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</style><title>An Identity-Based Blind Signature Scheme Using Lattice with Provable Security</title><div class="original-url"><br><a href="https://www.hindawi.com/journals/mpe/2020/7528571/">https://www.hindawi.com/journals/mpe/2020/7528571/</a><br><br></div><div id="article" role="article" style="text-rendering: optimizeLegibility; font-family: -apple-system-font; font-size: 1.2em; line-height: 1.5em; margin: 0px; padding: 0px;" class="system exported">
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<div class="page" style="text-align: start; word-wrap: break-word; max-width: 100%;"><h1 class="title" style="font-weight: bold; font-size: 1.95552em; line-height: 1.2141em; margin-top: 0px; margin-bottom: 0.5em; text-align: start; display: block; max-width: 100%;">An Identity-Based Blind Signature Scheme Using Lattice with Provable Security</h1><div style="max-width: 100%;"><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">Abstract</h4><p style="max-width: 100%;">With the rapid development of quantum computing and quantum information technology, the universal quantum computer will emerge in the near decades with a very high probability and it could break most of the current public key cryptosystems totally. Due to the ability of withstanding the universal quantum computer’s attack, the lattice-based cryptosystems have received lots of attention from both industry and academia. In this paper, we propose an identity-based blind signature scheme using lattice. We also prove that the proposed scheme is provably secure in the random oracle model. The performance analysis shows that the proposed scheme has less mean value of sampling times and smaller signature size than previous schemes. Thus, the proposed scheme is more suitable for practical applications.</p><span style="max-width: 100%;"></span><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">1. Introduction</h4><p style="max-width: 100%;">Currently, the emergence of quantum computing causes a potential threat to the traditional cryptosystems. In 2011, the first commercial quantum computer “D-Wave One” was worked out, which provided the application of certain cracking algorithms to the traditional public key cryptography with feasible condition. Furthermore, it is because most of mathematical hard problems in the traditional cryptosystems are vulnerable to the strong computing power of quantum computers. Therefore, it is obvious that the influence quantum computers bring to the traditional cryptosystem will permeate into the information security and Internet security of all areas of a country, such as politics, economy, culture, and military.</p><p style="max-width: 100%;">Specifically, it can be explained from two main aspects: Firstly, for the integer factorization problem, the conjecture that an <span style="max-width: 100%;">-</span>bit integer can be decomposed by the <span style="max-width: 100%;">-</span>qubit quantum computer easily is proposed by Beauregard [<a style="text-decoration: none; max-width: 100%;">1</a>]. As for the discrete logarithm problem, Proos and Zalka [<a style="text-decoration: none; max-width: 100%;">2</a>] pointed out that <span style="max-width: 100%;">-</span>bits elliptic curve discrete logarithmic problem [<a style="text-decoration: none; max-width: 100%;">3</a>, <a style="text-decoration: none; max-width: 100%;">4</a>] can be solved by <span style="max-width: 100%;">-</span>qubit quantum computer. Secondly, the valid length of the secret key in traditional cryptosystem will be half of the original length under the attack of quantum adversary.</p><p style="max-width: 100%;">Blind signature was first proposed by Chaum [<a style="text-decoration: none; max-width: 100%;">5</a>] to make electronic money in an electronic cash system. In general, the user can get a valid signature of any message through a blind signature scheme, where the signer knows nothing about the actual message. This special property makes the blind signature used widely. Therefore, a plenty of blind schemes were worked out after the work of Chaum, such as [<a style="text-decoration: none; max-width: 100%;">6</a>, <a style="text-decoration: none; max-width: 100%;">7</a>]. However, those schemes had the significant problem on certificates, which is the core problem in public key infrastructure (PKI) cryptosystem. In 1984, the identity-based (ID-based) public key cryptosystem was worked out by Shamir [<a style="text-decoration: none; max-width: 100%;">8</a>], which is useful to eliminate the serious defect of the PKI cryptosystem. Since then, lots of ID-based blind signature schemes were proposed with efficient performance.</p><p style="max-width: 100%;">As is known to all, most of the above blind signature schemes cannot resist the attack of quantum algorithms. It is because the computational power of the quantum computers is so strong that the hard problems in those schemes are easy to be broken. In order to remove this threat, the postquantum cryptography appears in the vision of cryptographers, which is that the traditional cryptosystem still holds its security under the attack of the quantum adversary. In the postquantum cryptography systems, the lattice-based cryptography is the most promising. Currently, lots of cryptographic protocols have been devised on the lattice, such as [<a style="text-decoration: none; max-width: 100%;">9</a>–<a style="text-decoration: none; max-width: 100%;">11</a>].</p><p style="max-width: 100%;">There are several advantages of the lattice-based cryptography which are worth noting. Firstly, this cryptosystem has got widespread attention in the last decade. Then, this cryptography currently cannot be broken by any algorithms, including quantum algorithms. Moreover, lattice-based cryptography has the same level of security in the average case and the worst case. Finally, the designs of lattice-based schemes are very simple and efficient, including mainly matrix-vector multiplication, linear summation operation, and modulo operation.</p><p style="max-width: 100%;">Taking advantage of these benefits, some blind signature schemes were designed, but several problems included in these schemes make them inapplicable in the real environment. For example, some blind signature schemes lack the formal security proof or describe the ability of the adversary incorrectly. Besides, the efficiency shortcomings in other schemes are too serious to be neglected, such as the scheme proposed by Rückert [<a style="text-decoration: none; max-width: 100%;">12</a>] and the work of Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>]. The main reason for this is that complex algorithms are used in the process of signing or the efficient aborting technology is not involved in these blind signature schemes.</p><p style="max-width: 100%;">In order to improve the practicability of blind signature, a new ID-based scheme on lattice is proposed in this paper, which is more efficient and secure. Specifically, the main contributions of this paper are as follows:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;">Firstly, our blind signature scheme can resist the attack of the malicious quantum adversaries, because it is based on lattice. Meanwhile, we prove that our scheme is secure based on SIS problem in the random oracle model. The lattice cryptosystem also makes it more efficient due to the simple operations involved in lattice-based algorithms.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Secondly, we use the bimodal Gaussian rejection sampling in our scheme to prevent the leakage of critical information, such as the signer’s secret key. Using this aborting technology, it makes the mean value of sampling times needed to generate a valid signature smaller. Additionally, we can get the blind signature with smaller size under this novel technology.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">Finally, because the framework of ID-based cryptosystem is used in our scheme, it means that the additional cost is not needed to manage lots of certificates in our scheme. Therefore, the proposed scheme under this cryptosystem is more practical in the real application.</span></span></span></p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">2. Related Work</h4><p style="max-width: 100%;">In this section, we will mainly talk about the related works on the blind signature schemes. Due to its excellent concealment, blind signature has been studied widely and put into the applications where important data needs to hold its privacy, such as electronic cash (e-cash) [<a style="text-decoration: none; max-width: 100%;">14</a>], electronic voting [<a style="text-decoration: none; max-width: 100%;">15</a>], and oblivious transfer [<a style="text-decoration: none; max-width: 100%;">16</a>].</p><p style="max-width: 100%;">In order to design electronic money used in the e-cash system, Chaum [<a style="text-decoration: none; max-width: 100%;">5</a>] proposed the first blind signature scheme. After the work of Chaum, lots of blind signature schemes were worked out based on PKI cryptosystem, the hardness of which is mostly based on the integer factorization problem or discrete logarithm problem [<a style="text-decoration: none; max-width: 100%;">17</a>–<a style="text-decoration: none; max-width: 100%;">19</a>]. However, as we all know, the issue of certificates’ management is an apparent defect in this cryptosystem. Fortunately, the identity-based public key cryptography was proposed by Shamir [<a style="text-decoration: none; max-width: 100%;">8</a>] to eliminate this drawback.</p><p style="max-width: 100%;">Owing to the good advantages of ID-based cryptosystem, the first ID-based blind signature scheme was worked out by Zhang and Kim [<a style="text-decoration: none; max-width: 100%;">20</a>]. Later, Huang et al. [<a style="text-decoration: none; max-width: 100%;">21</a>] proposed another ID-based blind signature scheme in 2005. In 2008, a generalized ID-based blind signature with bilinear pairings was designed by Kalkan et al. [<a style="text-decoration: none; max-width: 100%;">22</a>]. Then, in 2010, Rao et al. [<a style="text-decoration: none; max-width: 100%;">23</a>] constructed a blind signature scheme on the basis of ID-based digital signature framework proposed by Hess [<a style="text-decoration: none; max-width: 100%;">24</a>]. Following the work of Rao et al, a provably secure randomized blind signature scheme was constructed by Fan et al. [<a style="text-decoration: none; max-width: 100%;">25</a>] using bilinear pairings. Furthermore, there were two other new ID-based blind signature schemes based on bilinear pairings designed by Zhang et al. [<a style="text-decoration: none; max-width: 100%;">26</a>] and Shakerian et al. [<a style="text-decoration: none; max-width: 100%;">27</a>], respectively, in the same year.</p><p style="max-width: 100%;">However, in 2011, He et al. [<a style="text-decoration: none; max-width: 100%;">28</a>] proposed a novel ID-based blind signature scheme using no bilinear pairings. Their work opened up a new direction in the design of the ID-based blind signature scheme, because the new blind signature scheme constructed by them guaranteed both high efficiency and anonymity. Later, a new provably secure and pairing-free ID-based partially blind signature scheme was worked out by Islam et al. [<a style="text-decoration: none; max-width: 100%;">29</a>] in 2016, which was used in an online e-cash system. Besides, this scheme was provably secure in the random oracle model. In 2017, an untraceable ID-based blind signature scheme without pairing for e-cash payment system was proposed by Kumar et al. [<a style="text-decoration: none; max-width: 100%;">30</a>]. Then, James et al. [<a style="text-decoration: none; max-width: 100%;">31</a>] proposed an efficient pairing-free ID-based blind signature scheme with message recovery in 2018.</p><p style="max-width: 100%;">Although ID-based cryptosystem can solve the efficiency drawback of schemes in PKI cryptosystem, it cannot resist the attack of quantum algorithms. In 2010, the first lattice-based blind signature scheme was proposed by Rückert [<a style="text-decoration: none; max-width: 100%;">12</a>], which was provably secure in the random oracle model. Later, in 2017, a novel round-optimal lattice-based blind signature scheme used in the cloud services was constructed by Zhu et al. [<a style="text-decoration: none; max-width: 100%;">32</a>]. Similarly, a new postquantum blind signature scheme on lattice was proposed by Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>] in 2018, in which the unimodal rejection sampling technology was used to improve the probability of generating a valid signature.</p><p style="max-width: 100%;">Unfortunately, the efficiency problem still existed in these schemes because they were designed under the PKI cryptosystem. So some ID-based blind signature schemes were worked out to deal with this disadvantage of previous schemes. In 2014, Zhang and Ma [<a style="text-decoration: none; max-width: 100%;">33</a>] proposed a lattice-based proxy blind signature scheme based on ID-based cryptosystem, whose security was held in the standard model. Then, another ID-based blind signature scheme on lattice was constructed by Gao et al. [<a style="text-decoration: none; max-width: 100%;">34</a>] in 2016, which was based on the standard model. Interestingly, a two-round ID-based blind signature scheme on lattice was still proposed by Gao et al. [<a style="text-decoration: none; max-width: 100%;">35</a>] on the random oracle model in 2017. In addition, this scheme was proved to have the power to resist the selective identity and chosen message attacks to remain unforgeable and unconditionally blind based on the SIS problem.</p><p style="max-width: 100%;">However, no aborting technology or only unimodal rejection sampling was used in these schemes. In 2013, Ducas et al. [<a style="text-decoration: none; max-width: 100%;">36</a>] proposed a modified aborting technology based on the original rejection sampling, called bimodal Gaussians rejection sampling, which reduces the rejecting field between the actual sampling distribution function and the expected sampling distribution function. This means that the signer can generate a valid signature with fewer samples. Additionally, this new aborting technology still keeps the basic ability to prevent the leakage of information of the signer’s secret key. Therefore, using the bimodal Gaussian rejection sampling, a new ID-based blind signature scheme on lattice is constructed in this paper based on the work of Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>]. Our scheme has the excellent ability to resist quantum algorithm and high efficiency, combining the advantages of lattice-based cryptosystem with that of ID-based cryptosystem.</p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3. Preliminaries</h4><p style="max-width: 100%;">In this section, the basic knowledge about lattices will be described firstly. Next, we introduce the Gaussian distribution in detail.</p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.1. Lattices</h5><p style="max-width: 100%;">A lattice is defined as a discrete additive subgroup of n-dimensional Euclidean vector space <span style="max-width: 100%;">.</span> Namely, if are linearly independent vectors in <span style="max-width: 100%;">,</span> a lattice is the set of all integer combinations of these vectors:<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>and the matrix is one base of <span style="max-width: 100%;">.</span> Normally, is described as its corresponding dimension.</p><p style="max-width: 100%;">In particular, the following two types of lattices should be paid more attention, called module lattice:<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></p><h6 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.1.1. Small Integer Solution (SIS) Problem</h6><p style="max-width: 100%;">Given a positive integer <span style="max-width: 100%;">,</span> a matrix <span style="max-width: 100%;">,</span> and a real number <span style="max-width: 100%;">,</span> the SIS problem is to find a nonzero vector such that <span style="max-width: 100%;">,</span> and <span style="max-width: 100%;">.</span> This kind of SIS problem is homogeneous. As for inhomogeneous SIS problem, it is to find a nonzero preimage satisfying <span style="max-width: 100%;">,</span> where <span style="max-width: 100%;">.</span></p><p style="max-width: 100%;">Then, there are two important algorithms used in our protocol, which are applied to generate the secret keys of the trusted third party and the signer.</p><h6 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.1.2. Trapdoor Generation Algorithm</h6><p style="max-width: 100%;">An integer <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">;</span> there is an effective algorithm <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> that can generate a matrix and a basis of the lattice <span style="max-width: 100%;">.</span> Besides, the distribution of the matrix is the uniform distribution in approximately and the orthogonal matrix <span style="max-width: 100%;">.</span></p><h6 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.1.3. General Preimage Sampling Algorithm</h6><p style="max-width: 100%;">There are an integer <span style="max-width: 100%;">,</span> a prime number <span style="max-width: 100%;">,</span> and a positive integer <span style="max-width: 100%;">.</span> The lattice is defined by the matrix <span style="max-width: 100%;">.</span> Additionally, the matrix is a base of the lattice <span style="max-width: 100%;">.</span> If the parameter of the Gaussian distribution <span style="max-width: 100%;">,</span> there is a polynomial-time algorithm <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where is a random matrix defined in <span style="max-width: 100%;">,</span> sampling a matrix in a distribution closing to <span style="max-width: 100%;">,</span> such that <span style="max-width: 100%;">.</span></p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.2. Gaussian Distribution and Bimodal Gaussian Rejection Sampling</h5><h6 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.2.1. Gaussian Distribution</h6><p style="max-width: 100%;">In statistics, the distribution function of continuous Gaussian distribution is <span style="max-width: 100%;">,</span> where is the center and is the standard deviation. Furthermore, if <span style="max-width: 100%;">,</span> we usually make the equation simpler, writing it as <span style="max-width: 100%;">.</span> In the case of lattice <span style="max-width: 100%;">,</span> the function is <span style="max-width: 100%;">.</span> So the discrete Gaussian distribution over is written as <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Meanwhile, the discrete Gaussian distribution defined over is normally described as <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> If the center <span style="max-width: 100%;">,</span> we usually write these two symbols as and <span style="max-width: 100%;">.</span></p><p style="max-width: 100%;">In the following, some theorems on the discrete Gaussian distribution are shown.</p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">Theorem 1. </span><i style="max-width: 100%;">We assume that <span style="max-width: 100%;">,</span> so the following formula holds:<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></i></span></p><p style="max-width: 100%;">Furthermore, if we have <span style="max-width: 100%;">,</span> and for any element <span style="max-width: 100%;">,</span> the following conclusion is made out:<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">Theorem 2. </span><i style="max-width: 100%;">It is described that we have <span style="max-width: 100%;">,</span> where <span style="max-width: 100%;">,</span> and is an element in <span style="max-width: 100%;">.</span> We have<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></i></span></p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">Theorem 3. </span><i style="max-width: 100%;">If the matrix is chosen randomly and <span style="max-width: 100%;">,</span> we have that <span style="max-width: 100%;">,</span> whose distribution is uniform approximately in <span style="max-width: 100%;">.</span></i></span></p><h6 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">3.2.2. Rejection Sampling</h6><p style="max-width: 100%;">The rejection sampling is a useful aborting skill in lattice-based schemes. Speaking in formal terms, when the positive constant and a special distribution are given, we need to find another distribution <span style="max-width: 100%;">,</span> which makes <span style="max-width: 100%;">.</span> So we can say that the distribution is seen as another distribution with the probability <span style="max-width: 100%;">.</span> In general, is the mean value of times to get an effective sample.</p><p style="max-width: 100%;"><i style="max-width: 100%;">(1) Rejection Sampling Lemma.</i> It is assumed that is a distribution whose preimage is <span style="max-width: 100%;">,</span> where <span style="max-width: 100%;">,</span> and maps to <span style="max-width: 100%;">.</span> When <span style="max-width: 100%;">,</span> we can have a constant to give the distribution of the following output:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;"><span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;"><span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">The <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> is given out with the probability is within the statistical distance of another distribution:</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">The output <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> is sent with the probability <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span></span></span></span></p><p style="max-width: 100%;"><i style="max-width: 100%;">(2) Bimodal Gaussian Rejection Sampling.</i> In the original Gaussian rejection sampling, the mean value of repetitions of the sampling is <span style="max-width: 100%;">,</span> when the standard deviation <span style="max-width: 100%;">,</span> where the Gaussian “tail-cut” factor is proportional to the square root of the security parameter.</p><p style="max-width: 100%;">In this paper, we introduce the bimodal Gaussian rejection sampling in our scheme to get a smaller rejection area and signature size. As the paper in [<a style="text-decoration: none; max-width: 100%;">13</a>] mentioned, is considered as signer’s signature. But the form of the signature must be changed if we need to use the bimodal Gaussian rejection sampling in the scheme. Before the signer begins to sign the message, a random bit is sampled. Then, the relevant signature is <span style="max-width: 100%;">.</span> Thus, the probability distribution of is <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> According to the requirement of rejection sampling, firstly, the inequality must be held. Secondly, we need to make as small as possible. For the sake of interpretation, we give out the following formula:<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></p><p style="max-width: 100%;">Because there is a fact in math that the inequality is always true for any <span style="max-width: 100%;">,</span> we can get the value by making get the value instead of the value <span style="max-width: 100%;">.</span> It is easy to see that the mean value in the bimodal Gaussian rejection sampling is smaller than that of original Gaussian rejection sampling. Besides, we know that the size of the final signature on lattice is roughly so that this size of the signature produced by using this rejection sampling is much shorter.</p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">4. Security Model</h4><p style="max-width: 100%;">In this section, the security model of the blind signature will be introduced in detail. Normally, in addition to all kinds of security attributes that a general signature scheme has, a blind signature should have two more security attributes:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(i)</span><span style="max-width: 100%;">Blindness: the signer does not know the specific content of the actual message signed by itself.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(ii)</span><span style="max-width: 100%;">Unforgeability: after a message is signed, the signer who gets the signature of this message cannot link this signature to the details of the corresponding process.</span></span></span></p><p style="max-width: 100%;">In fact, blindness means that a malicious signer can only get information independent of the actual message. In particular, there is a formal game used to describe the blindness.</p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">4.1. Blindness Game</h5><p style="max-width: 100%;">If any probabilistic polynomial-time algorithm cannot win the following game, we will consider the corresponding ID-based signature protocol as blind. In this game, there are two honest users and <span style="max-width: 100%;">.</span> In addition, is considered to be a malicious signer. The game of blindness is defined as follows:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;"> gets the public parameters by querying <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;"> performs <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Namely, can get the secret key of the identity by using Key Extract algorithm.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;"> chooses a random bit secretly. Then it sends a pair of messages <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> to and <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(4)</span><span style="max-width: 100%;"> executes the signature scheme with and <span style="max-width: 100%;">,</span> respectively. The messages input by and are and <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(5)</span><span style="max-width: 100%;">The outputs <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> and <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> received by and are sent to in arbitrary order.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(6)</span><span style="max-width: 100%;"> outputs a bit <span style="max-width: 100%;">.</span></span></span></span></p><p style="max-width: 100%;">It is worth noting that wins the game of blindness if and only if <span style="max-width: 100%;">.</span> Moreover, we consider as the advantage of to win this game.</p><p style="max-width: 100%;">Next, another security game aimed at unforgeability will be defined as below. In this game, acts as the challenger and is an adversary playing as a user.</p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">4.2. Unforgeability Game</h5><p style="max-width: 100%;">We think that can break the unforgeability of an ID-based blind signature scheme, if makes extract queries and issue queries during the time and the corresponding advantage of is at least. Otherwise, this scheme is unforgeability. The game of unforgeability is defined as follows:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;">Setup: after inputting the security parameter <span style="max-width: 100%;">,</span> runs the Setup algorithm to generate the systematic public parameter and the master secret key <span style="max-width: 100%;">.</span> Then, the public parameter is sent to <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Query: there are three kinds of queries that can choose to send to <span style="max-width: 100%;">.</span><span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(a)</span><span style="max-width: 100%;">Hash query: after getting this query, would select a random value and return it to <span style="max-width: 100%;">.</span> It is worth noting that random oracle queries are responded by the challenger consistently.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(b)</span><span style="max-width: 100%;">Extract query: after receiving this query, would run the Key Extract algorithm to get the relevant secret key and give it back to <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(c)</span><span style="max-width: 100%;">Issue query: after obtaining this query, executes the sign algorithm with cooperatively to get the signature <span style="max-width: 100%;">.</span> But before this operation, would get the ’s secret key by performing the extract query. Finally, the signature is given back to <span style="max-width: 100%;">.</span></span></span></span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">Forgery: after the above query phase, will use the useful information to forge a signature corresponding to the message of the user, of which identity is <span style="max-width: 100%;">.</span> Additionally, outputs valid signature pairs <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> If the following conditions are satisfied by these signature pairs, we can conclude that wins this game. Furthermore, is the advantage of to get final success in this game.<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(a)</span><span style="max-width: 100%;">For any and <span style="max-width: 100%;">,</span> we have that <span style="max-width: 100%;">,</span> where and <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(b)</span><span style="max-width: 100%;"><span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(c)</span><span style="max-width: 100%;"> never uses the extract query to get the secret key of the user whose identity is <span style="max-width: 100%;">.</span></span></span></span></span></span></span></p><p style="max-width: 100%;">Generally, an ID-based blind signature is considered to have blindness and unforgeability, if and of any polynomial time adversary are negligible.</p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">5. Our Scheme</h4><p style="max-width: 100%;">In this section, we will introduce our ID-based blind signature scheme in detail. Notably, there are two important algorithms used in our scheme, which are and [<a style="text-decoration: none; max-width: 100%;">37</a>, <a style="text-decoration: none; max-width: 100%;">38</a>]. Meanwhile, public key generator (PKG) is the trusted third party.</p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">5.1. System Setup</h5><p style="max-width: 100%;">After getting the safety parameter and <span style="max-width: 100%;">,</span> PKG performs the following four steps:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;">Choosing a prime number <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">,</span> and <span style="max-width: 100%;">,</span> where <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Executing the algorithm <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where <span style="max-width: 100%;">.</span> These matrices and are the public key and the secret key of PKG, respectively.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">Selecting two secure hash functions <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> and <span style="max-width: 100%;">.</span> Actually, <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(4)</span><span style="max-width: 100%;">Making the parameters <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> public and keeping as a secret.</span></span></span></p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">5.2. Key Extraction Phase</h5><p style="max-width: 100%;">In our scheme, PKG uses the following method to generate the user’s key pair. The key extract phase is shown in Figure <a href="fig1/" target="_blank" style="color: rgb(65, 110, 210); max-width: 100%; text-decoration: underline;">1</a>.<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;">Computing and performing the algorithm <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> We can know that and <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Owing to <span style="max-width: 100%;">,</span> getting <span style="max-width: 100%;">,</span> and choosing rows of randomly and then computing the relevant transposed matrix and finally calculating <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">Computing <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> and <span style="max-width: 100%;">,</span> where is the unit matrix, and then making the computation <span style="max-width: 100%;">.</span> So, and are the user’s public keys and is the corresponding secret key.</span></span></span></p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">5.3. Sign Phase</h5><p style="max-width: 100%;">Essentially, this phase is an interactive three-move identification scheme over lattice based on SIS problem. It is assumed that is the actual message needed to be signed. The specific interaction process is as follows:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;">The signer selects a random vector and calculates <span style="max-width: 100%;">.</span> Then, is transmitted to the user.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Blind: the user chooses two blind factors and and computes <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Noticeably, is the message to be signed and is a random value. Besides, the function is a commitment. Then, is worked out, where <span style="max-width: 100%;">.</span> Finally, is sent to the signer by using the bimodal Gaussian rejection sampling to stop from leaking some information of <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;">BSign: the signer selects randomly. Upon that, it can compute <span style="max-width: 100%;">.</span> Similarly, is sent to the user in the same way to hide relevant information of <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(4)</span><span style="max-width: 100%;">Unblind: the user can get the value of <span style="max-width: 100%;">.</span> Then, is output by the unimodal Gaussian rejection sampling. If <span style="max-width: 100%;">,</span> we make <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where is the rejection region of Gaussian sampling. Otherwise, we have <span style="max-width: 100%;">.</span> Finally, is given to the signer.</span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(5)</span><span style="max-width: 100%;">After holding <span style="max-width: 100%;">,</span> the signer checks whether the value of is or not. If it holds, the blind signature <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> is valid. On the contrary, if <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> and <span style="max-width: 100%;">,</span> the signer restarts the sign phase with the user.</span></span></span></p><p style="max-width: 100%;">The sign phase is shown in Figure <a href="fig2/" target="_blank" style="color: rgb(65, 110, 210); max-width: 100%; text-decoration: underline;">2</a>.</p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">5.4. Verify Phase</h5><p style="max-width: 100%;">In this phase, the verifier should check whether the following conditions are right or not:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;"></span></span></span></p><p style="max-width: 100%;">Actually, <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> is the final signature pair. If the two above conditions are satisfied, we have <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span></p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">5.5. Correctness Analysis Phase</h5><p style="max-width: 100%;">In this section, we mainly talk about the correctness and repetition of our blind signature. Firstly, we have<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></p><p style="max-width: 100%;">Thus, we have the fact that <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> = <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Additionally, on the basis of Theorem <span class="converted-anchor" style="max-width: 100%;">1</span> and rejection sampling lemma, there is with overwhelming probability.</p><p style="max-width: 100%;">Next, because the bimodal Gaussian rejection sampling is used in two places in our scheme, the mean value of repetitions is smaller than that of the original scheme. According to the introduction of Gaussian distribution, we have that<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span></p><p style="max-width: 100%;">In the rejection sampling lemma, we need to keep <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> as small as possible. Therefore, the value of is worked out, where <span style="max-width: 100%;">,</span> and <span style="max-width: 100%;">.</span> Obviously, and are both less than the original values in the general rejection sampling, but not <span style="max-width: 100%;">.</span> Therefore, it means that a valid blind signature can be generated successfully in lesser repetitions, whose specific number is <span style="max-width: 100%;">.</span></p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">6. Security Proof</h4><p style="max-width: 100%;">In this section, we mainly prove that our scheme is blind and unforgeable by using the security model defined in Section <span class="converted-anchor" style="max-width: 100%;">4</span>. In fact, we need a malicious adversary to play games of security with a challenger.</p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">6.1. Blindness</h5><p style="max-width: 100%;">We mainly prove the blindness of our scheme from the indistinguishability of views. Normally, views are the information conveyed between the signer and the users, as the following theorem shows.</p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">Theorem 4. </span><i style="max-width: 100%;">The proposed ID-based blind signature scheme on lattice has blindness.</i></span></p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;"><i style="max-width: 100%;">Proof. </i></span>As the game of blindness shows, a dishonest signer <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> selects two messages and <span style="max-width: 100%;">.</span> Then these messages are sent to two honest users <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> and <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where <span style="max-width: 100%;">.</span> Then this malicious signer plays the game of blindness with and in the interactive way, respectively. Therefore, we can prove that our ID-based blind signature scheme is blind to by showing all outputs of the users are independent of the relevant messages signed. We can see from the proposed scheme that the outputs are and the final signature <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Because we have that <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> is always a random number in the view of <span style="max-width: 100%;">.</span> Therefore, we can only consider two values and <span style="max-width: 100%;">.</span><br style="max-width: 100%;">Firstly, we consider about <span style="max-width: 100%;">.</span> We assume that and are generated in the game of blindness. is held by <span style="max-width: 100%;">.</span> Similarly, is corresponding to <span style="max-width: 100%;">.</span> In our scheme, we can know that <span style="max-width: 100%;">,</span> where can be seen as a random value. Besides, is transmitted by using the bimodal Gaussian rejection sampling. Therefore, after getting and <span style="max-width: 100%;">,</span> cannot link and to their respective messages or <span style="max-width: 100%;">.</span> It is because the distribution of and is both <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> but the output distribution of them is the same as that of under the bimodal Gaussian rejection sampling, which is <span style="max-width: 100%;">.</span> In fact, the mean value of the distribution of should be different from that of <span style="max-width: 100%;">.</span> However, we know these mean values can be considered as a random number. So we set the mean value as uniformly for sake of simplicity. So we can say that the statistical distance between and is 0; namely, <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span><br style="max-width: 100%;">Next, we talk about <span style="max-width: 100%;">.</span> In the proposed scheme, we have that <span style="max-width: 100%;">,</span> where is a blind factor. However, the output way of is different from that of the above challenge <span style="max-width: 100%;">,</span> because the Gaussian rejection sampling used in this place is unimodal rather than bimodal. But this cannot make an influence on the blindness. Similarly, we assume that is the final signature of and is related signature received. Similarly, we set the mean value of distribution of and as <span style="max-width: 100%;">.</span> It is because the value is computed by the signer that the distribution of and is both <span style="max-width: 100%;">.</span> According to the Gaussian rejection sampling, the output distribution of and is the same as that of <span style="max-width: 100%;">,</span> which is <span style="max-width: 100%;">.</span> Therefore, cannot determine the corresponding messages of and from their output distribution. That is, the relevant statistical distance <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span><br style="max-width: 100%;">On the contrary, we assume that gets the corresponding parameters and the secret key by playing the game of blindness with and <span style="max-width: 100%;">.</span> Besides, and are information has after playing this game. It is worth noting that if uses the method of guessing a random value of without any help to win the game of blindness, this probability is <span style="max-width: 100%;">.</span><br style="max-width: 100%;">In addition, for <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">,</span> and are the data exchanged between the signer and the user, when the issue query is performed by the user. Finally, <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> and <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> are returned to the dishonest signer <span style="max-width: 100%;">.</span> For each <span style="max-width: 100%;">,</span> there are two random blind factors that map to <span style="max-width: 100%;">.</span> Thus, and <span style="max-width: 100%;">.</span> Since <span style="max-width: 100%;">,</span> where is the unit matrix, we have<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>In the above equations, are two elements in one view and are two data in another view. Besides, two blind factors and are always included in the equation, which result in the indistinguishability to <span style="max-width: 100%;">.</span> Therefore, the probabilistic polynomial time adversary makes out the right value of successfully with probability <span style="max-width: 100%;">.</span><br style="max-width: 100%;">In a word, our ID-based blind signature scheme on lattice has the security attribute of blindness.</span></p><h5 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">6.2. Unforgeability</h5><p style="max-width: 100%;">In fact, unforgeability ensures that valid signatures can be output by a malicious user at most. is the maximum of queries that this adversary can make to the challenger. As the process of the game of unforgeability, we will give out the specific steps of this game on the basis of the proposed scheme.</p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">Theorem 5. </span><i style="max-width: 100%;">If is a probabilistic polynomial time adversary, it can break our ID-based blind signature on lattice with the nonnegligible probability. So, we can construct a polynomial-time algorithm using as its subroutine to solve the SIS problem with overwhelming probability.</i></span></p><p style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;"><i style="max-width: 100%;">Proof. </i></span>We assume that and are the maximum of queries that can make to the random oracle and the signer. Furthermore, the values of responses of the random oracle are determined in advance. Thus, we have <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where <span style="max-width: 100%;">,</span> because the adversary would make a query to before sending a signature query. As shown in the following content, plays the game of unforgeability with the challenger <span style="max-width: 100%;">:</span><span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(i)</span><span style="max-width: 100%;">Setup: after inputting the security parameter <span style="max-width: 100%;">,</span> the challenger picks a random matrix and a hash function <span style="max-width: 100%;">.</span> Additionally, the random oracle is controlled by <span style="max-width: 100%;">.</span> Then, these systematic public parameters are opened to <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(ii)</span><span style="max-width: 100%;">Query: can make four types of queries to <span style="max-width: 100%;">:</span> query, query, extract query, and issue query. It is worth noting that could maintain four empty lists before answering to those queries, namely, <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">,</span> and <span style="max-width: 100%;">.</span> The specific processes of the answers to these queries will be displayed as follows:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;"> query: as mentioned above, holds an empty list in advance, whose form of item is <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> After receiving an query about the identity <span style="max-width: 100%;">,</span> searches the corresponding item in firstly. If there is an element <span style="max-width: 100%;">,</span> gives to as its response. Otherwise, chooses a matrix at random, whose columns obey the distribution <span style="max-width: 100%;">.</span> Then, computes <span style="max-width: 100%;">.</span> According to Theorems <span class="converted-anchor" style="max-width: 100%;">1</span> and <span class="converted-anchor" style="max-width: 100%;">3</span> in the Gaussian Distribution section, and a random matrix are held with nonnegligible probability. Finally, the new item <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> is inserted into <span style="max-width: 100%;">.</span> Besides, is returned to <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Extract query: after acquiring this query, looks for the corresponding item <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> in firstly. Then gets a random matrix from the matrix <span style="max-width: 100%;">,</span> where <span style="max-width: 100%;">.</span> Moreover, can compute the transposed matrix of the matrix and we assign the value of to <span style="max-width: 100%;">.</span> In the end, is inserted in and is given back to <span style="max-width: 100%;">.</span> If a corresponding item does not exist, picks a random matrix and makes an query. Similarly, the new item <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> is added in the list and the relevant matrix is transmitted to <span style="max-width: 100%;">.</span> Furthermore, calculates <span style="max-width: 100%;">.</span> Then, can compute <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> and give to the adversary as the public key of the user whose identity is <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(3)</span><span style="max-width: 100%;"> query: similarly, the challenger maintains an empty list <span style="max-width: 100%;">,</span> of which item is <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> When launches an query, searches the corresponding item in <span style="max-width: 100%;">.</span> If there is a related item in <span style="max-width: 100%;">,</span> the element is given back to <span style="max-width: 100%;">.</span> Otherwise, the answer to the adversary is a random that is not used yet, <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Besides, the new item is added in <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(4)</span><span style="max-width: 100%;">Issue query: after acquiring this query, searches for the corresponding item <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> on the basis of in <span style="max-width: 100%;">.</span> Then, uses as the secret key to execute the sign algorithm. Indeed, we can get the final signature <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Finally, sends <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> to as a response to this issue query.</span></span></span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(iii)</span><span style="max-width: 100%;">Forgery: after completing valid issue queries, gives out valid message-signature pairs <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> with nonnegligible probability <span style="max-width: 100%;">.</span> It is worth noting that we always have <span style="max-width: 100%;">.</span></span></span></span>If the response to an query is predetermined, namely, <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> can make as the answer of the random oracle with the probability <span style="max-width: 100%;">.</span> Here, is the size of output set of the random oracle <span style="max-width: 100%;">.</span> In other words, is one element in <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> with probability <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Therefore, can make a successful forgery <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> with the probability <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> where comes from <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> As mentioned previously, the query can take place in two places, so we need to talk about the specific scheme in two different scenarios:<span style="max-width: 100%;"><span style="max-width: 100%;"><span style="max-width: 100%;">(1)</span><span style="max-width: 100%;">Scenario 1: is generated by during responding to an issue query made by <span style="max-width: 100%;">.</span> Because is the response of a signature on <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;">,</span></span> we can have <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Thus, we must have that and <span style="max-width: 100%;">.</span> If not, it means that we can find a collision of the hash function <span style="max-width: 100%;">.</span> So, we can make a conclusion that <span style="max-width: 100%;">.</span> Besides, we have <span style="max-width: 100%;">,</span> because <span style="max-width: 100%;">.</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;">(2)</span><span style="max-width: 100%;">Scenario 2: is an answer of the random oracle <span style="max-width: 100%;">.</span> In order to solve the SIS problem, replays the game of unforgeability with <span style="max-width: 100%;">.</span> However, there is something different from the first process of this game. changes the values of response of random oracle <span style="max-width: 100%;">,</span> which is <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span></span> are different random values. According to the General Forking Lemma, can forge a new signature <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> of <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> with the probability <span style="max-width: 100%;">,</span> such that and <span style="max-width: 100%;">.</span> Additionally, the probability of is</span></span></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>Because of <span style="max-width: 100%;">,</span> we can have<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>In addition, and are held. Thus, with overwhelming probability, we have<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>If <span style="max-width: 100%;">,</span> then a valid solution of the SIS hard problem is found. Thus, we need to prove that the probability of is overwhelming. According to the property of minimum entropy of the preimage, we can know that there is another secret key such that <span style="max-width: 100%;">,</span> which is different from <span style="max-width: 100%;">.</span> Additionally, the adversary cannot distinguish the secret key from <span style="max-width: 100%;">,</span> after getting the view <span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"></span><span style="max-width: 100%;">.</span></span> Currently, we assume the secret key used in this game is <span style="max-width: 100%;">.</span> Furthermore, we calculate by the following way:<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>As mentioned above, we have <span style="max-width: 100%;">,</span> so the equation is set up. Then we can get <span style="max-width: 100%;">.</span> Therefore, the event that wants to tell the relevant secret on the basis of could not take place. Then, we have<span style="max-width: 100%;"><span style="max-width: 100%;"></span><span style="max-width: 100%;"><span style="max-width: 100%;"></span></span><span style="max-width: 100%;"></span></span>So, if <span style="max-width: 100%;">,</span> we can get the conclusion that is satisfied. Simultaneously, the adversary cannot tell which secret key is used by the challenger in the game. Thus, is set up with the probability at least.</span></p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">7. Performance Evaluation</h4><p style="max-width: 100%;">In this section, the performance analysis of our scheme is talked about in detail. Generally, we will give out the result of comparison between our scheme and two other representative articles in terms of communication complexity and computing complexity. Specifically, these data are mainly derived from the signature’s size and computational cost of generating system parameters of the relevant scheme. Currently, most of the lattice-based blind signature schemes are proved secure in the random oracle model, where the scheme in [<a style="text-decoration: none; max-width: 100%;">39</a>] proposed by Rückert was the most authoritative in 2010. Besides, another blind scheme was proposed by Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>] in 2018, on which we design our new ID-based blind signature scheme. It is worth noting that we do not consider the computational cost of the key extraction phase of our scheme when we make a comparison of the performance between our scheme and Zhang et al.’s [<a style="text-decoration: none; max-width: 100%;">13</a>] signature scheme because their scheme is a PKI-based blind signature scheme on lattice.</p><p style="max-width: 100%;">Firstly, we show a detailed comparison between our scheme and the blind signature scheme proposed by Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>]. To keep the reasonability, we will use the same way in Zhang et al.’s scheme [<a style="text-decoration: none; max-width: 100%;">13</a>] to choose the public system parameters. Namely, the security level of our scheme is the same as that of the scheme [<a style="text-decoration: none; max-width: 100%;">13</a>] proposed by Zhang et al. owing to the Hermite factor defined in [<a style="text-decoration: none; max-width: 100%;">40</a>], which can reach 80 bits. Table <a href="tab1/" target="_blank" style="color: rgb(65, 110, 210); max-width: 100%; text-decoration: underline;">1</a> shows some important system parameters in these two schemes, where are used to keep the hardness of SIS problem.</p><div data-section="tab1" style="max-width: 100%;"><div style="max-width: 100%;"><div class="scrollable" style="max-width: 100%; overflow-x: scroll; word-wrap: normal;"><table style="max-width: none; font-size: 0.9em; text-align: start; word-wrap: break-word; border-collapse: collapse;"><tbody style="max-width: 100%;"><tr style="max-width: 100%;"><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><div style="max-width: 100%;"><div class="scrollable" style="max-width: 100%; overflow-x: scroll; word-wrap: normal;"><table style="max-width: none; font-size: 0.9em; text-align: start; word-wrap: break-word; border-collapse: collapse;"><tbody style="max-width: 100%;"><tr style="max-width: 100%;"><td colspan="5" style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><hr style="max-width: 100%; background-color: rgba(0, 0, 0, 0.2); height: 1px; border: 0px;"></td></tr><tr style="max-width: 100%;"><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"></td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Zhang et al.’s scheme</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Value</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Our scheme</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Value</td></tr><tr style="max-width: 100%;"><td colspan="5" style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><hr style="max-width: 100%; background-color: rgba(0, 0, 0, 0.2); height: 1px; border: 0px;"></td></tr><tr style="max-width: 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579L65 516L85 496C110 533 150 575 205 575C258 575 300 543 300 481C300 407 232 369 141 339L147 310C163 315 188 321 211 321C268 321 338 284 338 192C338 94 288 40 217 40C160 40 119 68 93 91C85 98 77 97 69 91C60 84 47 71 46 58C44 46 48 35 62 22C75 10 116 -12 162 -12C234 -12 424 62 424 224C424 297 373 359 285 376V378Z" style="max-width: 100%;"></path></g><g transform="matrix(.013,0,0,-0.013,40.516,0)" style="max-width: 100%;"><path d="M226 -163V703H56V676C162 667 165 662 165 584V-43C165 -122 162 -126 56 -136V-163H226Z" style="max-width: 100%;"></path></g></svg></span></td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">1.1</td></tr><tr style="max-width: 100%;"><td colspan="5" style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><hr style="max-width: 100%; background-color: rgba(0, 0, 0, 0.2); height: 1px; border: 0px;"></td></tr></tbody></table></div></div></td></tr></tbody></table></div></div></div><p style="max-width: 100%;">In Table <a href="tab1/" target="_blank" style="color: rgb(65, 110, 210); max-width: 100%; text-decoration: underline;">1</a>, the parameters of the rejection sampling in our scheme are smaller than those of the scheme [<a style="text-decoration: none; max-width: 100%;">13</a>] proposed by Zhang et al. obviously. This is because the bimodal Gaussian rejection sampling is used in our scheme. Specifically, the size of a challenge is determined by the parameter <span style="max-width: 100%;">.</span> In general, should satisfy the condition to keep the correctness error at <span style="max-width: 100%;">.</span> According to rejection sampling lemma, we need to keep the rejection area between the actual distribution and objective distribution as small as possible. In this way, the signature algorithm can generate a valid signature using as few repetitions as possible. On the basis of the property of bimodal Gaussian rejection sampling, we only need to require that instead of <span style="max-width: 100%;">.</span> In this case, the bimodal Gaussian rejection sampling can work with the minimum mean value <span style="max-width: 100%;">.</span> Normally, because and are the mean values of the bimodal Gaussian rejection sampling, they are independent of <span style="max-width: 100%;">,</span> and <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">.</span> For <span style="max-width: 100%;">,</span> we have that when <span style="max-width: 100%;">.</span> This can hold because we have and <span style="max-width: 100%;">.</span> Similarly, we can get the optimal value of while we make that <span style="max-width: 100%;">.</span> However, because is the mean value of the unimodal rejection sampling, we need to only require that <span style="max-width: 100%;">.</span> In addition, the distribution of the final signature is <span style="max-width: 100%;">.</span> By Theorem <span class="converted-anchor" style="max-width: 100%;">1</span>, we can determine that the size of ’s every coefficient is with the probability at least <span style="max-width: 100%;">.</span> Because the value of in our scheme is far less than that in Zhang et al.’s scheme [<a style="text-decoration: none; max-width: 100%;">13</a>], we can get the smaller valid signature that is equal to bits approximately. Additionally, our <span style="max-width: 100%;">,</span> <span style="max-width: 100%;">,</span> and are smaller than those of the scheme proposed by Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>]. This means our blind signature scheme can use less time to generate a valid signature in the same security level. In the end, what we need to emphasize is that our blind signature scheme on lattice is based on the ID-based cryptosystem, which has already stronger efficiency than the PKI cryptosystem.</p><p style="max-width: 100%;">Next, we will give out the comparison between our scheme and the classical scheme proposed by Rückert et al. simply [<a style="text-decoration: none; max-width: 100%;">39</a>]. Here, and are the common system parameters in these two schemes. Moreover, is the bit size of the user’s identity and is the expansive Gaussian parameter in Rückert et al.’s scheme [<a style="text-decoration: none; max-width: 100%;">39</a>]. We can know that the size of final signature in our scheme is <span style="max-width: 100%;">.</span> According to the explanation of Rückert et al. [<a style="text-decoration: none; max-width: 100%;">39</a>], the size of signature is <span style="max-width: 100%;"><span style="max-width: 100%;"></span></span> in their ID-based blind scheme. Obviously, it is easy to make the conclusion that the signature’s size in our scheme is smaller than that of Rückert et al.’s scheme [<a style="text-decoration: none; max-width: 100%;">39</a>] in the random oracle model. In terms of computing complexity, there are only some simple operators involved in our sign algorithm and verify algorithm, such as scalar-multiplication on vector, addition on vector, matrix-vector multiplication, and hash function. However, in sign algorithm and verify algorithm of Rückert et al.’s scheme [<a style="text-decoration: none; max-width: 100%;">39</a>], the complex algorithms are included to generate a valid signature, such as ExtBasis algorithm and SamplePre algorithm. So our scheme is simpler and more efficient than the scheme proposed by Rückert et al. [<a style="text-decoration: none; max-width: 100%;">39</a>].</p><p style="max-width: 100%;">Based on the above result, we can make a conclusion that our scheme has less communicational and computational cost, compared with the latest blind signature scheme on lattice proposed by Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>] and the most authoritative blind signature scheme on lattice proposed by Rückert et al. [<a style="text-decoration: none; max-width: 100%;">39</a>]. Thus, our scheme has more efficient and practical value in applications. Table <a href="tab2/" target="_blank" style="color: rgb(65, 110, 210); max-width: 100%; text-decoration: underline;">2</a> shows the result of relevant comparison in detail.</p><div data-section="tab2" style="max-width: 100%;"><div style="max-width: 100%;"><div class="scrollable" style="max-width: 100%; overflow-x: scroll; word-wrap: normal;"><table style="max-width: none; font-size: 0.9em; text-align: start; word-wrap: break-word; border-collapse: collapse;"><tbody style="max-width: 100%;"><tr style="max-width: 100%;"><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><div style="max-width: 100%;"><div class="scrollable" style="max-width: 100%; overflow-x: scroll; word-wrap: normal;"><table style="max-width: none; font-size: 0.9em; text-align: start; word-wrap: break-word; border-collapse: collapse;"><tbody style="max-width: 100%;"><tr style="max-width: 100%;"><td colspan="4" style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><hr style="max-width: 100%; background-color: rgba(0, 0, 0, 0.2); height: 1px; border: 0px;"></td></tr><tr style="max-width: 100%;"><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Schemes</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Signature size</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Security level (bits)</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Cryptosystem</td></tr><tr style="max-width: 100%;"><td colspan="4" style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><hr style="max-width: 100%; background-color: rgba(0, 0, 0, 0.2); height: 1px; border: 0px;"></td></tr><tr style="max-width: 100%;"><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Zhang et al.</td><td style="max-width: 100%; padding: 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0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">76</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">ID-based</td></tr><tr style="max-width: 100%;"><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">Our scheme</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><span style="max-width: 100%;"><svg height="12.6721pt" version="1.1" viewBox="-0.0498162 -9.28833 63.7354 12.6721" width="63.7354pt" xmlns="http://www.w3.org/2000/svg" xmlns:xlink="http://www.w3.org/1999/xlink" style="max-width: 100%;"><g transform="matrix(.013,0,0,-0.013,0,0)" style="max-width: 100%;"><path d="M766 88L752 113C719 83 690 64 681 64C674 64 672 74 679 103L724 292C758 436 724 448 701 448C680 448 666 442 639 429C594 407 514 350 441 252H439L447 289C476 423 450 448 419 448C398 448 379 441 355 427C307 400 234 344 162 249H160L180 324C203 409 197 448 170 448C144 448 82 413 23 349L35 321C57 343 96 374 108 374C115 374 117 371 111 341C87 227 57 112 24 -6L32 -12C53 -4 81 4 108 6C119 68 134 128 149 171C177 229 309 383 364 383C387 383 388 355 373 282C354 190 330 92 303 -6L309 -12C332 -4 356 3 386 6C396 63 411 122 424 171C458 236 590 383 642 383C658 383 664 369 652 315L603 91C587 20 593 -12 619 -12C642 -12 708 23 766 88Z" style="max-width: 100%;"></path></g><g transform="matrix(.013,0,0,-0.013,14.612,0)" style="max-width: 100%;"><path d="M238 0V26C174 32 166 38 166 104V712C132 700 70 683 18 677V653C81 647 87 645 87 577V104C87 38 78 32 15 26V0H238Z" style="max-width: 100%;"></path></g><g transform="matrix(.013,0,0,-0.013,17.901,0)" style="max-width: 100%;"><path d="M257 449C165 449 37 374 37 209C37 98 119 -12 256 -12C355 -12 473 65 473 226C473 349 381 449 257 449ZM244 416C333 416 380 320 380 204C380 67 329 21 267 21C184 21 130 115 130 241C130 354 184 416 244 416Z" style="max-width: 100%;"></path></g><g transform="matrix(.013,0,0,-0.013,24.531,0)" style="max-width: 100%;"><path 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234C359 143 319 30 219 30C172 30 114 75 114 178C114 275 163 343 195 378C212 397 241 415 269 425C305 382 359 313 359 234Z" style="max-width: 100%;"></path></g><g transform="matrix(.0091,0,0,-0.0091,54.049,3.132)" style="max-width: 100%;"><path d="M290 377C321 398 342 415 358 430C378 450 389 473 389 502C389 578 329 635 238 635H237C184 635 137 610 109 578L64 515L88 493C112 529 154 573 208 573S303 542 303 482C303 409 233 370 141 341L149 308C165 313 190 319 215 319C272 319 341 283 341 193C342 98 292 43 222 43C163 43 122 72 96 94C88 101 79 100 70 94C61 87 47 73 46 60C44 47 48 37 62 23C76 10 118 -12 165 -12C238 -12 430 62 430 223C430 297 379 359 290 375V377Z" style="max-width: 100%;"></path></g><g transform="matrix(.013,0,0,-0.013,58.995,0)" style="max-width: 100%;"><path d="M275 270C275 450 212 609 64 710L45 686C145 604 203 442 203 270S147 -63 45 -147L64 -170C213 -68 275 89 275 270Z" style="max-width: 100%;"></path></g></svg></span></td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">80</td><td style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);">ID-based</td></tr><tr style="max-width: 100%;"><td colspan="4" style="max-width: 100%; padding: 0.25em 0.5em; border: 1px solid rgb(216, 216, 216);"><hr style="max-width: 100%; background-color: rgba(0, 0, 0, 0.2); height: 1px; border: 0px;"></td></tr></tbody></table></div></div></td></tr></tbody></table></div></div></div><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">8. Conclusion</h4><p style="max-width: 100%;">Integrating the advantage of ID-based cryptosystem with lattice-based cryptosystem, we construct an efficient and secure ID-based blind signature scheme in this paper to protect the privacy of confidential data, which can be widely applied to the e-cash and electronic voting system. Moreover, a useful aborting technology, bimodal Gaussian rejection sampling, is used in our scheme to accelerate the speed of generating a valid blind signature. Meanwhile, our scheme is provably secure in the random oracle model, which is on the basis of the SIS problem. By showing the comparison with the scheme of Zhang et al. [<a style="text-decoration: none; max-width: 100%;">13</a>] and that of Rückert et al. [<a style="text-decoration: none; max-width: 100%;">39</a>], we demonstrate the superiority of our scheme in communicational and computational efficiency.</p><p style="max-width: 100%;">To extend our scheme to get other useful properties and complete an original model of evaluating the extended scheme in the real application environment is the future work executed by us.</p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">Data Availability</h4><p style="max-width: 100%;">The data used to support the findings of this study are available from the corresponding author upon request.</p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">Conflicts of Interest</h4><p style="max-width: 100%;">The authors declare that they have no conflicts of interest.</p><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">Acknowledgments</h4><p style="max-width: 100%;">This work was supported by the National Natural Science Foundation of China (nos. 61772224, 61932016, and 61972294).</p></div><div style="max-width: 100%;"><h4 style="font-weight: bold; font-size: 1em; margin: 1em 0px; max-width: 100%;">References</h4><ol style="max-width: 100%;"><li style="max-width: 100%;">S. 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This is an open access article distributed under the <a rel="license" href="http://creativecommons.org/licenses/by/4.0/" target="_blank" style="color: rgb(65, 110, 210); max-width: 100%; text-decoration: underline;"> Creative Commons Attribution License</a>, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.</p></div></div></div></div></body></html>