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Universal forgery on a group signature scheme using self-certified public keys

Universal forgery on a group signature scheme using self-certified public keys. Author : Guilin Wang Source : Information Processing Letters Vol. 89 , 2004 , pp. 227-231 Speaker : Pay-Chai Chang ( 張培才 ). Outline. Introduction Tseng-Jan scheme review

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Universal forgery on a group signature scheme using self-certified public keys

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  1. Universal forgery on a group signature scheme using self-certified public keys Author : Guilin Wang Source : Information Processing Letters Vol. 89 , 2004 , pp. 227-231 Speaker : Pay-Chai Chang (張培才)

  2. Outline • Introduction • Tseng-Jan scheme review • Ateniese, Joye and Tsudik attack • The Attack • Conclusions

  3. Introduction (1/1) • Group signatures • A secure group signature scheme must satisfy the following properties :(1) Unforgeability(2) Anonymity(3) Unlinkability(4) Exculpability(5) Traceability(6) Coalition-resistance

  4. Tseng-Jan scheme review(1/7) • The scheme involves four parties : • TA (a trusted authority) • GM (a group manager) • Ui (group members) • Verifiers

  5. Tseng-Jan scheme review(2/7) TA(1) n:= p qwith p:=2 +1 and q:=2 +1 where p , q , , are all primes. (2) Selects an elementof orderv:=and satisfying ed = 1 mod v (3) Chooses a publicly known hash function and publishes public key ( n , e , g , )secret key( p , q , d )

  6. Tseng-Jan scheme review(3/7) GM with identity information GDwants to establish a group (1) chooses a secret key x(2) computes z:= gx mod n(3) sends z to the TA Then TA (1)evaluatesGID := f (GD)(2)calculatesy : = zGID-1 mod n , sG = z -d mod n(3)sends y and sGto GM

  7. Tseng-Jan scheme review(4/7) GM chooses a publicly known hash function h(·) and publishespublic key ( y , h(·))secret key( x , sG) GM checks the validity of his key pair bysG e y -GID mod n A User Ui, with identity information Di, wants to join the group : (1) selects his secret keysi

  8. Tseng-Jan scheme review(5/7) (2) computeszi = gsi mod nand sendszito the TA(3) TA sends backpi := (zi) IDi-1·d mod n whereIDi : = f (Di )(4) Ui checks whetherpiIDi e zi mod n. Ifpiis correct, User Ui sendspi to GM (5) GM returns xi to Ui ,xi : = piIDi ·x • sG mod n(6)Uichecks whether xie yGID • (si-1) mod nholds.If the answer is yes, the Ui stores his membership certificate (si, xi)

  9. Tseng-Jan scheme review(6/7) • User Ui signs a message m with his certificate( si , xi ) • Randomly selects three numbers r1 , r2 , r3 • computes his signature (A , B , C , D , E) A : = r1si B : = r2-e A mod n C : = y GID • A• r3mod n D : = si • h (m || A || B || C ) + r3C E : = xi • r2 h(m || A || B || C || D ) mod n • To verify the validity of signature (A, B, C, D, E) on message m, a verifier checks whether yGID • A• D(EeA Bh(m || A || B || C || D ) yGID • A) h (m || A || B || C)•Cc mod n

  10. (4) In case of disputes, the group manager’s checking: • (xi) eA B-h(m || A || B || C || D )EeA mod n • Verify the correctness • (1) xi = piIDi • x • sG = (zi ) dx • sG = (gxd ) si • sG = sG -si+1 mod n • (2) xi = sG -si+1 = ( yGID ) d(si – 1 )mod n • (3) • ( EeA Bh yGID • A ) h • C c = ( y GID • A (si – 1 )• y GID • A ) h • y GID • A • r3Cmod n= y GID • A (sih+ r3C)mod n = y GID • A • D mod n Tseng-Jan scheme review(7/7)

  11. Assume that two colluding group members U1 and U2 • have certificates (s1, x1) and (s2, x2) , respectively. • Let c: = gcd (s1-1, s2-1) (the case of c=1) • By using extended Euclidean algorithm, they can find , Z such that c = (s1-1) + (s2-1) • Fromxi = piIDi • x • sG = (zi ) dx • sG = (gxd ) si • sG = sG -si+1 mod n , they can find : sG c = Ateniese, Joye and Tsudik attack (1/2)

  12. Ateniese, Joye and Tsudik attack (2/2) (3) Choose a random number r, then define respectively : : = cr + 1 and : = (sG c) -rmod n ( , ) is a valid but illegal membership certificate = (sG c) -r= sG ( -cr-1 )+1 = sG -s+1 mod n

  13. The attack (1/3) • yGID • A• D(EeA Bh(m || A || B || C || D ) yGID • A) h (m || A || B || C)•Cc mod n • Choose four random numbers a1, a2, a3, A, then define:B : = ya1 mod n C : = ya2 mod n E : = ya3 mod n • From verification equation, we get the condition for D :GID ·A ·D = [a3eA + a1 ·h(m||A||B||C ||D)] h(m||A||B||C ) + GID ·A · h ( m||A||B||C ) + a2C mod v • Let a3eA + a1 ·h(m||A||B||C ||D) = 0GID ·A ·D = GID ·A · h(m||A||B||C) + a2C

  14. We choose two random numbers a1, a2 and re-define a1, a2a1 : = a1eA a2 = a2 ·GID·Athen • D = h(m||A||B||C ) + a2C Za3 = -a1 ·h(m||A||B||C ||D) Z • Summarize of attack • Select three random numbers a1, a2 and A • Then define :B : = ya1eA mod nC : = ya2·GID ·A mod nD : = h(m||A||B||C ) + a2C ZE : = y -a1 · h(m||A||B||C ) mod n The attack (2/3)

  15. (3) Output (A, B, C, D, E) as group signature for message m Prove that the forgery is successful. ( EeA Bh yGID • A ) h • C c = y -a1heAh • y a1eA hh • y GID • Ah • y a2 • GID • AC mod n = y GID • A ( h+ a2 C ) mod n = y GID • A • D mod n The attack (3/3)

  16. Tseng-Jan group signature scheme is insecure • Anybody can forge a valid group signature on any message such that the group manager is unable to determine the identity of the signer • Universally forgeable • ~ Thanks all ~ Conclusions (1/1)

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