowledge (zkpk) transmission For all positions in the vector there is a valid padding, except for at most one position.

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The robust DC-nets protocol 3/4 inizialization In this phase: each participant computes a vector that contains a padding and a unique message that cannot be distinguished from the padding. transmission reconstruction

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In this phase: if a proof of knowledge does not verify then the vector of the dishonest participant is reconstructed using trheshold cryptography reconstruction After this phase, we are left with a set of valid vectors, that is : For all positions in the vector there is a valid padding, except for at most one position.

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The robust DC-nets protocol 4/4 inizialization transmission reconstruction recuperation

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In this phase: All vectors are correct (honest participants or recovered vectors). Messages are recuperated by multiplication. recuperation n 2 1 n 2 1 n 2 1 ** … Vector Party 1 Vector Party n = n 2 1 m1 m2 … mn

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What does exactly the word robust assure? If the vector is correct, then there is a unique message in the vector An adversary may violate the slot reservation protocol to intentionally produce a collision For each collision, one honest message is not delivered

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ROBUSTNESS PROPERTY We propose to state this formally by definning a:

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Sender robustness, t-n SR:= M,N A0 m := M++N; S P[A]( m ) if (#(MПS) < 2t-n) then b:=1 else b:=0 |Pr[SR; b=1] is negligible on the security parameter

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Sender Robustness Violation 1 Example for 2 paticipants: n=2 * Vector Party 1Vector Party 2 = 2 1 1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 ???? m2 transmission result

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Sender Robustness Violation 2 Example for 2 paticipants: n=2 * Vector Party 1Vector Party 2 = 2 1 e(H(s||2), y2)^x1*m2 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 ???? m2 transmission result

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Sender Robustness Example for 2 paticipants: n=2 * Vector Party 1Vector Party 2 = 2 1 e(H(s||2), y2)^x1*m2 e(H(s||2), y2)^x1 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1*m2 m2 transmission result This is considered secure!

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A stronger robustness property Confusion resistant t-n CR:= M,N A0 m := M++N; S P[A( m )] if honest received < honest- dishonest then b:=1 else b:=0 |Pr[CR; b=1] is negligible on the security parameter

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A stronger robustness property Confusion resistant t-n CR:= M,N A0 m := M++N; S P[A( m )] if honest not received+dishonest received > dishonest. then b:=1 else b:=0 |Pr[CR; b=1] is negligible on the security parameter

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A stronger robustness property Confusion resistant t-n CR:= M,N A0 m := M++N; S P[A( m )] if (#(S\M) + #(M\S) > n-t) then b:=1 else b:=0 |Pr[CR; b=1] is negligible on the security parameter

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Confussion Resistant Violation Example for 2 paticipants: n=2 * Vector Party 1Vector Party 2 = 2 1 e(H(s||2), y2)^x1*m2 e(H(s||2), y2)^x1 2 1 e(H(s||1), y1)^-x2 *m1 e(H(s||2), y1)^-x2 2 1 m1*m2 m2 transmission result

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Theorems and Remarks Theo: DC-Nets is sender anonymous Theo: DC-Nets is sender robust Remark: DC-Nets is not confussion resistant

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Theorems and Remarks Theo: DC-Nets is sender anonymous Theo: DC-Nets is sender robust Remark: DC-Nets is not confussion resistant Solution? : messages should be sealed in such a way that multiplication of two seals produces another seal only with negligible probability

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Conclusions We have a proposed 2 properties to formally specify robustness of sender anonymous protocols We have detected GJ protocol satisfies only a weak form of robustness, and proposed a stronger version of the protocol Open questions: how to implement the stronger GJ?, how all these definitions extend to other forms of anonymity? generic conversion to stronger robustness?

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