A Novel Anonymous Proxy Signature Scheme

Abstract

Recently, several studies about proxy signature schemes have been conducted. In 2009, Yu et al. proposed an anonymous proxy signature scheme attempting to protect the proxy signer's privacy from outsiders. They claimed that their scheme can make the proxy signer anonymous. However, based on our research, we determined that this was not the case and the proxy signer's privacy was not anonymous. Hence, in this paper, we propose a new anonymous proxy signature scheme that truly makes the proxy signer anonymous while making it more secure and efficient when compared with Yu et al.'s scheme. Our proxy signature scheme consists of two contributions. First, we mainly use random numbers and bilinear pairings to attain the anonymous property. Secondly, we increase the security and efficiency of our proxy in the design.

1. Introduction

Proxy signature schemes can be used in many business applications such as signing important documents when the original signer is not present. For example, an important document needs to be signed by the CEO, but the CEO is out of the office or not immediately available. At this time, the CEO can use the proxy signature scheme to designate the general manager or business executive to sign the document on his or her behalf. The signed document will be valid and can be verified by everyone without the CEO actually signing it. Any proxy signature scheme has to meet the identifiability, undeniability, verifiability, and unforgeability security requirements. It may be necessary to protect the proxy signer’s privacy from outsiders or third parties. In 1996, Mambo et al. [1] first proposed the concept of proxy signature. In their proposal, there are three parties: a user also called original signer, a proxy signer whom is delegated to sign a message on behalf of the original signer, and a verifier who verifies whether a signed message is legal or not.

Since Mambo et al.’s 1996 scheme, many proxy signature schemes have been proposed [1–27] (some other schemes though are signature schemes whereas not proxy signatures such as [28–33]). Generally speaking, there are two main categories of proxy signature schemes, the first category is one-to-one and the other is one-to-many. In the former, there is one original signer and one proxy signer, but in the latter, except for the original signer, there are a group of proxy signers. The one-to-one schemes are [4, 7, 10, 12, 13, 15–17, 25–27] and the proxy blind signature [2], which is based on a special digital signature scheme first introduced by Chaum [34] in 1983. In the one-to-many, there are two subsets, one is the proxy multisignature and the other is the (𝑡,𝑛) threshold proxy signature. In the proxy multisignature [5, 6, 9, 19–22], the original signer has an authorized proxy signer group, each proxy signer has to generate a partial proxy signature. If all partials of signatures are correct, the proxy signature will be generated by summation or multiplication operations of the partial proxy signatures. In the (𝑡,𝑛) threshold proxy signature [3, 11, 18, 23, 24], the original signer can choose the threshold and a proxy signing key is shared by 𝑛  proxy signers. Any 𝑡 of proxy signers can cooperatively derive the proxy signing key to sign the message.

In any proxy signature, the following four security properties are required.

(i) Unforgeability
Only a designated proxy signer can create a valid proxy signature for the original signer. In other words, nobody can forge a valid proxy signature without the delegation of the original signer.

(ii) Verifiability
After checking and verifying the proxy signature, a verifier can be convinced that the received message is signed by the proxy signer authorized by the original signer.

(iii) Undeniability
The proxy signer cannot repudiate the signature he produced.

(iv) Identifiability
Anyone including the original signer can determine the corresponding proxy signer’s identity from the proxy signature. That is, from the proxy signature any verifier can determine the proxy signer’s identity.

Although proxy signatures incorporate the above-mentioned security functions, they still face many threats such as man-in-the-middle, replay, frame, and public-key substitute attacks. In frame attacks [23], the malicious original signer can forge a signature after intercepting sent information and the forged signature can be accepted by the verifier. In public-key substitute attacks [24], the attacker can be either the original signer or any proxy signer. By changing their public keys, he can forge a valid proxy signature [11]. This indicates that when designing a proxy signature scheme, care should be taken to avoid these kinds of attacks.

Researchers, Shum and Wei’s [26] and Yang, and Peng [10], presented two one-to-one anonymous proxy signature (APS) schemes. They point that an APS scheme should possess not only the security features of unforgeability, verifiability, and undeniability, but also the properties of anonymity and anonymity revocation. The anonymity means that only one of the proxy signers can sign the message in the proxy signer group, other proxy signers cannot know who the signer is. And the anonymity revocation indicates that once required, the proxy signer can assure the others that he is the real signer. However, N. Y. Lee and M. F. Lee [27] indicate that Shum and Wei’s scheme [26] violates the property of the unforgeability. Yang and Peng [10] therefore proposed a modified one-to-one APS scheme. In 2009, Yu et al. [8] first proposed a one-to-many APS scheme. In their scheme, there is a group of proxy signers, but only one proxy signer can anonymously signs the message. By using a group of signers, Yu et al. want to provide privacy and anonymous protection for the real proxy signer. They claim that their scheme is provably secure. However, based on our research by just using some of the transmitted data along with public information, we were able to isolate and identify the proxy signer. More details of the analysis are described in Section 3.2.

The rest of the paper is organized as follows. In Section 2, we present the basic concepts of bilinear pairings and some related mathematical problems. In Section 3, we review and show the weakness of Yu et al.’s scheme. Section 4 shows the proposed scheme, and Section 5 makes comparison of computation efficiency between Yu et al.’s scheme and ours. Finally, a conclusion is given in Section 6.

2. Background

In this section, we describe the concept of bilinear pairings which is used as the mathematical basis for this design.

Let 𝐺₁ be a cyclic additive group of order 𝑞 generated by a base point 𝑃 on Elliptic curve and 𝐺₂ a cyclic multiplicative group with the same order. It is assumed that solving the Elliptic curve discrete logarithm problem (ECDLP) in 𝐺₁ and discrete logarithm problem (DLP) problem in 𝐺₂ is difficult. A bilinear map 𝑒 is defined as 𝑒∶𝐺₁×𝐺₁→𝐺₂, which has the following properties:(1)bilinearity: 𝑒(𝑎𝑃,𝑏𝑄)=𝑒(𝑃,𝑄)^𝑎𝑏, where 𝑃,𝑄∈𝐺₁ and all 𝑎,𝑏∈𝑍^∗𝑞;(2)nondegeneracy: there exists 𝑃,𝑄∈𝐺₁ such that 𝑒(𝑃,𝑄)≠1; in other words, the map does not send all pairs in 𝐺₁×𝐺₁ to the identity in 𝐺₂;(3)computability: there is an efficient algorithm to compute 𝑒(𝑃,𝑄) for all 𝑃,𝑄∈𝐺₁.

3. Review of Yu et al.’s Scheme

In this section, we review Yu et al.’s APS scheme [8] and demonstrate that the original APS cannot satisfy the anonymous property in Section 3.2.

3.1. Yu et al.’s APS Scheme

There are six phases in Yu et al.’s APS scheme: (1) the parameter generation phase, (2) the key generation phase, (3) the delegation signing phase, (4) the delegation verification phase, (5) the APS generation phase, and (6) the APS verification phase. We describe them as follows.(1)In the parameter generation phase, on input of security parameter 𝑘, a system parameter generation algorithm outputs a cyclic additive group 𝐺₁ of order 𝑞, a multiplicative group 𝐺₂ of the same order, a bilinear map 𝑒∶𝐺₁×𝐺₁→𝐺₂, and a generator 𝑃 of 𝐺₁. This algorithm also outputs two cryptographic hash functions: 𝐻₀∶{0,1}^∗×𝐺₁→𝑍^∗𝑞 and 𝐻₁∶{0,1}^∗→𝐺₁. (2)In the key generation phase as shown in Figure 1, the original signer  𝐴𝑙𝑖𝑐𝑒 selects 𝑥_𝑜∈𝑍^∗𝑞 as her private key and computes her public key as 𝑌_𝑜=𝑥_𝑜𝑃. Each proxy signer  𝑢_𝑖∈𝒰 randomly selects 𝑥_𝑖∈𝑍^∗𝑞 as his/her private key and sets the corresponding public key as 𝑌_𝑖=𝑥_𝑖𝑃. (3)In the delegation signing phase, 𝐴𝑙𝑖𝑐𝑒 firstly generates a warrant 𝑚_𝑤 which contains some explicit descriptions about the delegation relation such as the identities of both 𝐴𝑙𝑖𝑐𝑒 and the proxy signers, the expiration time of the delegation, and the signing power in the warrant. Then, 𝐴𝑙𝑖𝑐𝑒 randomly picks a number 𝑟∈𝑍^∗𝑞 and computes 𝑅=𝑟𝑃 and 𝑠=𝑟+𝑥_𝑜𝐻₀(𝑚_𝑤,𝑅)mod𝑞. Finally, 𝐴𝑙𝑖𝑐𝑒 sends (𝑚_𝑤,𝑅,𝑠) to the proxy signers in set 𝒰={𝑢₁,…,𝑢_𝑛}.(4)Upon receiving (𝑚_𝑤,𝑅,𝑠), each proxy signer  𝑢_𝑖 checks if the equation 𝑠𝑃=𝑅+𝐻₀(𝑚_𝑤,𝑅)𝑌_𝑜 holds. If it does not, the delegation will be rejected. Otherwise, it will be accepted and each proxy signer  𝑢_𝑖 computes his/her proxy secret key as psk_𝑖=𝑠+𝑥_𝑖𝐻₀(𝑚_𝑤,𝑅)mod𝑞. (5)In the APS generation phase as shown in Figure 2, proxy signer  𝑢_𝑠∈𝒰 signs on a message 𝑚 with his proxy secret key  psk_𝑠 on behalf of the original signer, 𝐴𝑙𝑖𝑐𝑒, in an anonymous way. 𝑢_𝑠 first chooses random numbers 𝑟_𝑖∈𝑍^∗𝑞, where 𝑖∈{1,2,…,𝑛} and 𝑖≠𝑠, computes both 𝜎_𝑖=𝑟_𝑖𝑃 and 𝜎_𝑠=(1/psk_𝑠)(𝐻₁(𝑚‖𝑚_𝑤)−∑_𝑖≠𝑠𝑟_𝑖(𝑅+𝐻₀(𝑚_𝑤,𝑅)(𝑌_𝑜+𝑌_𝑖))), and sends 𝜎=(𝜎₁,𝜎₂,…,𝜎_𝑛,𝑚,𝑚_𝑤,𝑅) to the verifier.(6)In the APS verification phase, given public keys 𝑌_𝑜,𝑌₁,…,𝑌_𝑛 and a received anonymous proxy signature 𝜎, the verifier can examine the validity of the signature 𝜎 by checking whether the following expression holds: ^𝑛_𝑖=1=𝑒𝑅_𝑛+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝜎_𝑖_𝑖=_{1,𝑖≠𝑠}⋅𝑒𝑅+𝑒𝐻𝑅₀+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝜎_𝑖=_𝑛𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑠,𝜎_𝑠_𝑖=_{1,𝑖≠𝑠}⋅𝑒𝑅𝑒+𝑟𝐻_𝑖₀𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑠,ps1k_𝑠=×𝐻₁𝑚‖𝑚_𝑤−_𝑖≠𝑠𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖_𝑛𝑖=_{1,𝑖≠𝑠}⋅𝑒𝑃,𝑒𝐻𝑟_𝑖1𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃=𝑒𝑃,𝐻₁𝑚‖𝑚_𝑤−_𝑖≠𝑠𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖𝑚‖𝑚_𝑤.(1)

Figure 1 

Key generation, delegation signing, and delegation verification phases of Yu et al.’s scheme.

Figure 2 

APS generation phase and the APS verification phase of Yu et al.’s scheme.

3.2. Weakness of Yu et al.’s Scheme

After reviewing Yu et al.’s scheme above, we now explain the violation of the scheme’s anonymous property which they emphasized as follows.

Since 𝑅, 𝐻₀(𝑚_𝑤,𝑅), and (𝑌_𝑜+𝑌_𝑠) are public, we can obtain psk_𝑠𝑃 by deducing psk_𝑠𝑃=𝑅+𝐻₀(𝑚_𝑤,𝑅)(𝑌_𝑜+𝑌_𝑠) because psk_𝑠𝑃==𝑠𝑟++𝑥𝑥_𝑖𝑜𝐻₀𝑚_𝑤,𝑅𝑃=𝑟+𝑥𝐻_0𝑜𝑚_𝑤,𝑅+𝑥_𝑖𝐻₀𝑚_𝑤,𝑅𝑃=𝑟𝑃+𝑥+_𝑜𝑥_𝑖𝐻₀𝑚_𝑤,𝑅𝑃=𝑅+𝐻₀+𝑥_𝑖𝐻₀𝑚_𝑤,𝑅𝑃𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑠.(2)

Next, we define an inspector 𝐗 to be 𝑒(psk_𝑥𝑃,𝜎_𝑗), where psk_𝑥 is 𝑢_𝑥’s secret proxy key, 𝜎_𝑗 is a specific subsignature in 𝜎, and 𝑥,𝑗∈{1,…𝑛}. In addition, we define 𝐘 to be ∏^{𝑛𝑖=1,𝑖≠𝑥}𝑒((𝑅+𝐻₀(𝑚_𝑤,𝑅)(𝑌_𝑜+𝑌_𝑖)),𝜎_𝑖). Then, if there exist some 𝑥 and 𝑗 satisfying 𝐗⋅𝐘=𝑒(𝑃,𝐻₁(𝑚‖𝑚_𝑤)), we can determine that 𝑥 should be equal to 𝑗, and 𝑢_𝑗 is then the right proxy signer. This is because if 𝑢_𝑗 is the right proxy signer, then the corresponding subsignature 𝜎_𝑗 must have the factor 1/psk_𝑗, and therefore only applying the right psk_𝑥𝑃, that is, 𝑥=𝑗, can cancel the factor result in the holing of the end. Otherwise, we continue to examine next possible 𝑥 or 𝑗. By doing this way, we can deduce the right proxy signer at most 𝑛² times.

For more clarity, we take three proxy signers, 𝑢₁, 𝑢₂, 𝑢₃, as an example. Suppose 𝑢₂ is the real proxy signer, then 𝜎₁=𝑟₁𝑃, 𝜎₂=(psk₂)⁻¹(𝐻₁(𝑚‖𝑚_𝑤)−∑^{3𝑖=1,𝑖≠1}𝑟_𝑖(𝑅+𝐻₀(𝑚_𝑤,𝑅)(𝑌_𝑜+𝑌_𝑖))) and 𝜎₃=𝑟₃𝑃.

If we first try 𝜎₁ with different 𝑥=1,2,3, then we have three tries as in the following. (1.1) When 𝑥=1 and thus 𝐗=𝑒(psk₁𝑃,𝜎₁), the value 𝐗⋅𝐘 should be 𝑒psk₁𝑃,𝜎₁⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₁𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝜎₁⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₁𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑟_𝑖𝑃⋅𝑒𝑅+𝐻⋅₀𝑟₁𝑃⋅𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁,𝜎₂≠𝑒𝑃,𝐻₁𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃,𝜎₃𝑚‖𝑚_𝑤.(3)(1.2) When 𝑥=2 and thus 𝐗=𝑒(psk₂𝑃,𝜎₁), the value 𝐗⋅𝐘 should be 𝑒psk₂𝑃,𝜎₁⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₂𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝜎₁⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₂𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑟_𝑖𝑃⋅𝑒𝑅+𝐻⋅₀𝑟₁𝑃⋅𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁,𝜎₂≠𝑒𝑃,𝐻₁𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃,𝜎₃𝑚‖𝑚_𝑤.(4)(1.3) When 𝑥=3 and thus 𝐗=𝑒(psk₃𝑃,𝜎₁), the value 𝐗⋅𝐘 should be 𝑒psk₃𝑃,𝜎₁⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₃𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝜎₁⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₃𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑟_𝑖𝑃⋅𝑒𝑅+𝐻⋅₀𝑟₁𝑃⋅𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₂,𝜎₂≠𝑒𝑃,𝐻₁𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁,𝜎₃𝑚‖𝑚_𝑤.(5)

Secondly, if we try 𝜎₂ with different 𝑥=1,2,3, then we have three tries as in the following. (2.1) When 𝑥=1 and thus 𝐗=𝑒(psk₁𝑃,𝜎₂), the value 𝐗⋅𝐘 should be 𝑒psk₁𝑃,𝜎₂⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₁𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝜎₂⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₁𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑟_𝑖𝑃⋅𝑒𝑅+𝐻⋅₀𝑟₂𝑃⋅𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁,𝜎₁≠𝑒𝑃,𝐻₁𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃,𝜎₃𝑚‖𝑚_𝑤.(6)(2.2)When 𝑥=2 and thus 𝐗=𝑒(psk₂𝑃,𝜎₂), the value 𝐗⋅𝐘 should be𝑒psk₂𝑃,𝜎₂⋅³_𝑖=_1,𝑖≠1=𝑒𝑃,psk₂𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝜎₂⋅³_𝑖=_1,𝑖≠1=𝑒𝑃,psk₂𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃⋅1psk₂𝐻¹𝑚‖𝑚_𝑤−_𝑖≠𝑠⋅𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖_3𝑖=_1,𝑖≠1=𝑒𝑃,𝐻𝑒₁𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃⋅𝑚‖𝑚_𝑤−_𝑖≠1𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖₃_𝑖=_1,𝑖≠1=𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑒𝑌𝑃_𝑜,+𝐻𝑌_𝑖1,𝑃𝑚‖𝑚_𝑤⋅𝑒𝑒𝑃P,,𝑟𝑟₁₁𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁⋅𝑒𝑃,𝑟₃𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃=𝑅+𝐻₀𝑚_𝑤,𝑅𝑒𝑌_𝑜𝑃+,𝑌𝐻₁₁𝑒𝑃,𝑟₃𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃𝑚‖𝑚_𝑤⋅𝑒𝑒𝜎𝜎₁₁,𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁⋅𝑒𝜎₃,𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃=𝑒𝑃,𝐻,₁𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁𝑒𝜎₃,𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃𝑚‖𝑚_𝑤.(7)(2.3) When 𝑥=3 and thus 𝐗=𝑒(psk₃𝑃,𝜎₂), the value 𝐗⋅𝐘 should be 𝑒psk₃𝑃,𝜎₂⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₃𝑒𝑟_𝑖𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑃𝜎₂⋅³_𝑖=_1,𝑖≠2=𝑒𝑃,psk₃𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌_𝑖,𝑟_𝑖𝑃⋅𝑒𝑅+𝐻⋅₀𝑟₂𝑃⋅𝑒𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜+𝑌₁,𝜎₁≠𝑒𝑃,𝐻₁𝑚_𝑤,𝑅𝑌_𝑜+𝑌₃,𝜎₃𝑚‖𝑚_𝑤.(8)

From the above demonstration, for inspector 𝐗=𝑒(psk_𝑥𝑃,𝜎_𝑗), only when the subscript 𝑥=𝑗=2, the result of 𝐗⋅𝐘 is 𝑒(𝑃,𝐻₁(𝑚‖𝑚_𝑤). Therefore, we determined that 𝑢₂ is the right proxy signer and the anonymous property that they emphasized is broken.

4. Proposed Scheme

In this section, we propose a new one-to-many APS scheme to correct the anonymous flaw as discovered in Section 3. Our scheme is the same as theirs in the first two phases. The differences are in the last four phases, the delegation signing, delegation verification, APS generation, and APS verification phase. More details of our APS are shown in Section 4.1. Its correctness is demonstrated in Section 4.2 and the APS requirements are analyzed in Section 4.3. Before describing our protocol, we define some basic notations listed in Table 1.

Table 1 

The definitions of used notations

4.1. The New Proposed APS Scheme

In our APS scheme, there also exist an original signer  𝐴𝑙𝑖𝑐𝑒 and a proxy signer group  {𝒫₁,𝒫₂,…,𝒫_𝑛}, and only one proxy signer in the proxy signers group can sign the message. For more clarity, we show our scheme in detail as follows. The proposed scheme consists of six phases: (1) the parameter generation phase, (2) key generation phase, (3) delegation signing phase, (4) delegation verification phase, (5) APS generation phase, and (6) APS verification phase. Phases (1) and (2) are the same as in Yu et al.’s scheme, which has been delineated in Section 3.1. We omit these phases in the following but show phases (3) and (4) in Figure 3 and phases (5) and (6) in Figure 4.(3) In the delegation signing phase, as shown in Figure 3, the original signer randomly selects a number 𝑟∈𝑍^∗𝑞 and uses 𝑟 to compute 𝑅=𝑟𝑃 and 𝑣=𝑟+𝑥₀𝐻₀(𝑚_𝑤,𝑅). Then, the original signer sends (𝑚_𝑤,𝑅,𝑣) to each proxy signer  𝒫_𝑖∈{𝒫₁,𝒫₂,…,𝒫_𝑛} with warrant 𝑚_𝑤, where warrant contains the records of the original signer’s and proxy signer’s identities, delegation, authorization period, valid period, and so forth.(4) In the delegation verification phase, after receiving (𝑚_𝑤,𝑅,𝑣) the proxy signer  𝒫_𝑖 first checks whether the equation 𝑣𝑃?=𝑅+𝐻₀(𝑚_𝑤,𝑅)𝑌_𝑜 holds. If it does not, stop the protocol, otherwise, he stores (𝑚_𝑤,𝑅). Second, when signing message 𝑚,𝒫_𝑖 chooses random numbers 𝑟_𝑖∈𝑍^∗𝑞, 𝑖=1 to 𝑛, and 𝑉=𝑣𝑃 computes 𝑐=𝐻₁(𝑟₁‖⋯‖𝑟_𝑛), 𝑈=𝑐𝑃, and the proxy secret key, psk_𝑖=𝑟_𝑖−1∗𝑥_𝑖−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈).(5) In the APS generation phase, as shown in Figure 4, let 𝒫_𝑠 be the real proxy signer. He computes 𝜎_𝑖=𝑟_𝑖𝑉, where 𝑖∈{1,2,…,𝑛} and 𝑖≠𝑠 and computes 𝐿=𝑐∗𝑥_𝑠−1∗𝑉, then sets 𝑌,𝜎_𝑠,𝑝𝜎sum=∑^𝑛𝑖=1𝜎_𝑖,𝐴,𝐵,𝐶, and 𝐷, as 𝑌=∑^𝑛𝑖=1𝑌_𝑖, 𝜎_𝑠=psk_𝑠∗𝑌=𝑟_𝑠−1∗𝑥_𝑠−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)^∗𝑌, 𝐴=𝑟_𝑠∗𝑐∗psk_𝑠𝑃, 𝐵=𝑟_𝑠𝜎_𝑠, 𝐶=𝑟_𝑠∗𝑝𝜎sum, and 𝐷=𝑟_𝑠∗𝑐∗𝑉, respectively. Finally, 𝒫_𝑠 outputs 𝜎=(𝜎₁,𝜎₂,…,𝜎_𝑛,𝑚,𝑚_𝑤,𝑐,𝐴,𝐵,𝐶,𝐷,𝐿,𝑈,𝑉) as the anonymous proxy signature and sends 𝜎 to the verifier. (6) In APS verification phase, upon receiving the proxy signature the verifier computes ∑^𝑛𝑖=1𝑌_𝑖=𝑌 and checks whether the equation 𝑒(𝐷,∑^𝑛𝑖=1𝜎_𝑖)⋅𝑒(𝐴,𝑌)?=𝑒(𝑐𝑉,𝐶−𝐵)⋅𝑒(𝐿,𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)𝑌)⋅𝑒(𝑈,𝐵) holds. If it holds, the verifier accepts the signature, otherwise rejects it.

Figure 3 

The delegation signing and delegation verification phases of our scheme.

Figure 4 

Anonymous proxy signature generation phase and the verification phase of our scheme.

4.2. Correctness

In the delegation verification phase, each proxy signer can check whether the equation 𝑣𝑃?=𝑅+𝐻₀(𝑚_𝑤,𝑅)𝑌_𝑜 holds as follows.

Proof (first proof). 𝑣𝑃?=𝑅+𝐻₀𝑚_𝑤,𝑅𝑌_𝑜𝑣𝑃==𝑟𝑃𝑟++𝑥𝑥_𝑜𝑜𝐻₀𝑚_𝑤,𝑅𝑃=𝑅+𝐻₀𝐻₀𝑚_𝑤,𝑅𝑃𝑚_𝑤,𝑅𝑌_𝑜.(9)

If it holds, the proxy signer can know that the message is sent from the original signer. Because in the verification equation, he use the original signer’s public key 𝑌_𝑜 to examine it. If any adversary intercepts the message and modify it, it cannot pass the verification equation.

In the proxy signature verification phase, the following equation gives the correctness of the verification.

Proof (second proof). ^𝑛_𝑖=1=𝑒𝑒(𝑐𝑉𝐷,,𝐶𝜎−_𝑖𝐵)⋅⋅𝑒𝑒(𝐴𝐿,𝑌,𝐻)?₂𝑚_𝑤‖𝑚,𝑉,𝑈𝑌⋅𝑒(𝑈,𝐵)_𝑛_𝑖=1=𝑒𝐷,𝜎_𝑖⋅𝑒(𝐴,𝑌)_𝑛𝑖=_{1,𝑖≠𝑠}=𝑒𝑐𝑟_𝑠𝑉,𝜎_𝑖⋅𝑒𝑐𝑟_𝑠𝑉,𝜎_𝑠⋅𝑒𝑟_𝑠∗𝑐∗psk_𝑠𝑃,𝑌_𝑛_𝑖=_{1,𝑖≠𝑠}𝑒𝑐𝑟_𝑠𝑉,𝜎_𝑖⋅𝑒𝑐𝑟∗_𝑠𝑌𝑉,)𝑟⋅𝑒_𝑠−1𝑐∗𝑃𝑥,𝑟_𝑠−1𝑠∗𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈=_𝑛psk_𝑠𝑌_𝑖=_{1,𝑖≠𝑠}𝑒𝑐𝑟_𝑠𝑉,𝜎_𝑖⋅𝑒𝑐𝑟∗_𝑠𝑌𝑉,)𝑟⋅𝑒_𝑠−1𝑐∗𝑃𝑥,𝑟_𝑠−1𝑠∗𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈=_𝑛𝜎_𝑠_𝑖=_{1,𝑖≠𝑠}=⋅𝑒(𝑈𝑒,𝐵𝑐)_𝑛𝑟_𝑠𝑉,𝜎_𝑖⋅𝑒𝑥_𝑠−1∗𝑐𝑉,𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈∗𝑌_𝑖=_{1,𝑖≠𝑠}=𝑒𝑐𝑟_𝑠𝑒𝑐𝑟_𝑠𝑉,𝜎_𝑖⋅𝑒𝐿,𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈𝑌⋅𝑒(𝑈,𝐵)𝑉,^𝑛_𝑖=_{1,𝑖≠𝑠}𝜎_𝑖⋅𝑒𝐿,𝐻²=𝑒𝑐𝑟_𝑠𝑚_𝑤‖𝑚,𝑉,𝑈𝑌⋅𝑒(𝑈,𝐵)=⋅𝑒𝑒(𝑐𝑈𝑉,,𝑉𝐵𝑟,_𝑠)𝑝𝜎sum−𝜎_𝑠⋅𝑒𝐿,𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈𝑌=⋅𝑒𝑒(𝑐𝑈𝑉,,𝐵𝑟_𝑠)𝑝𝜎sum−𝜎_𝑠⋅𝑒𝐿,𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈𝑌=⋅𝑒𝑒((𝑐𝑈𝑉,,𝐶𝐵)−𝑝𝐵𝜎)s⋅u𝑒m−𝐿,𝜎𝐻_𝑠₂⋅𝑒𝐿,𝐻₂𝑚_𝑤‖𝑚,𝑉,𝑈𝑌𝑚_𝑤‖𝑚,𝑉,𝑈𝑌⋅𝑒(𝑈,𝐵).(10)

4.3. Security Analyses

In this section, we demonstrate that our APS scheme can satisfy the security properties as discussed in Section 1 for (1) verifiability, (2) unforgeability, (3) undeniability, (4) anonymity, and (5) anonymity revocation. Now, we demonstrate why our scheme can satisfy these five security properties as follows.

(1) Verifiability
In APS verification phase, after checking and verifying the proxy signature 𝜎, where 𝜎=(𝜎₁,𝜎₂,…,𝜎_𝑛,𝑚,𝑚_𝑤,𝑐,𝐴,𝐵,𝐶,𝐷,𝐿,𝑈,𝑉), the verifier can calculate to check whether the verification equation (∏^𝑛𝑖=1𝑒(𝐷,𝜎_𝑖))⋅𝑒(𝐴,𝑌)?=𝑒(𝑐𝑉,𝐶−𝐵)⋅𝑒(𝐿,𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)𝑌)⋅𝑒(𝑈,𝐵) holds. If it does, the verifier can be convinced that the received message is signed by one of the proxy signer members authorized by the original signer because 𝑌(=∑^𝑛𝑖=1𝑌_𝑖) and 𝑉(=𝑣𝑃=𝑅+𝐻₀(𝑚_𝑤,𝑅)𝑌_𝑜) are used in the verification equation.

(2) Unforgeability
It means that any entity (other than the real proxy signer  𝒫_𝑠), including the original signer, cannot generate a valid proxy signature. Only an authorized proxy signer  𝒫_𝑠 can create a valid proxy signature 𝜎. If any attacker wants to forge a proxy signature, he must be authorized by the original signer signing on a warrant 𝑚_𝑤 and use the proxy signer’s proxy secret key  psk_𝑠 to compute 𝜎_𝑠. However, this is impossible since the identity of the attacker wasn not in 𝑚_𝑤 signed by the original signer. Not to mention, he does not know psk_𝑠. Under this situation, even if he want to (1) fake the proxy signer key as psk^𝑠, (2) change value 𝑐 to 𝑐^, or (3) randomly select 𝑟^𝑠∈𝑍^∗𝑞, trying to counterfeit the proxy signature, we demonstrate that his attempt deems to fail. We demonstrate the reasons for the failures of these three cases in the following.

Case 1. If an attacker does not know the proxy secret key  psk_𝑠, he cannot generate valid 𝜎_𝑠(=psk_𝑠∗𝑌), 𝑝𝜎sum(=∑^𝑛_𝑖=1𝜎_𝑖), 𝐴(=𝑟_𝑠∗𝑐∗psk_𝑠𝑃), 𝐵(=𝑟_𝑠𝜎_𝑠), and 𝐶(=𝑟_𝑠∗𝑝𝜎sum). Even if he uses a random psk^𝑠 to sign the message, since psk_𝑠=𝑟_𝑠−1∗𝑥_𝑠−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈), he cannot evaluate the right value 𝑥_𝑠−1 for computing 𝐿 to be successfully verified in the verification equation.

Case 2. Because 𝑐 is changed to 𝑐^, this results in at least one of the random numbers 𝑟_𝑖 should also be modified. Without loss of generality, we let 𝑟_𝑖=𝑟₁≠𝑟_𝑠. Accordingly, all the parameters 𝑈(=𝑐𝑃), psk_𝑠(=𝑟_𝑠−1∗𝑥_𝑠−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)), 𝜎_𝑠(=psk_𝑠∗𝑌), 𝑝𝜎sum(=∑^𝑛𝑖=1𝜎_𝑖), 𝐴(=𝑟_𝑠∗𝑐∗psk_𝑠𝑃), 𝐵(=𝑟_𝑠𝜎_𝑠), 𝐶(=𝑟_𝑠∗𝑝𝜎sum), 𝐷(=𝑟_𝑠∗𝑐∗𝑉), and 𝐿(=𝑐∗𝑥_𝑠−1∗𝑉) are changed as well. That is 𝜎^=(𝜎^1,𝜎₂,…,𝜎^𝑠,𝜎_𝑠+1,…,𝜎_𝑛,𝑚,𝑚_𝑤,𝑐^,𝐴^,𝐵^,𝐶^,𝐷^,𝐿^,𝑈^,𝑉). Apparently, the verification equation (∏^𝑛𝑖=1𝑒(𝐷,𝜎_𝑖))⋅𝑒(𝐴,𝑌)=𝑒(𝑐𝑉,𝐶−𝐵)⋅𝑒(𝐿,𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)𝑌)⋅𝑒(𝑈,𝐵) cannot hold. Below, we only show the inequality of portion of the verification equation 𝑒(𝐴^,𝑌)=𝑒(𝑈^,𝐵^): 𝑒𝐴^,𝑌==𝑒𝑒𝑐𝑟^𝑠_∗𝑐^∗psk^𝑠𝑃,𝑌=𝑒𝑐_𝑃,𝑟^𝑠psk^𝑠𝑌≠𝑒(𝑈,𝑃𝐵,)𝑟.^𝑠𝜎_𝑠(11)

Case 3. In this case, if any attacker randomly selects 𝑟^𝑠∈𝑍^∗𝑞, trying to generate the valid proxy signature 𝜎′. Accordingly, the parameters 𝑈(=𝑐𝑃), psk_𝑠(=𝑟_𝑠−1∗𝑥_𝑠−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)), 𝜎_𝑠(=𝑟_𝑠−1∗𝑥_𝑠−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)∗𝑌), 𝑝𝜎sum(=∑^𝑛𝑖=1𝜎_𝑖), 𝐴(=𝑟_𝑠∗𝑐∗psk_𝑠𝑃), 𝐵(=𝑟_𝑠𝜎_𝑠), 𝐶(=𝑟_𝑠∗𝑝𝜎sum), 𝐷(=𝑟_𝑠∗𝑐∗𝑉), and 𝐿(=𝑐∗𝑥_𝑠−1∗𝑉) are all changed. Therefore, the signature now becomes 𝜎^=(𝜎^1,𝜎^2,…,𝜎^𝑠,𝜎_𝑠+1,…,𝜎^𝑛, 𝑚,𝑚_𝑤,𝑐^,𝐴^,𝐵^,𝐶^,𝐷^,𝐿^,𝑈^,𝑉). As in Case 1, the verifier checks whether 𝑒(𝐴′,𝑌)=𝑒(𝑈,𝐵′) holds or not. Apparently, it cannot pass the verification.

(3) Undeniability
As in Section 4.2 proof (second proof), the verifier uses the verification equation: (∏^𝑛𝑖=1𝑒(𝐷,𝜎_𝑖))⋅𝑒(𝐴,𝑌)=𝑒(𝑐𝑉,𝐶−𝐵)⋅𝑒(𝐿,𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)𝑌)⋅𝑒(𝑈,𝐵) to check whether the proxy signature comes from one of the members in the proxy signer group. Since the equation 𝑉(=𝑣𝑃=𝑅+𝐻₀(𝑚_𝑤,𝑅)𝑌_𝑜) includes the original signer’s public key 𝑌_𝑜 and 𝑌=∑^𝑛𝑖=1𝑌_𝑖, it means the original signer and the proxy signer group cannot repudiate their participations in the signature generation.

(4) Anonymity
In the APS generation phase, all the parameters A, B, C, D, and 𝐿 have to be multiplied by 𝑟_𝑠∈𝑍^∗𝑞 to make the proxy signature 𝜎 anonymous. If any attacker wants to know who is the real proxy signer, he must know the value 𝑟_𝑠 to use 𝑟_𝑠−1 for unrandomizing all parameters to get 𝐴^(=𝑐^∗psk^𝑠𝑃), 𝐵(=𝜎^𝑠), 𝐶^(=𝑝𝜎sum^), 𝐷^(=𝑐^∗𝑉), and 𝜎^𝑠(=𝑥_𝑠−1∗𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)∗𝑌). But now 𝜎_𝑖=𝑟_𝑖𝑉,𝑖≠𝑠, even the attacker knows 𝑟_𝑠, without the knowledge of 𝑟_𝑖 and 𝑥_𝑠, he cannot know who the real signer is. Not to mention, he cannot know the value of 𝑟_𝑠. It means that anyone cannot know who signs the signature. Hence, the anonymity holds.

(5) Anonymity Revocation
In our scheme, only the proxy signer knows 𝑟_𝑠−1 and the secret 𝑥_𝑠−1. He can convince the others that he is the real proxy signer by just showing them 𝑟_𝑠−1and the holdness of the equation𝑟_𝑠∗𝑥_𝑠∗𝜎_𝑠=𝐻₂(𝑚_𝑤‖𝑚,𝑉,𝑈)∗𝑌 without revealing 𝑥_𝑠 in polynomial time.

5. Comparisons

In this section, we compare the computational cost between Yu et al.’s APS scheme and ours and summarize the result in Table 2. We denote by 𝑒 the pairing operation, Pm and Pa the point multiplication and point addition on 𝐺₁ respectively, and by 𝑛 the number of proxy signers. In Yu et al.’s APS scheme, the generation and verification of psk should be (2𝑛+1)Pm+𝑛Pa instead of (𝑛+1)Pm operations. Because in Yu et al.’s scheme, the generation and verification of psk are 𝑅=𝑟𝑃 and 𝑠𝑃=𝑅+𝐻₀(𝑚_𝑤,𝑅)𝑌₀, the 𝑠𝑃 should be computed by 𝑛  proxy signers. The APS verification should be (𝑛+1)𝑒+𝑛Pm+2𝑛Pa rather than the original (𝑛+1)𝑒+𝑛Pm+(𝑛+1)Pa as listed in the table of [8]. From Table 2, we can see that our scheme is more efficient than Yu et al.’s protocol.

Table 2 

Comparison of computational costs of our scheme and Yu et al.'s scheme.

6. Conclusions

In 2009, Yu et al. first proposed a one-to-many APS scheme attempting to protect the proxy signer’s privacy while maintaining secrecy to outsiders. However, after analyses, we determined that Yu et al.’s original protocol could not satisfy the anonymous property. Accordingly, we proposed a novel one-to-many APS scheme to reach the goal. Our construction makes use of a random number 𝑟_𝑠, one-way hash function and bilinear pairings to make the proxy signature anonymous. After comparisons, we conclude that our new protocol is a significant improvement against attackers trying to reveal the identity of the real signer and is more efficient in computational cost as demonstrated in Table 2.