Short Traceable Signatures Based on

Bilinear Pairings

Seung Geol Choi

Columbia Universityjoint work with

Moti YungColumbia University

Kunsoo ParkSeoul National

University

Contents

Overview of Traceable Signatures Motivation Preliminaries ZK for SDH Representation Construction Security Conclusion

Overview of Traceable Signatures

Traceable Signatures

Can be regarded as an extension of group signatures. Provides all the operations of group

signatures- setup, join, sign, verify, open

Provides stronger revocation of anonymity- tracing (reveal, trace)

Provides claiming (claim, claim_verify)

Why do we need traceable sig.? Consider following setting:

Anonymous Users Typical Abstract Large System:

Many users Many remote verification points. Users issue signatures that get aggregated and

verified in remote points.

Scenario #1:

Authority

Verification Points

Tracing Request:“open signature”

Scenario #2

Authority

Verification Points

TracingRequest:

“USER X” needs to be traced

Using the opening mechanism from scenario #1:all signatures must be aggregated and the Authoritywill have to Open all to discover the ones signed by user X

Shortcomings of group sig. Signatures from remote verification points must

be aggregated. Load Balancing Concerns Authority must open all signatures thus

severely (and unnecessarily) violating the privacy of many users. Privacy Concerns

Authority is typically a distributed entity so that opening requires the collaboration of many agents. Efficiency Concerns

Outcome: Scenario #1 is insufficient for dealing with the above tracing request.

Scenario #3

User wantsto claim a signature as his

Features of Traceable Sig.(1) Anonymity

A user (group member) signs on behalf of the group.

Verification is done using the group’s public-key.

Claiming A user can claim his own signature.

Features of Traceable Sig.(1) Revocation of Anonymity

The group manager can open a problematic signature and find out who signed it.

The tracing agents can trace all the signatures of a suspicious user.

Motivation

Motivation

Previous constructions were quite long. [KTY04] : 1206 bytes [NS04]: 917 bytes

Adapt the short group signature [BBS04] to traceable signature. Ours: 362 bytes

1.5 ~ 3 times the length of the RSA sig.

Basic Tools

Three main basic tools

Bilinear Pairings One more SDH (Strong Diffie-Hellman)

representation problem Linear Encryption Scheme

Basic Tools – Bilinear Pairings G1, G2, GT : cyclic groups of prime order p

P1 , P2 : generator of G1 , G2

ψ: G2 G1 (isomorphism mapping)

Def: Bilinear pairing e : G1 x G2 GT is: Bilinear:

e(aP1, bP2) = e(P1, P2)ab for all a, b Z Non-degenerate:

e(P1, P2) ≠ 1

Efficiently computable

Basic Tools – One More SDH Representation Problem (1)

SDH Representation Given P1, P2, Q, R where Q G1 , R = γP2 SDH Representation:

(A, x, t) s.t. A = (xP1 + Q)/ (t+γ) or equivalently e(A, tP2+R) = e(xP1 + Q,

P2)

One more SDH representation problem Given K SDH representations, output another

valid SDH representation

Basic Tools – One More SDH Representation Problem (2)

Under q-SDH Assumption, One more representation problem is hard.

q-SDH Assumption [BB04] The following q-SDH problem is hard:

P1, P2, γP2, γ2P2, …, γqP2 ? (A, x) s.t. (γ +x)A = P1 where A G1 , x Zp

Basic Tools – Linear Encryption [BBS04] (1) Keys:

Encryption Key: X, Y, Z G1

Decryption Key: ξ1, ξ2 s.t. ξ1 X = Z, ξ2 Y=Z

Encryption/Decryption E(M) = ( r1X, r2Y, M+(r1+r2)Z ) D(C1, C2, C3) = C3 – ξ1C1 – ξ2C2

Basic Tools – Linear Encryption [BBS04] (2) Semantic Security:

Under DLDH (Decisional Linear Diffie-Hellman) Assumption [BBS04], linear encryption is semantically secure.

DLDH Assumption The following problem is hard:

Given X, Y, Z, aX, bY, cZc = a + b? or c is randomly chosen?

ZK for SDH Representation

Basic Idea

Why do we need this? Come up with zk proof for the rep,

and use the proof as a sig (FS transform) Anonymity

The rep is a witness of a proof a signing key

Basic Setting

Proof: PK{(A,x,t): e(xP1 + Q, P2) = (A, tP2+R)} Instance: P1, Q, P2, R

Where P1 (gen. of G1), Q (random point) P2 (gen. of G2), R (= γP2)

Prover’s aux input (SDH rep./witness): (A, x, t) s.t. e(xP1 + Q, P2) = (A, tP2+R)

Other Public Parameters For linear enc. : X, Y, Z (gen. of G1) Etc. : W (gen. of G2)

ZK for SDH Representation (1) Prover constructs T1, … T5:

T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z (linear enc. of A)

T4 = r3W, T5 = e(P1, T4)x (DLP of x )

Sub-proof PK{(a1, a2, b1, b2, u, v):

T1 = a1X, T2 = a2Y, uT1 = b1X, uT2 = b2Y, T5 = e(P1, T4)v , e (T3, P2)u e(T3, R) = e(Z, P2) (b1+b2) e(Z, R) (a1+a2) e(P1, P2)v e(Q, P2) }

ZK for SDH Representation (2) Exists a Simulator (i.e. it is ZK)

T1, …, T5 : From semantic security of linear enc:

- Pick a random A’- T1 = r1X, T2 = r2Y, T3 = A’ + (r1+r2)Z

From DDH:- pick a random x’ - T4 = r3W, T5 = e(P1, T4)x’

Indistinguishable from the original transcript Sub-Proof:

Runs the simulator of Sub-Proof

ZK for SDH Representation (3) Exists an extractor (i.e. it is POK)

Sub-Proof: Simple 3-move honest verifier DLP ZK-POK exists an extractor for the Sub-Proof

Using the extractor of DLP proof, we can also extract an SDH Rep.

Specifically Let (a1, a2, b1, b2, u, v) be the extracted witness. b1 + b2 = u(a1 + a2)

ZK for SDH Representation (4)

e (T3, P2)u e(T3, R) = e(Z, P2) (b1+b2) e(Z, R) (a1+a2) e(P1, P2)v e(Q, P2)

e(T3, uP2+R) = e(Z, (b1+b2)P2+(a1+a2) R) • e(vP1+Q, P2)

e(T3, uP2+R) / e(Z, u(a1+a2)P2+(a1+a2) R) = e(vP1+Q, P2)

e(T3, uP2+R) / e((a1+a2)Z, uP2+ R) = e(vP1+Q, P2)

e(T3 - (a1+a2)Z, uP2+ R) = e(vP1+Q, P2)

If we Let A = T3 – (a1+a2)Z, e(A, uP2+ R) = e(vP1+Q, P2)

(A, u, v) is an SDH rep.

Construction

Procedures of Traceable sig. Setup Join/Iss Sign/Verify Open Reveal/Trace Claim/Claim_Verify

Construction - Setup

Generate public parameters for ZK for SDH Rep. P1, Q, P2, R, X, Y, Z, W

For SDH rep. : P1, Q, P2, R For linear enc. : X, Y, Z s.t. ξ1 X = Z, ξ2 Y=Z Etc. : W

The group manager’s private key:(γ, ξ1, ξ2) γ : for the generation of SDH rep (join proc.) ξ1, ξ2 : dec. key for linear enc. (opening)

Construction – Join/Iss

Interactive Protocol between a user (Join) and the group manager (Iss)

Ui (user i) GM : xiP1

GM Ui: (Ai, ti) s.t. e(Ai, tiP2+ R) = e(xiP1+Q, P2)

Note that GM can generate (Ai, ti) without knowing the value xi. Let Ci = xiP1

A = (Ci + Q)/ (t+γ)

Ui now has an SDH rep: (Ai, xi, ti) GM stores the joining record: (Ai, Ci, ti)

Construction – Sign/Verify (1) Big Picture of ZK Protocol for SDH Rep:

3 move honest verifier proof for DLP Instance: T1, …, T5

P (Prover) V (Verifier): B1, …, B6

V P : c P V : sa1, sa2, sb1, sb2, su, sv V : checks if sa1, sa2, sb1, sb2, su, sv are consistent.

Construction – Sign/Verify (2)

Details T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z

T4 = r3W, T5 = e(P1, T4)x

d1 = r1t , d2 = r2t B1 = br1X, B2 = br2X,

B3 = btT1 – bd1X, B4 = btT2 – bd2YB5 = e(P1, T4) bx B6 = e(T3, P2)bt e(Z, P2)-bd1-bd2 e(Z, R)-br1-br2 e(P1, P2)-bx

sr1 = br1 + cr1, sr2 = br2 + cr2,

sd1 = bd1 + cd1, sd2 = bd2 + cd2,

sx = bx + cx, st = bt + ct,

Construction – Sign/Verify (3) Apply the variant of Fiat-Shamir to the

protocol (Schnorr type sig.) Sign:

Replace B1, …, B6 of the verifier with hash function: c = H(m, T1, …, T5, B1, …, B6)

The signature will be:(T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx )

362 bytes: T5 = 1024 bits, all others 170 bits. Verification:

construct B’1, …, B’6 from the signature. check if H(m, T1, …, T5, B’1, …, B’6) =? c.

Construction – Open

Given a signature: (T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx )

The GM use his decryption key for linear enc. to recover A from T1, T2, T3. T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z Dec(T1, T2, T3) = T3 – ξ1T1 – ξ2T2 = A Look up the user j from the join records

{(Ai, Ci, ti)} such that Aj = A

Construction – Tracing a user (Reveal/Trace)

Reveal Given the identity j of a certain user Uj ,

returns an information to be used for tracing The GM returns Cj from his join record (Aj, Cj, tj).

Trace Given Cj (from Reveal) the tracing info of Uj,

and a sig. (T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx ), decides whether it’s Uj’s sig. or not. e(Cj, T4) =? T5 ( Note that T5 = e(P1, T4)x )

Construction - Claiming a Sig.(Claim/Claim_Verify) Claim:

Given a sig. (T1, … ,T5, c, sr1, sr2, sr1, sr2, st, sx )

The signer returns a NIZK proof.PK{ y: T5 = e(P1, T4)y}

Claim_Verify: Verify the proof.

Security

Security Model [KTY04]

There are three kind of attacks Misidentification: the adv. forges a valid

signature that is opened/traced to no one. Framing: the adv. forges a valid signature

that is opened/traced to an innocent user even if the adv. corrupts the GM.

Anonymity: the adv. distinguishes a sig. of user A from a sig. of user B.

The adv. is allowed to access oracles.

Oracles

Returns thePublic-key

Returns theGM’s privatekey

Executes a joindialog internally

Executes a Iss procedure.(Adv is playing the role of user. Oracle is playing the role of GM.)

Executes a Join procedure.(Adv is playing the role of GM. Oracle is playing the role of user.)

Given <i>, returns the tracing info. Ci.

Given <i, m>, returnsa signature on m by the i-th user

QY

Qs

Qp-join

Qa-join

Qb-join

Qsig Qreveal

Misidentification attack

Adv Oracles

Forges a sig. satisfying • it opens to none of the controlled group or• it traces to none of the controlled group.

Represents the system

collectively: good users and GM

Secure against Misidentification from the hardness of one-more SDH rep.

problem

QY, Qp-join, Qa-join, Qsig, Qreveal

Framing attack

Adv Oracles

Forges a sig. satisfying• it opens to an innocent user or• it traces to an innocent user.

Represents the system

collectively: good users and GM

Secure against Framing from the hardness of DLP.

QY, QS, Qb-join, Qsig

T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z T4 = r3W, T5 = e(P1, T4)x

Anonymity attackAdv Oracles

Selects two usersi0 i1 (by name) Pick b randomly from

{0,1}Generate a sig. σ of ib

σGuess b

i0, i1

QY, Qp-join, Qa-join, Qsig, Qreveal

• The adv is not allowed to call Qreveal(i0) or Qreveal(i1) before or after i0 and i1 are chosen.

Secure against Anonymity from semantic security of linear encryption

and the DDH

T1 = r1X, T2 = r2Y, T3 = A + (r1+r2)Z T4 = r3W, T5 = e(P1, T4)x

Security of Our scheme

Theorem : Under the q-SDH and DLDH assumption, our scheme is secure in the random oracle model.

Conclusion

Conclusion

Invented a New Technical Tool One more SDH rep. problem based on q-

SDH assumption Constructed a Short Scheme

Ours: 362 bytes 1.5 ~ 3 times the length of the RSA sig.

[KTY04] : 1206 bytes, [NS04]: 917 bytes Proved the security formally