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Key Substitution Attacks on Some Provably Secure Signature Schemes

Key Substitution Attacks on Some Provably Secure Signature Schemes. Author: Chik-How Tan Source: IEICE Trans. Fundamentals, Vol.E87-A, No.1 Jan. 2004 Speaker: Su Sheng-Yao. Outline. Introduction Two Provably Secure Signature Scheme Fischlin Signature Scheme

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Key Substitution Attacks on Some Provably Secure Signature Schemes

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  1. Key Substitution Attacks on Some Provably Secure Signature Schemes Author: Chik-How Tan Source: IEICE Trans. Fundamentals, Vol.E87-A, No.1 Jan. 2004 Speaker: Su Sheng-Yao

  2. Outline • Introduction • Two Provably Secure Signature Scheme • Fischlin Signature Scheme • Camenisch-Lysyanskaya Signature Scheme • Cryptoanalysis • Conclusion

  3. Introduction • Provable Security • Security could be proved under standard and well-believed complexity theoretic assumptions • Definition, Protocol, Proof • Provably Secure Signature Schemes • Key Substitution Attack • U’s public key and signature s on m • adversary A tries to produce a new public key s.t. s is also a valid A’s signature on m

  4. Application • e-lottery • the gambler uses his/her secret key to sign on the e-lottery to ensure that he owns the e-lottery • e-coupon (禮卷) • require be signed by the buyer and later signed by the shop

  5. History • (1998) Goldwasser, Micali and Rivest introduced the security notion of existential unforgeability against adaptive chosen-message attacks • (1999) Blake-Wilson and Menezes introduced a duplicate-signature key selection attacks • (2004) Menezes and Smart analyzed the security of some signature schemes against this attack, named as key substitution attacks

  6. Fischlin Signature Scheme (1/2) • Key Generation: N=pq ( p=2p’+1, so does q ) three random quadratic residues h1, h2, XZN* • Signature Generation: compute (l-bit) H(m), H(.): collision resistant hash fun. compute y=(Xh1ah2a XOR H(m))1/e mod N e: random (l+1)-bit prime a: l-bit long Public key (N, X, h1, h2) Private key (p, q) Signature (y, a, e)

  7. Fischlin Signature Scheme (2/2) • Signature Verification: check e : (l+1)-bit odd integer a: l-bit ye= (Xh1ah2a XOR H(m)) mod N

  8. Camenisch-Lysyanskaya Signature Scheme (1/2) • Key Generation: N=pq ( p=2p’+1, so does q ) three random quadratic residues h1, h2, XZN* • Signature Generation: compute y=(Xh1sh2m)1/e mod N e >2lm+1: random prime of length le=lm+2 s: random number st. ls=lN+lm+l Public key (N, X, h1, h2) Private key (p, q) Signature (y, s, e)

  9. Camenisch-Lysyanskaya Signature Scheme (2/2) • Signature Verification: • check e: 2le-1 < e < 2le ye= (Xh1sh2m) mod N

  10. Cryptanalysis (1/2) • Weak-key substitution attack (stronger) • produce public/private key • Strong-key substitution attack • public key (without knowing private key) Weak-Key Substitution Attack the same form X = yeh1-sh2-t mod N signature (y, a, e) where s=a, t=a XOR H(m) in Fischlin sheme t=m in C-L scheme

  11. Cryptanalysis (2/2) • choose two new primes st. • choose two random quadratic residues compute Then public key is valid with secret key and signature (y, a, e) of m

  12. Conclusion • Attack the two schemes by weak-key substitution attack • A signature scheme secure against existential forgery under adaptive chosen-message attack is inadequate • A scheme should be against key substitution attacks or rather under multi-user setting

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