Threshold Signature Explained— Bringing Exciting Applications with TSS

10 min readFeb 12, 2020

— A deep dive into threshold signature without mathematics by ARPA’s cryptographer Dr. Alex Su

Threshold signature is a distributed multi-party signature protocol that includes distributed key generation, signature, and verification algorithms.

In recent years, with the rapid development of blockchain technology, signature algorithms have gained widespread attention in both academic research and real-world applications. Its properties like security, practicability, scalability, and decentralization of signature are pored through.

Due to the fact that blockchain and signature are closely connected, the development of signature algorithms and the introduction of new signature paradigms will directly affect the characteristics and efficiency of blockchain networks.

In addition, institutional and personal account key management requirements stimulated by distributed ledgers have also spawned many wallet applications, and this change has also affected traditional enterprises. No matter in the blockchain or traditional financial institutions, the threshold signature scheme can bring security and privacy improvement in various scenarios. As an emerging technology, threshold signatures are still under academic research and discussions, among which there are unverified security risks and practical problems.

This article will start from the technical rationale and discuss about cryptography and blockchain. Then we will compare multi-party computation and threshold signature before discussing the pros and cons of different paradigms of signature. In the end, there will be a list of use cases of threshold signature. So that, the reader may quickly learn about the threshold signature.

I. Cryptography in Daily Life

Before introducing threshold signatures, let’s get a general understanding of cryptography. How does cryptography protect digital information? How to create an identity in the digital world? At the very beginning, people want secure storage and transmission. After one creates a key, he can use symmetric encryption to store secrets. If two people have the same key, they can achieve secure transmission between them. Like, the king encrypts a command and the general decrypts it with the corresponding key.

But when two people do not have a safe channel to use, how can they create a shared key? So, the key exchange protocol came into being. Analogously, if the king issues an order to all the people in the digital world, how can everyone proves that the sentence originated from the king? As such, the digital signature protocol was invented. Both protocols are based on public key cryptography, or asymmetric cryptographic algorithms.

“Tiger Rune” is a troop deployment tool used by ancient emperor’s, made of bronze or gold tokens in the shape of a tiger, split in half, half of which is given to the general and the other half is saved by the emperor. Only when two tiger amulets are combined and used at the same time, will the amulet holder get the right to dispatch troops.

Symmetric and asymmetric encryption constitute the main components of modern cryptography. They both have three fixed parts: key generation, encryption, and decryption. Here, we focus on digital signature protocols. The key generation process generates a pair of associated keys: the public key and the private key. The public key is open to everyone, and the private key represents the identity and is only revealed to the owner. Whoever owns the private key has the identity represented by the key. The encryption algorithm, or signature algorithm, takes the private key as input and generate a signature on a piece of information. The decryption algorithm, or signature verification algorithm, uses public keys to verify the validity of the signature and the correctness of the information.

II. Signature in the Blockchain

Looking back on blockchain, it uses consensus algorithm to construct distributed books, and signature provides identity information for blockchain. All the transaction information on the blockchain is identified by the signature of the transaction initiator. The blockchain can verify the signature according to specific rules to check the transaction validity, all thanks to the immutability and verifiability of the signature.

For cryptography, the blockchain is more than using signature protocol, or that the consensus algorithm based on Proof-of-Work uses a hash function. Blockchain builds an infrastructure layer of consensus and transaction through. On top of that, the novel cryptographic protocols such as secure multi-party computation, zero-knowledge proof, homomorphic encryption thrives. For example, secure multi-party computation, which is naturally adapted to distributed networks, can build secure data transfer and machine learning platforms on the blockchain. The special nature of zero-knowledge proof provides feasibility for verifiable anonymous transactions. The combination of these cutting-edge cryptographic protocols and blockchain technology will drive the development of the digital world in the next decade, leading to secure data sharing, privacy protection, or more applications now unimaginable.

III. Secure Multi-party Computation and Threshold Signature

After introducing how digital signature protocol affects our lives, and how to help the blockchain build identities and record transactions, we will mention secure multi-party computation (MPC), from where we can see how threshold signatures achieve decentralization. For more about MPC, please refer to our previous posts which detailed the technical background and application scenarios.

MPC, by definition, is a secure computation that several participants jointly execute. Security here means that, in one computation, all participants provide their own private input, and can obtain results from the calculation. It is not possible to get any private information entered by other parties. In 1982, when Prof. Yao proposed the concept of MPC, he gave an example called the “Millionaires Problem” — two millionaires who want to know who is richer than the other without telling the true amount of assets. Specifically, the secure multiparty computation would care about the following properties:

Privacy: Any participant cannot obtain any private input of other participants, except for information that can be inferred from the computation results.
Correctness and verifiability: The computation should ensure correct execution, and the legitimacy and correctness of this process should be verifiable by participants or third parties.
Fairness or robustness: All parties involved in the calculation, if not agreed in advance, should be able to obtain the computation results at the same time or cannot obtain the results.

Supposing we use secure multi-party computation to make a digital signature in a general sense, we will proceed as follows:

Key generation phase: all future participants will be involved together to do two things: 1) each involved party generates a secret private key; 2) The public key is calculated according to the sequence of private keys.
Signature phase: Participants joining in a certain signature use their own private keys as private inputs, and the information to be signed as a public input to perform a joint signature operation to obtain a signature. In this process, the privacy of secure multi-party computing ensures the security of private keys. The correctness and robustness guarantee the unforgeability of the signature and everyone can all get signatures.
Verification phase: Use the public key corresponding to the transaction to verify the signature as traditional algorithm. There is no “secret input” during the verification, this means that the verification can be performed without multi-party computation, which will become an advantage of multi-party computation type distributed signature.

The signature protocol constructed on the idea of secure multiparty computing is the threshold signature. It should be noted that we have omitted some details, because secure multiparty computing is actually a collective name for a type of cryptographic protocol. For different security assumptions and threshold settings, there are different construction methods. Therefore, the threshold signatures of different settings will also have distinctive properties, this article will not explain each setting, but the comparative result with other signature schemes will be introduced in the next section.

IV. Single Signature, Multi-Signature and Threshold Signature

Besides the threshold signature, what other methods can we choose?

Bitcoin at the beginning, uses single signature which allocates each account with one private key. The message signed by this key is considered legitimate. Later, in order to avoid single point of failure, or introduce account management by multiple people, Bitcoin provides a multi-signature function. Multi-signature can be simply understood as each account owner signs successively and post all signatures to the chain. Then signatures are verified in order on the chain. When certain conditions are met, the transaction is legitimate. This method achieves a multiple private keys control purpose.

So, what’s the difference between multi-signature and threshold signature?

Several constraints of multi-signature are:

The access structure is not flexible. If an account’s access structure is given, that is, which private keys can complete a legal signature, this structure cannot be adjusted at a later stage. For example, a participant withdraws, or a new involved party needs to change the access structure. If you must change, you need to complete the initial setup process again, which will change the public key and account address as well.
Less efficiency. The first is that the verification on chain consumes power of all nodes, and therefore requires a processing fee. The verification of multiple signatures is equivalent to multiple single signatures. The second is performance. The verification obviously takes more time.
Requirements of smart contract support and algorithm adaptation that varies from chain to chain. Because multi-sig is not naturally supported. Due to the possible vulnerabilities in smart contracts, this support is considered risky.
No anonymity, this is not able to be trivially called disadvantage or advantage, because anonymity is required for specific conditions. Anonymity here means that multi-signature directly exposes all participating signers of the transaction.

Correspondingly, the threshold signature has the following features:

The access structure is flexible. Through an additional multi-party computation, the existing private key sequence can be expanded to assign private keys to new participants. This process will not expose the old and newly generated private key, nor will it change the public key and account address.
It provides more efficiency. For the chain, the signature generated by the threshold signature is not different from a single signature, which means the following improvements : a) The verification is the same as the single signature, and needs no additional fee; b ) the information of the signer is invisible, because for other nodes, the information is decrypted with the same public key; c) No smart contract on chain is needed to provide additional support.

In addition to the above discussion, there is a distributed signature scheme supported by Shamir secret sharing. Secret sharing algorithm has a long history which is used to slice information storage and perform error correction information. From the underlying algorithm of secure computation to the error correction of the disc. This technology has always played an important role, but the main problem is that when used in a signature protocol, Shamir secret sharing needs to recover the master private key.

As for multiple signatures or threshold signature, the master private key has never been reconstructed, even if it is in memory or cache. this short-term reconstruction is not tolerable for vital accounts.

V. Limitations

Just like other secure multi-party computation protocols, the introduction of other participants makes security model different with traditional point-to-point encrypted transmission. The problem of conspiracy and malicious participants were not taken into account in algorithms before. The behavior of physical entities cannot be restricted, and perpetrators are introduced into participating groups.

Therefore, multi-party cryptographic protocols cannot obtain the security strength as before. Effort is needed to develop threshold signature applications, integrate existing infrastructure, and test the true strength of threshold signature scheme.

VI. Scenarios

1. Key Management

The use of threshold signature in key management system can achieve a more flexible administration, such as ARPA’s enterprise key management API. One can use the access structure to design authorization pattern for users with different priorities. In addition, for the entry of new entities, the threshold signature can quickly refresh the key. This operation can also be performed periodically to level up the difficulty of hacking multiple private keys at the same time. Finally, for the verifier, the threshold signature is not different from the traditional signature, so it is compatible with old equipments and reduces the update cost. ARPA enterprise key management modules already support Elliptic Curve Digital Signature Scheme secp256k1 and ed25519 parameters. In the future, it will be compatible with more parameters.

2. Crypto Wallet

Wallets based on threshold signature are more secure because the private key doesn’t need to be rebuilt. Also, without all signatures posted publicly, anonymity can be achieved. Compared to the multi-signature, threshold signature needs less transaction fees. Similar to key management applications, the administration of digital asset accounts can also be more flexible. Furthermore, threshold signature wallet can support various blockchains that do not natively support multi-signature, which reduces the risk of smart contracts bugs.

Conclusion

This article describes why people need the threshold signature, and what inspiring properties it may bring. One can see that threshold signature has higher security, more flexible control, more efficient verification process. In fact, different signature technologies have different application scenarios, such as aggregate signatures not mentioned in the article, and BLS-based multi-signature. At the same time, readers are also welcomed to read more about secure multi-party computation. Secure computation is the holy grail of cryptographic protocols. It can accomplish much more than the application of threshold signatures. In the near future, secure computation will solve more specific application questions in the digital world.

About Author

Dr. Alex Su works for ARPA as the cryptography researcher. He got his Bachelor’s degree in Electronic Engineering and Ph.D. in Cryptography from Tsinghua University. Dr. Su’s research interests include multi-party computation and post-quantum cryptography implementation and acceleration.

About ARPA

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