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Blind Signature Schemes.Pdf A simple forward secure blind signature scheme based on master keys and blind signatures Yeu-Pong Lai † Chin-Chen Chang †Department of Computer Science and Information Engineering, National Defence University, Chung Cheng Institute of Technology, Tauyuan, Taiwan, 335. Email: [email protected] Department of Computer Science and Information Engineering, National Chung Cheng University, Chaiyi, Taiwan, 621. Email: [email protected] Abstract tackers can be prevented, these cryptosystems can per- form correctly and securely. However, some cryptosys- The forward secure property is important in digi- tems may not be secure if the secret information is re- tal signature schemes. Many schemes have been pro- vealed or detected. Furthermore, these cryptosystems posed for the forward secure signature. Among them, may cause confusion. For instance, in digital signature only a few have been proposed for the forward secure schemes, when the secret key of a signer is revealed or blind signature. The blind signature is, however, a crit- stolen, others who know the secret key can counterfeit ical technique in e-business and other applications and the digital signatures of this signer. The signer, how- , thus, more research should be devoted on it. This pa- ever, cannot withdraw all the digital signatures indis- per focuses on the design of forward secure blind sig- criminately, because some signatures are indeed signed nature schemes. Digital signature schemes have been by the signer. The forward secure property ensures that proposed and discussed for years. Some of them are ef- these signatures, signed before the secret key was re- ficient and secure. Even specific computing hardware vealed, are valid. In other words, the forward secure is designed for these schemes. Our idea is, there- property makes it impossible to forge signatures valid fore, to combine two existing schemes, Koyama’s mas- in previous time periods even if the current secret key of ter key scheme and Chaum’s blind signature scheme, the signer is compromised. Many forward secure signa- so that a forward secure blind signature scheme re- ture schemes have been proposed recently [1][4]. These sults. This proposed scheme is also fully examined for schemes all have the forward secure property. the forward secure property and the blind signing prop- In addition, there are several signature schemes, erty. Since these two schemes are both based on the RSA called blind signature schemes, have been proposed for cryptosystem, the security of the proposed scheme de- untraceable applications. The untraceable property en- pends on the difficulty of solving the factoring prob- sures that signature requesters cannot be identified by lem. a signer so that the signature schemes can be applied Keywords: Forward secure property, blind signa- to e-commerce applications. The most popular blind ture, RSA cryptosystem, master key signature scheme was proposed by David Chaum in 1982[2]. This blind signature scheme and its variations play important roles in cryptography as well as in prac- 1. Introduction tical applications such as e-cash, e-voting systems, etc. However, these schemes do not have the forward se- Nowadays, many cryptosystems are believed to be cure property. In contrast, in 2003, Duc et al. proposed quite secure, and can be proven to be secure under very a forward secure blind signature scheme based on the reasonable assumptions. As long as secret information OGQ blind signature scheme [3]. is stored safely and accessed by internal or external at- As mentioned above, the requirement for forward se- Proceedings of the 19th International Conference on Advanced Information Networking and Applications (AINA’05) 1550-445X/05 $20.00 © 2005 IEEE curity is that digital signatures that are valid in previ- untraceability of the proposed scheme. The properties ous time periods should be unforgeable . Moreover, if are separately discussed in subsections 5.1, 5.2, 5.3, and the current secret key of a signer is compromised, at- 5.4, respectively. Finally, conclusions are given in Sec- tackers cannot forge signatures valid in previous time tion 6. periods to obligate a signer to the forged signatures. Intuitively, this requirement can be satisfied by using 2. A master key scheme different public key pairs for the signing procedures in each time period. However, the public key should be Master key schemes are public key cryptosystems. In changed and announced frequently. this scheme, there is a special key pair, called master The shortcoming can be overcome by applying the key pair, which can decrypt (or encrypt) messages en- master key schemes instead of the public key schemes. crypted (or decrypted) by other participants’ keys. Ev- The master key schemes use a public key and sev- ery participant has a public key and a secret key. The eral secret keys (or several public keys and one se- master public key can decrypt messages encrypted by cret key). Thus, each secret key can be assigned for the participants with their secret keys. Also, messages the signing procedure in one time period. The pub- encrypted with the master secret key can be decrypted lic key is not changed anyway. In 1982, Koyama pro- by participants using their secret keys. posed a master key scheme based on the RSA public- Koyama proposed a master key scheme for the RSA key cryptosystem[5]. The master encryption and de- public-key cryptosystems[5]. The master key pair is cryption keys in this scheme can be generated by using generated from several public key and secret key pairs. the Euclidean algorithm. Thus, the computation for the The master secret key is the multiplicative inverse of master keys is not complex. In addition, this scheme is the master public key modulo to the least common mul- compatible to Chaum’s blind signature scheme for gen- tiplier of these moduli for participants’ key pairs. Thus, erating a forward secure blind signature scheme. the master key of a cryptosystem can be generated if The most popular application of blind signature certain conditions are satisfied. schemes is the use of e-cash. There are three partic- For k participators in the cryptosystem, there are ipants in the scheme, a bank, customers, and a verifier. k keypairs,(e1,d1), (e2,d2), ...,(ek,dk), which sat- The bank signs the e-cash and maintains customers’ isfy eidi =1mod Li,wherei is from 1 to k. Li is the bank accounts. The untraceablility of customers is an least common multiplier of pi − 1andqi − 1, where pi important requirement for e-business, since customers and qi are secret and the multiplier ni = piqi is pub- like to keep their privacy when spending money. Be- lic. The master key pair (eh,dh) satisfies the equa- cause of the anonymity of customers, the forward se- tion ehdh =1mod L,whereL = lcm(L1,L2,...,Lm). cure property in these e-cash schemes is very impor- tant. When the secret key of the bank is compromised, Koyama’s master key scheme: attackers can generate e-cash. The bank cannot repudi- Let e1,i represent the master encryption key for the ate the e-cash, because legal customers should not pay keys, e1,e2,...,ei.Fork participants, the master en- for the mistake of the bank losing the secret key. For cryption key eh is e1,k and the master decryption key this reason, the forward secure blind signature is pro- dh is d1,k. The master encryption key e1,1 is the en- posed and discussed. cryption key e1. The master encryption key satis- This paper proposes a way to combine a master fies: key scheme with a blind signature scheme to obtain e1,i = e1,i−1 mod L1,i, a forward secure blind signature scheme. The follow- where L1,i = lcm(L1,L2,...,Li), ing section introduces a master key scheme for the for- and ward secure property. Actually, any RSA-based master e1,i = ei mod Li. key scheme might be tried and fitted for the proposed scheme[6]. A numerical example is also provided to Deriving from the previous equations, we get show the performance of this master key scheme. Sec- xi−1 × L1,i−1 − yi × Li = ei − e1,i−1. tion 3 presents a very popular blind signature scheme. It is widely employed in e-commerce for its simplic- Since both the numbers xi−1 and yi are natu- ity and security. It is also an RSA-based cryptosystem. ral numbers, they can be derived by using the Eu- The combination of the above two schemes is then pro- clidean algorithm when the numbers L1,i−1, Li, ei, posed in Section 4. The proposed scheme is a forward and e1,i−1 are given. Thus, e1,i can be determined af- secure blind signature scheme. Section 5 discusses the ter the numbers xi−1 and yi have been computed. The properties of correctness, blindness, unforgeability and master decryption key dh is then equal to d1,k satisfy- Proceedings of the 19th International Conference on Advanced Information Networking and Applications (AINA’05) 1550-445X/05 $20.00 © 2005 IEEE ing e1,kd1,k =1mod L1,k. After this phase, the signer makes the key (e, n) pub- lic and keeps his/her secret key (d, p, q) secret. The Example: numbers p and q are two distinct large primes. Other Let parameters satisfy n = pq, ed =1mod Φ(n), where (e1,d1,p1,q1,n1) = (23, 7, 3, 17, 51), Φ(n)=(p − 1)(q − 1). The signing phase presents (e2,d2,p2,q2,n2)=(7, 17, 11, 13, 143), the protocol between the signer and a requester for (e3,d3,p3,q3,n3)=(3, 29, 5, 23, 115). the signer to blindly sign a hashed message, H(m).
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