An efficient certificateless blind signature scheme without bilinear pairing

Abstracts

Recently, the certificateless public key cryptography (CLPKC) has been studied widely since it could solve both of the certificate management problem in traditional public key cryptography (TPKC) and the key escrow problem in the identity-based public key cryptography (ID-based PKC). To satisfy requirements of different applications, many certificateless blind signature (CLBS) schemes using bilinear pairing for the CLPKC setting have been proposed. However, the bilinear pairing operation is very complicated. Therefore, the performance of those CLBS schemes is not very satisfactory. To solve the problem, we propose an efficient CLBS scheme without bilinear pairing. Performance analysis shows that the proposed scheme could reduce costs of computation and storage. Security analysis shows the proposed scheme is provably secure against both of two types of adversaries.

blind signature; certificateless cryptography; bilinear pairing; random oracle model

Recentemente, a criptografia de chave pública sem certificado (CLPKC) tem sido amplamente estudada, uma vez que poderia resolver o problema de gerenciamento de certificados na criptografia de chave pública tradicional (TPKC) e o problema chave de escrow da criptografia de chave pública baseada em identidade (ID-based PKC). Para atender aos requisitos de diferentes aplicações, têm sido propostos muitos sistemas de assinatura cega sem certificado (CLBs), sistemas que utilizam o emparelhamento bilinear para a configuração de CLPKC. No entanto, a operação de emparelhamento bilinear é muito complicada. Portanto, o desempenho desses regimes CLBs não é muito satisfatório. Para resolver o problema, propomos um esquema de CLBS eficiente sem emparelhamento bilinear. Uma análise de desempenho mostra que o esquema proposto poderia reduzir os custos de computação e armazenamento. Uma análise de segurança mostra que o esquema proposto é comprovadamente seguro contra dois tipos de adversários.

assinatura cega; criptologia sem certificado; emparelhamento bilinear; modelo de oráculo aleatório

INTRODUCTION

The blind signature (BS) scheme is a variation of digital signature scheme, which was first proposed by Chaum (1983). In the BS scheme, a user could get a signature for any message but the signer does not know the content of the message. Due to such properties, BS schemes are widely used in electronic voting, electronic payment and electronic cash.

After Chaum's work, many BS schemes in the traditional public key cryptography (TPKC) were proposed for different applications. However, the TPKC faces with the certificate management problem since a certificate generated by a trusted third party is needed to bind the user's identity and his public key. To solve the problem, Shamir (1984)introduced the concept of the identity-based public key cryptography (ID-based PKC). In the ID-based PKC, no certificate is required since the user's identity is his public key. However, the ID-based PKC faces with the key escrow problem since the user's private key is generated by the key generation centre (KGC) and the KGC knows all users' private keys. To solve the certificate management problem in the TPKC and the key escrow problem in ID-based PKC, Al-Riyami and Paterson (2003)proposed the concept of the certificateless public key cryptosystem (CLPKC). In the CLPKC, a user's private key consists of two parties, i.e. a partial secret key generated by the KGC and a secret value generated by the user.

Since groundbreaking work of Al-Riyami and Paterson (2003), many certificateless encryption (CLE) schemes (Sun and Zhang 2010) (Yang and Tan 2011), certificateless signature (CLS) schemes (Tian and Huang 2012) (Tsai et al. 2012) (Gong and Li 2012) (He et al. 2012c, 2013a, b) and certificateless key agreement schemes (HE et al. 2011c, 2012a, d) have been proposed for applications in CLPKC setting. Several certificatless blind signature (CLBS) schemes (Zhang and Zhang 2008) (Wang and Lu 2008) (Yang et al. 2009) (Sun and Wen 2009) (Zhang and Gao 2010) (Zhang et al. 2011) also were proposed. Zhang and Zhang (2008) proposed the fist CLBS scheme using bilinear pairing. Then several other CLBS schemes (Wang and Lu 2008) (Yang et al. 2009) (Sun and Wen 2009) (Zhang and Gao 2010) (Zhang et al. 2011) using bilinear pairing were proposed to improve performance or security of Zhang et al.'s scheme. However, the bilinear pairing operation is very complicated. Theoretical analysis (Chen et al. 2007) (Hankerson et al. 2004) and experimental results (Cao and Kou 2010) (He et al. 2011a, 2012b) demonstrate that the computation cost of a bilinear pairing operation is similar to that of a dozen or so elliptic curve scalar multiplication operations. Therefore, the performance of those CLBS schemes is not very satisfactory and CLBS scheme without bilinear pairing is required for practical applications.

In this paper, we propose an efficient CLBS scheme without bilinear pairing operation and show it is secure against both of two kinds of various attacks. Section 2 gives some background about CLBS schemes. Section 3 proposes a new CLBS scheme without bilinear paring. Security analysis and performance analysis are proposed in Section 4 and Section 5 separately. Some conclusions are given in Section 6.

PRELIMINARIES

Notations

For convenience, some notations used in the paper are described as follows.

• p, n: two large prime numbers;

• F_p: a finite field;

• E(F_p): an elliptic curve defined by the equation y²= x³ + ax + b over F_p, where a, b ∈ F_p and Δ = 4a³ + 27b² ≠ 0 (Koblitz 1987);

• G: the group with order n consisting of points on E(F_p) and the point at infinity O;

• P: a generator of the group G;

• DLP: the discrete logarithm problem (DLP), whose task is to compute x for given Q = xP;

CLBS Scheme

There are six polynomial time algorithms (Zhang et al. 2011) in a CLBS scheme, i.e. Setup, PartialPrivateKeyExtract, SetSecretValue, SetPublicKey, Sign and Verify.

Setup: Taking a security parameter k as input, this algorithm is executed by the KGC to generate the system parameters params and the master key mk.

PartialPrivateKeyExtract: Taking the system parameters params, the master key mk and a users identity ID as inputs, this algorithm is executed by the KGC to generate the users partial private key ID_D.

SetSecretValue: Taking the system parameters params as input, this algorithm is executed by a user to generate his secret value x_ID.

SetPublicKey: Taking the system parameters params and a users secret value x_ID as inputs, this algorithm is executed by the user to generate his public key PK_ID.

Sign: Taking the system parameters params, the partial private key D_ID, the secret value x_ID and a message m as inputs, this algorithm is executed by the user to generate a signature. There are three sub-algorithms in the algorithm, i.e. Blind, BSign and Unblind.

1. Blind. Taking message m and a random string r as inputs, the sub-algorithm is executed by the user to generate a blinded message m′.

2. BSign. Taking a blind message m′, the signers private signing key and the system parameters params as inputs, the sub-algorithm is executed by the signer to generate a blind signature σ′.

3. Unblind. Taking a blind signature σ′, the previously generated random string r and the system parameters params as inputs, the sub-algorithm is executed by the user to generate an unblinded signature σ.

Verify: Taking the system parameters params, a signer's identity ID, a signer's public key PK_ID, a message m and a signature σ as inputs, this algorithm is executed by the verifier to verify the legality of σ. If σ is legal, 1 will be outputted; otherwise, 0 will be outputted.

Security Model for CLBS Scheme

There are two kinds of adversaries in the CLBS scheme, i.e. the Type I adversary A₁ and the Type II adversary A₂.A₁ could replace user's public keys with some value he chooses, but he cannot get the master key. A₂ represents a malicious KGC, who cannot replace users' public keys but he could use the master key to generate users' partial private keys. The type adversaries in CLBS could be divided into normal adversary, strong adversary, and super adversary according to their attacks power (Huang et al. 2007). In the security analysis of the proposed CLBS scheme, we just need to consider two strongest types of adversaries, i.e. the super Type I adversary and the super Type II adversary. The abilities of an adversary A ∈ {A₁, A₂} are formally modeled by queries issued by adversaries.

ExtractPartialPrivateKey(ID): The adversary A could get the partial private key D_ID through the query.

ExtractSecretValue(ID): The adversary A could get the secret value x_ID through the query.

RequestPublicKey(ID): The adversary A could get the public key PK_ID through the query. ReplacePublicKey(ID, ): The adversary A could replace the public key PK_ID with a new public key through the query.

SuperSign(ID, m): The adversary A could get a signature σ through the query such that 1 ← Verify (params, ID, , m, σ), where is the current public key and it may be replaced by A.

We consider the following games against the super Type I and the super Type II adversaries.

Game 1: The first game is performed between a challenger C and a super Type I adversary A₁ for a CLBS scheme as follows.

Initialization. C runs Setup algorithm and generates a master secret key mk, public system parameters params. C keeps mk secret and then gives params to A₁.

Queries. A₁ can adaptively issue the ExtractPartialPrivateKey, ExtractSecretValue, RequestPublicKey, ReplacePublicKey, and SuperSign queries to C.

Output. Eventually, A₁outputs (ID_t, m_t, σ_t). A₁ wins the game if

1. ExtractPartialPrivateKey(IDt) and SuperSign (ID_t, m_t) queries have never been queried.

2. 1 ← Verify(params, ID_t, , m, σ), where which may be replaced by A₁ is the current public key of ID_t

Game 2: The second game is performed between a challenger C and a super Type II adversary A₂ for a CLBS scheme as follows:

Initialization. C runs Setup algorithm and generates a master secret key mk, public system parameters params. C gives mk and params to A₂.

Queries. A₂ can adaptively issue the ExtractPartialPrivateKey, ExtractSecretValue, RequestPublicKey, and SuperSign queries to C.

Output. Eventually, A₂outputs (ID_t, m_t, σ_t). A₁ wins the game if

1. ExtractSecretValue(ID_t) and SuperSign (ID_t, mt) queries have never been queried.

2. 1 ← Verify(params, ID_t, , m, σ), where is the original public key of ID_t

Definition 1 (Blindness) (Zhang et al. 2011). Suppose two honest users U₀ and U₁ engage in the blind signature issuing protocol with a probabilistic polynomial-time adversary A on two messages m_b and m_1−b, and output two signatures σ and σ′ respectively, where b ∈ {0; 1} is a random bit chosen uniformly. At last, (m₀, m₁, σ_b, σ_{1− b}) are sent to A and then A outputs b′ ∈ {0, 1}. We call a signature scheme is blind if the inequation |Pr[b = b′] − 1/2| < n^–c holds for all such adversaries A, any constant c, and sufficiently large n.

Definition 2. We say that a certicateless blind signature scheme is secure against the super Type I adversary if for any polynomially bounded super Type I adversary A₁, is negligible, where denote the success probability that A wins in Game 1.

Definition 3. We say that a certicateless blind signature scheme is secure against the super Type II adversary if for any polynomially bounded super Type II adversary A₂, is negligible, where denote the success probability that A₁ wins in the Game 2.

We say a certificateless blind signature scheme is secure if it is blind and secure against two types of adversaries.

THE PROPOSED CLBS SCHEME

Based on He et al. work (He et al. 2011a, b), we propose a new CLBS scheme without bilinear pairing. The proposed scheme consists of six algorithms, i.e. Setup, PartialPrivateKeyExtract, SetSecretValue, SetPublicKey, Sign and Verify. The details of these algorithms are described as follows:

Setup: Given a security parameters k, KGC does the following steps to generate the system parameters and the mast key.

1. KGC chooses a k-bit prime p, generates an elliptic curve E(Fp) over finite feld F_p. KGC chooses a group G) with order n over E(F_p) and chooses a generator P) of the group G).

2. KGC chooses a random as the master mk and computes public key P_pub = sP.

3. KGC chooses three secure hash functions , and .

4. KGC publishes the system parameter params = {p, n, E(F_p), G, P, P_pub, H₁, H₂, H₃} and keeps the master key s secretly.

PartialPrivateKeyExtract: Given the system parameter params, the master key mk, and a user's identity ID, KGC generates a random number , computes R_ID = r_IDP, h_ID = H₁(ID, R_ID) and s_ID = r_ID + h_IDs mod n. Then KGC returns the partial private key D_ID = (s_ID, R_ID) to the user.

SetSecretValue: Given the system parameter params, the user with identity ID generates a random number , computes P_ID = x_IDP and sets x_ID as his secret value.

SetPublicKey: Given the system parameter params and the user's secret value x_ID, the user computes his public key P_ID = x_IDP.

Sign: Given a message m, the following three sub-algorithms are executed to generate a legal signature. First of all, the signer generates a random number , computes and sends and R_ID to the user.

1. Blind: Upon receiving the message and R_ID, the user generates three random numbers α, β, , computes h = H₂(m, R, ID, R_ID, PK_ID, P_pub), h_ID = H₁(ID, R_ID), , and . At last, the user sends to the signer.

2. BSign: Upon receiving the message , the signer computes and . At last, the signer sends to the user.

3. Unblind: Upon receiving the message , the user computes and outputs the signature σ = (R_ID, R, z).

Verify: To verify the legality of the signature σ = (R_ID, R, z) for message m and the signer with identity ID, the verifier computes h = H₂(m, R, ID, R_ID, PK_ID, P_pub), h_ID = H₁(ID, R_ID) and . The verifier checks whether zP and are equal. If they are equal, 1 is returned; otherwise, 0 is returned.

SECURITY ANALYSIS

In this section, we will analyze the security of the proposed CLBS scheme. We will show the proposed scheme is provably secure in the random oracle model (Bellare and Rogaway 1993). The following theorems are proposed for the security.

Theorem 1. The proposed CLBS scheme is blind.

Proof. Let (R_ID, R, z) be one of the two signatures given to the adversary A. Let (R_ID, , , ) be the message transmitted between the user and the signer. We just need to show that there are three random factors (α, β, γ) that could map (R_ID, , , ) to (R_ID, R, z). From the description of the proposed CLBS scheme, we could get that

Through equations (2) and (3), we could get that and . With the two above equations and equation (1), we could get that only a unique element exists. Then we could get β and γ also exist uniquely since and .

Then, we could conclude that three random factors (α, β, γ) always exist between (R_ID, R, z) and (R_ID, , , ). Therefore, A outputs a correct value b′ with probability exactly 1/2 and the proposed CLBS scheme is blind.

Theorem 2. The proposed CLBS scheme is secure against the super Type I adversary in random oracle model if the DLP is hard.

Proof. Suppose there is a super Type I adversary A₁ has non-negligible εadvantage in attacking the proposed CLBS scheme. We will show that an algorithm C could solve the DLP running A₁ as a subroutine.

Given a DLP instance Q = αPfor randomly chosen , C picks an identity ID^* at random as the challenged ID, sets P_pub= Q, chooses three secure functions and gives system parameters to A₁. C answers A₁'s queries as follows.

• H₁ query: A₁ maintains a list of tuples < ID, R_ID, h_ID >. Upon receiving a query on a message ID, R_ID, C returns h_ID to A₁ if contains a tuple < ID, R_ID, h_ID >, otherwise, C picks a random number h_ID ∈ Z^*_n, adds < ID, R_ID, h_ID > to and returns h_ID to A₁.

• H₂ query: A₁ maintains a list of tuples < m, R, ID, R_ID, PK_ID, P_pub, h >. Upon receiving a query on a message < m, R, ID, R_ID, PK_ID, P_pub>, C returns h to A₁ if contains a tuple < m, R, ID, R_ID, PK_ID, P_pub, h >; otherwise, C picks a random number , adds < m, R, ID, R_ID, PK_ID, P_pub, h > to and returns h to A₁.

• H₃ query: A₁ maintains a list of tuples < ID, R_ID, PK_ID, P_pub, h >. Upon receiving a query on a message < ID, R_ID, PK_ID, P_pub>, C returns h to A₁ if contains a tuple < ID, R_ID, PK_ID, P_pub, h >, otherwise, C picks a random number h∈ Z^*_n, adds < ID, R_ID, PK_ID, P_pub, h >to and returns h to A₁.

• ExtractPartialPrivateKey(ID) query. Upon receiving a query with the user's identity ID, C answers the query as follows.

  1)If ID ≠ ID^*, C generates two random numbers a_ID, , sets R_ID ← a_IDP − b_IDP_pub, h_ID = H₁(ID, R_ID) b_ID and s_ID← a_ID. C adds < ID, R_ID, h_ID > and < ID, s_ID, R_ID to > and separately. C returns < s_ID, R_ID > to A₁.

  2)Otherwise, C generates two random numbers a_ID, , sets R_ID ← a_IDP, h_ID = H₁(ID, R_ID) ← b_ID and s_ID← ⊥. C adds < ID, R_ID, h_ID> and < ID, s_ID, R_ID to > and separately. C returns < s_ID, R_ID to A₁.

• RequestPublicKey(ID) query. Upon receiving a query with the user's identity ID, C returns PK_ID to A₁.

• ExtractSecretValue(ID) query. Upon receiving a query with the user's identity ID, C picks a random number , computes PK_ID = x_IDP, adds < ID, x_ID, PK_ID> to and returns x_ID to A₁.

• ReplacePublicKey(ID, ) query. Upon receiving a query with the message (ID; ), C sets , and if the list contains < ID, x_ID, PK_ID >, otherwise, C makes a ExtractSecretV alue query with ID, sets , and .

• SuperSign(ID, m) query. Upon receiving a query with the message (ID, m), Clooks up and for the tuples< ID, s_ID, R_ID > and < ID, x_ID, PK_ID > and performs as follows:

  1)If x_ID ≠ ⊥, then C performs according to the description of the scheme, and returns the generated (R_ID, R, z) to A₁.

  2)Otherwise, C generates two random a; , sets z ← a, H₂(m, R, ID, R_ID, PK_ID, P_pub) ← b, , where . C returns σ = (R_ID, R, s) to A₁ and adds < m, R, PK_ID, R_ID, P_pub, b > to .

Eventually, A₁ outputs a valid signature (ID, mt, σ_t), where . If ID_t ≠ ID^*, C stops the simulation; otherwise, C finds < ID_t, , > and < ID_t, , > in and respectively. From the forgery lemma (David and Jacque 2000), we know that A₁ could output another legal signature if the same random number replayed but with different choice of the random oracle H₂.

The following equation holds because the signature is valid

and

Let l, , and α denote discrete logarithms of R; , and P_pub respectively, i.e. R = lP, , and P_pub = P. Then we could get the following two equations.

and

Because only l and α are unknown to C in the above two equations, he could solve those equations and outputs α as the solution of the DLP. Since ID^* is randomly chosen, then we have , where is the number of H₁ query A₁ has made. C could solve the DLP with a non-negligible probability . This contradicts the hardness of the DLP. Therefore, the proposed scheme is secure against the super Type I adversary.

Theorem 3. The proposed CLBS scheme is secure against the super Type II adversary in random oracle model if the DLP is hard.

Proof. Suppose there is a super Type II adversary A₂ has non-negligible advantage ε in attacking the proposed CLBS scheme. We will show that there is an algorithm C could solve the DLP running A₂ as a subroutine.

Given a DLP instance Q = αPfor randomly chosen , C picks an identity ID^* at random as the challenged ID, chooses a random number , sets P_pub = sP, and chooses three secure functions. C give the master key s and system parameters to A₂. C answer H₁ query, H₂ query, H₃ query, ExtractPartialPrivateKeyquery, ReplacePublicKey query and SuperSign(ID, m) query like he does in the above theorem. C simulates other oracle queries of A₂ as follows:

1. ExtractPartialPrivateKey(ID) query. Upon receiving a query with the user's identity ID, C generates a random number , computes R_ID = r_IDP, h_ID = H₁(ID, R_ID) and s_ID = r_ID+ h_IDs mod n. C adds < ID, R_ID, h_ID> and < ID, s_ID, R_ID > to and separately. Then, C returns < s_ID, R_ID > to A₂.

2. ExtractSecretValue(ID) query. Upon receiving a query with the user's identity ID, C does as follows. If ID ≠ ID^*, C picks a random number and computes PK_ID = x_IDP, returns x_ID to A₂ and adds < ID, x_ID, PK_ID > to ; otherwise, C sets PK_ID = Q, adds < ID, ⊥, PK_ID > to .

Eventually, A₂ outputs a valid signature (ID, m_t, σ_t), where . If ID_t ≠ ID^*, C stops the simulation; otherwise, C finds < ID_t, , > and ID_t, , in and respectively. The public key is the original pub key is ID_t. From the forgery lemma (David and Jacque 2000), we know that A² could output another legal signature if the same random number replayed but with different choice of the random oracle H₂. The following equation holds because the signature is valid.

and

Let l, α, and s denote discrete logarithms of R, , and P_pub respectively, i.e. R = lP, , and P_pub = sP. Then we

and

Because only l and α are unknown to C in the above two equations, he could solve those equations and outputs as the solution of the DLP. Since ID^* is randomly chosen, then we have , where is the number of H₁ query A₂ has made. C could solve the DLP with a non-negligible probability . This contradicts the hardness of the DLP. Therefore, the proposed scheme is secure against the super Type II adversary.

PERFORMANCE ANALYSES

In this section, we will compare the efficiency of the proposed CLBS scheme with the three latest CLBS schemes, i.e. Sun et al.'s CLBS scheme (Sun and Wen 2009), Zhang et al.'s scheme (Zhang and Gao 2009) and Zhang et al.'s CLBS scheme (Zhang et al. 2011).

To achieve the security level of 1024 bits RSA, bilinear pairing-based CLBS scheme and ECC-based CLBS scheme, we have to use the Tate pairing defined over a supersingular elliptic curve on a finite field F_q and a secure elliptic curve on a finite field F_p separately, where the length of q and p are 512 bits and 160 bits respectively. We also assume the output of the hash function is 160 bits. Some notations are defined as follows.

• e: a bilinear pairing operation;

• E: a modular exponentiation operation;

• M: an ECC-based scale multiplication operation;

• Mpair: a bilinear pairing-based scale multiplication operation;

It is well known that the computational cost of a hash function operation could be ignored when it is compared with that of a bilinear pairing operation, a modular exponentiation operation, a ECC-based scale multiplication operation or a bilinear pairing-based scale multiplication operation. Therefore, we just need to counter the bilinear pairing operation, the modular exponentiation operation, the ECC-based scale multiplication operation and the bilinear pairing-based scale multiplication operation in performance comparisons. The comparisons are listed in Table I.

TABLE I
Performance comparisons of different schemes.

	Sun et al.'s scheme	Zhang and Gao's scheme	Zhang et al.'s scheme	Our scheme
Sign	3E + 1Mpair	3E + 2Mpair	2e + 2E+ 1Mpair	6M
Verify	1e + 1E+ 1Mpair	1e + 1E+ 1Mpair	3e + 1E	4M
Size	256B	148B	148B	100B

Theoretical analyses (Chen et al. 2007) and experimental results (Cao and Kou 2010) (He et al. 2011a, 2012b) demonstrate that the computation costs of a bilinear pairing operation, a modular exponentiation operation and a bilinear pairing-based scale multiplication operation are about 19, 3 and 3 times of that of a ECC-based scale multiplication operation. Therefore, we could get that the computational cost of the Sign algorithm in the proposed scheme is 60%, 54.55% and 10.71% of that in Sun et al.'s CLBS scheme (Sun and Wen 2009), Zhang et al.'s CLBS scheme (Zhang and Gao 2009) and Zhang et al.'s CLBS scheme (Zhang et al. 2011) separately. The computational cost of the Verify algorithm in the proposed scheme is 16%, 16%, 6.35% of that in Sun et al.'s CLBS scheme (Sun and Wen 2009), Zhang et al.'s CLBS scheme (Zhang and Gao 2009) and Zhang et al.'s CLBS scheme (Zhang et al. 2011) separately. Besides, the signature size in the proposed scheme is 39.06%, 39.06% and 67.57% of that in Sun et al.'s CLBS scheme (Sun and Wen 2009), Zhang and Gao's CLBS scheme (Zhang and Gao 2009) and Zhang et al.'s CLBS scheme (Zhang et al. 2011) separately. Therefore, the proposed CLBS scheme has better performance than those previous CLBS schemes.

CONCLUSIONS

Recently, the certicicateless public key cryptography without bilinear pairing operation attracted wide attention since such schemes have better performance than traditional ones. In this paper, we propose the first CLBS scheme without bilinear pairing operation. Performance analyses demonstrates that the proposed scheme has much better performance than previous CLBS schemes. We also show that the proposed scheme is provably secure against both of two types of adversaries in the random oracle. Therefore, the proposed scheme is more suitable for practical applications.

The authors thank Dr. Alexander Kellner and the anonymous reviewers for their valuable comments. This research was supported by the Application Foundation Research Project of Yunnan Science and Technology Department (No. 2011FZ168), the Key Scientific Research Projects of Yunnan Education Department (No. ZD201109) and the National Natural Science Foundation of China (Nos. 61202447 and 61201180).

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• Zhang L, Zhang F, Qin B and Liu S. 2011. Provably-secure electronic cash based on certificateless partially-blind signatures. Electron Commer R A 10: 545-552.

Publication Dates

• Publication in this collection
  June 2014

History

• Received
  27 May 2013
• Accepted
  10 Dec 2013