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With the rise of Bitcoin, blockchain which is the core technology of Bitcoin has received increasing attention. Privacy preserving and performance on blockchain are two research points in academia and business, but there are still some unresolved issues in both respects. An aggregate signature scheme is a digital signature that supports making signatures on many different messages generated by many different users. Using aggregate signature, the size of the signature could be shortened by compressing multiple signatures into a single signature. In this paper, a new signature scheme for transactions on blockchain based on the aggregate signature was proposed. It was worth noting that elliptic curve discrete logarithm problem and bilinear maps played major roles in our signature scheme. And the security properties of our signature scheme were proved. In our signature scheme, the amount will be hidden especially in the transactions which contain multiple inputs and outputs. Additionally, the size of the signature on transaction is constant regardless of the number of inputs and outputs that the transaction contains, which can improve the performance of signature. Finally, we gave an application scenario for our signature scheme which aims to achieve the transactions of big data on blockchain.

Since the emergence of Bitcoin [

There are still some flaws on blockchain where privacy preserving and performance are two important aspects. When achieving the characteristics of blockchain, preserving the privacy is the focus of academic research. In this field, Monero and Zcach are representative projects where ring signature, zero-knowledge proof, and other cryptographic technologies play important roles. In addition, achieving rapid trading to meet realistic demands is another challenge that blockchain faces. In this field, lightning network is widely recognized, but there are also some flaws in its theories and implement.

Meanwhile, we know big data has been used in many fields. However, there are still many flaws in the storage, transmission, transaction, and privacy preserving of big data. And blockchain was considered to be an ideal technology for solving these flaws. Thus, we applied our new signature scheme to the transactions of big data on blockchain.

(

(

(

CoinJoin technique.

In addition, Monroe proposed a

In Zcash, a noninteractive zero-knowledge proof [

Give the serial number.

Use zk-SNARK to prove that it holds the user’s private key to generate this commitment.

There,

Bilinear:

Nondegenerate: there exists

Computability: there is an efficient algorithm to compute

There,

The result of the aggregate signature is

Assume that

(

If

(

If

(

When transactions are generated on blockchain, cryptographic signatures are used to judge the legality of the transactions and the identities of the senders [

Without loss of generality, we deal with a single transaction, which is divided into inputs and outputs; the details are shown in Figure

Model of single transaction.

As shown in Figure

For each

According to (

For each

We can obtain that

The left side of (

A malicious attacker impedes

The sum of all the outputs is

Because we know that

In order to modify our basic scheme, this paper combines aggregate signature with the basic scheme to obtain a modified scheme.

Recall that elliptic curve on the finite group

(

(

The modified scheme greatly avoids the drawback in the basic scheme. If a malicious attacker impedes

In Section

We know that

Basic transaction structure.

As shown in Figure

Using the properties of the bilinear map, the left side of the verification equation expands:

Figure

Aggregate transaction structure.

As shown in Figure

New transaction structure.

It is easy to show that the security of our new signature scheme is equivalent to the traditional bilinear aggregate signature. As the aggregate chose-key security model which was proposed in [

The forger wins if the aggregate signature

An aggregate forger

Forged aggregate signature is by at most

An aggregate signature scheme is

Let

Besides, the security of the scheme which is used to hide the amount of the transactions has been analyzed in Section

Big data brings many benefits to our lives. At the same time, there are some drawbacks in big data. Firstly, the utilization of data is poor. Large amounts of data are in the idle state, occupying a lot of storage space. Secondly, there are a lot of drawbacks in the security and privacy of the data. The use of big data exposes personal privacy and other security problems, while big data may be used to do illegal activities by criminals. At the same time, there are some drawbacks in the transmission efficiency and transmission accuracy of data. Blockchain is considered to be an ideal solution to these problems. Based on this, we try to apply our signature scheme to the transactions of big data [

Here, we consider the transactions of big data on blockchain. The infrastructure is based on the P2P network which is the network model of blockchain [

Infrastructure of transaction of data.

We consider the inputs and outputs of a particular transaction, which consists of data inputs, data outputs, and the corresponding amount of outputs and amount of inputs which are described in Figure

Single transaction of data.

In this paper, we have proposed a new signature scheme for the transactions on blockchain based on aggregate signature and ECC. Through our new signature scheme, the amount will be hidden when the transactions contain multiple inputs and outputs [

There are still many interesting problems to be solved. For example, it would be valuable to explore the possibility of achieving a signature scheme which combines our scheme with ring signature. Using our scheme to construct a practical complete application is also another interesting problem [

Because we know that

The authors declare that there are no conflicts of interest regarding the publication of this paper.

This paper is supported by National Key Research and Development Program (nos. 2016YFB0800101 and 2016YFB0800100), State Key Laboratory of Mathematics and Advanced Computing Open Topic (no. 2015A14), and National Natural Science Foundation of China (no. 61602512).