Can Schnorr Aggregate Signatures Be Nested Inside Other Schnorr Aggregate Signatures?

Schnorr signatures are a type of digital signature used in various blockchain and cryptocurrency applications. They are designed to provide secure and efficient way for parties to verify the authenticity and integrity of messages without revealing their contents. One of the key features of Schnorr signatures is that they can combine multiple parties’ signatures into a single shared signature, known as an aggregate signature. In this article, we will explore whether it is possible to nest aggregate signatures inside other aggregate signatures using Schnorr protocols.

Understanding Schnorr Signatures

Schnorr signatures are based on the “blind signature” concept, which allows parties to sign messages without revealing their contents to others. They consist of three components: a private key (p), a public value (P), and a hash function (h). The public value P is derived from the private key p using the Schnorr hash function h(P) = r + p^k, where k is an integer parameter.

Aggregate Signatures

Schnorr aggregate signatures are used to combine multiple parties’ signatures into a single shared signature. An aggregate signature consists of a list of signatures (s_i), and the Schnorr hash function for each signature is combined using a hash function hAggregate(s_i).

Can Aggregate Signatures Be Nested Inside Other Aggregate Signatures?

In theory, it is possible to nest aggregate signatures inside other aggregate signatures using Schnorr protocols. However, there are some limitations and potential issues that need to be considered.

One approach to nesting aggregate signatures is to use a combination of the Schnorr hash function and a new hash function hAggregate2(s_i). The idea is to create a new public value P2 by combining two private keys p1 and p2 using the Schnorr hash function: P2 = hAggregate2(P1, P2). Then, we can derive an aggregate signature s2 from this public value using the same Schnorr hash function.

However, there are potential issues to consider:

• Security: Nesting aggregate signatures may introduce new security risks, such as increased complexity and vulnerability to attacks. The nested aggregate signature may be more difficult to verify or extract information from.

• Performance: Creating and verifying nested aggregate signatures can be computationally expensive, especially for large inputs. This may make them less practical for real-world applications.

Example Use Case

To illustrate this concept, let’s consider an example use case where we have two parties (Alice and Bob) who want to sign a message together using Schnorr signatures. We will create a nested aggregate signature inside another aggregate signature.

Let’s assume Alice has private key p1 = e1^p2^k1, where e1 is the public value for signing. She wants to combine her signature with Bob’s signature to form an aggregate signature s_nested(s1, s2), where s1 and s2 are individual Schnorr signatures.

We create a new public value P2 by combining Alice’s private key p1 = e1^p2^k1 with Bob’s private key p2: P2 = hAggregate2(P1, P2) = e1^hAggregate(s1, s2).

Then, we derive an aggregate signature s_nested using the Schnorr hash function: s_nested = hAggregate(s1, s2)

The public value for this nested aggregate signature is P3 = e1^P2.

Conclusion

In conclusion, it is possible to nest aggregate signatures inside other aggregate signatures using Schnorr protocols. However, there are potential security and performance issues that need to be considered when creating such nested signatures. It is essential to carefully evaluate the trade-offs and limitations of this approach before applying it in a real-world scenario.

Recommendations

• Carefully consider the security risks and potential vulnerabilities introduced by nesting aggregate signatures.

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