Table 2 Digital signatures scheme comparison

Aitzhan [41]

Problems of providing transaction security in decentralized smart grid energy trading

Blockchain, multi-signed, and anonymous encrypted traffic

To overcome the problem of General Byzantine and to defend against double payment attacks in any electronic payment system

Although the performance of centralized solutions will eventually outperform decentralized solutions, this does not negate the need for decentralized solutions.

Yuan [42]

Amount will be hidden in blockchain big data transactions with multiple inputs and outputs

Elliptic curve discrete logarithm and bilinear mapping.

According to the security analysis, the potential forgery of the attacker cannot be realized, and the security performance and the aggregate signature are basically the same.

The evaluation by the aggregated signature time, aggregate verification time, and signature space size proves to be superior to other signature schemes.

Bonneau et al. [49]

In the blockchain Bitcoin transaction, the modern enterprise deploys a complex internal control system

ECDSA compatible threshold signature

Integrate two-factor security measures to increase Bitcoin’s security potential and bring it closer to the widely used currency

The total execution time is small compared to the time required for Bitcoin transactions to be confirmed on the blockchain, with an average of 10 min. Therefore, this system is efficient enough to work well in practice.

AS

Dikshit and Singh [50]

All Bitcoin transactions are recorded and stored in the publicly available database of the blockchain

ECDSA and threshold signature

Potential to significantly increase Bitcoin’s security

The program is more practical than previously proposed, users can get different weights according to their needs

Qian et al. [56]

Solving the privacy protection and performance issues of the blockchain

Based on the discrete logarithm problem, instead of constructing a bilinear map-based.

Length of the aggregate signature was independent of the number of users

Reducing the computational overhead of the signature and verification process, reducing the storage overhead of the blockchain, and improving the communication efficiency

Li et al. [57]

Used to secure the blockchain network over existing classical channels.

Public and private keys are generated by the Bonsai trees with RandBasis algorithm from the root keys.

Secure against the adaptively chosen message attack in the random oracle

Not only ensure the randomness, but also construct the lightweight nondeterministic wallets

Zhu et al. [37]

In the process of building blockchain system

Elliptic curve pairs based on bilinear mapping group system

It can protect the owner’s unforgery and trader’s non-repudiation.

Using exponent times and element length calculation complexity of linear group and communication/storage costs indicate that the program performance is better than previous

GS

Benjamin [44]

Bitcoin transactions in the blockchain

ECDSA

Security of cryptosystem based on the elliptic curve is derived from the computation infeasibility of the discrete logarithm problem of elliptic curve.

Same as ECDSA

Guo et al. [55]

Guaranteeing validity of EHRs encapsulated in blockchain

Multiple authorities to generate and distribute public/private keys.

By sharing the secret pseudorandom function seeds among authorities, the collusion attack can be defended against.

Avoiding the escrow problem and conforms to the mode of distributed data storage in the blockchain

Shen and Ring [43]

In the transaction of the strongly dispersed anonymous password currency Monero

Ring signature

Hides the volume of transaction transactions between the sender and the receiver to protect privacy

The actual size of the signature is smaller than estimated

RS

Mercer et al. [46]

Compatible with blockchain library and implements Ethereum smart contract

Ring signature

Integrity, unforgeability, and anonymity

Since the compromise between anonymity guarantees and program availability is at the discretion of the user, it is more expensive to implement in Ethereum.

Ren et al. [60]

Suitable for the blockchain with lower bandwidth cost

Non-interactive zero-knowledge with Compact Linear Knowledge of Exponent Assumption

This scheme is anonymous and unforgeable in the standard model.

This scheme could reduce the signature size and pairing computations in the verification process. When the ring size is large, the effect of our improvements is obvious.

Tian et al. [38]

Fair contract signing protocol for privacy protection on blockchain

Based on blind signatures and verifiable signatures

By setting the three aspects of the security definition, verifying that the hypothesis is fulfilled, and having the ability to resist fraud, the security of the solution can be proved.

Evaluate the performance of signatures and protocols better than previous ones, from block generation time and the cost of communications through fair-contract-signing agreements

BS

Shentu [45]

In block chain transactions, a centralized coin mixer is prevented from mixing Bitcoins with multiple inputs and multiple outputs.

Blind signature based on elliptic curve

Resists super attackers

10.5 times faster based on Rabin than RSA-based version

Wu et al. [47]

Bitcoin transactions where multiple participants in the blockchain have Bitcoin accounts while maintaining the anonymity of multiple owners

Blind signature and threshold signature

Prevents attackers from altering transaction information

Calculate the computational complexity of the generated key, the modular multiplication time, the signature and verification of the use of the operation time to illustrate its efficiency

Andreev [48]

Bitcoin transactions in the blockchain

Blind signature

The signing party can provide services for storing private keys and authentication transactions without knowing the funds being transferred and has confidentiality.

Similar to the overhead of blind signatures

Cruz and Kaji [51]

E-voting system under blockchain

Bitcoin protocol and blind signature

Protects the privacy and anonymity of voters

The computational cost is very low and does not require much computing power

Fu et al. [52]

A transaction with a public key as an account address on a blockchain

Blind signature.

With anonymity and privacy protection

Introducing agents at the payment stage, shortening transaction confirmation time and improving transaction efficiency

PS

Lin et al. [54]

Deploying in e-business, cloud computing, and blockchain.

Bilinear groups as the underlying tool.

Secure against existential forgery on adaptively chosen message and ID attack

Avoiding the shortcomings of the use of public-key certificates

Sato and Matsuo [39]

When the underlying encryption algorithm (Hash function and digital signature) compromise.

Long-term signature scheme based on ETSI

When the compromise of the signature scheme occurs, this scheme can avoid the change of the key pair and hard fork.

When changing the hash algorithm, the consumption of the block size depends on the output length of the new hash function and the number of transactions in the block (the number of mutual references between the transaction’s hash values) and is therefore superior to other schemes.

Other Signatures

Chalkias et al. [53]

Applying specific chain/graph structures to decrease key generation, signing, and verification costs and signature size

Blockchain architecture and Merkle tree-based signature schemes.

BPQS outperforms existing hash-based algorithms when a key is reused for reasonable numbers of signatures.

BPQS supports a fallback mechanism to allow for a practically unlimited number of signatures if required.