Can there be multiple private keys for one public key

Question

I know that in public-key cryptography, each public key is mathematically linked to a private key. However, I’m wondering if it's possible to have multiple private keys corresponding to the same public key.

Would this break security, or is it actually possible in some cryptographic systems? If so, are there any real-world use cases where this might be necessary or beneficial?

Answer 1

In traditional public-key cryptography systems like RSA and ECDSA, each public key is uniquely associated with a single private key; this one-to-one relationship ensures that only the holder of the corresponding private key can decrypt messages encrypted with the public key or generate valid signatures. Introducing multiple private keys for a single public key is not standard practice and can compromise the security of the system.

Theoretical Possibility

Mathematically, in schemes like RSA, there can be multiple private keys corresponding to the same public key. This arises because the equation e⋅d≡1(modϕ(n)) can have multiple solutions for d (the private exponent) given e (the public exponent) and ϕ(n) (Euler's totient function of n). However, publishing multiple valid private keys corresponding to the same public key can lead to vulnerabilities, as it may provide attackers with additional information to compromise the system.

Alternative Approaches

While assigning multiple private keys to a single public key is not standard, there are cryptographic methods to achieve similar objectives:

1. Threshold Cryptography: In a t-out-of-n threshold scheme, a private key is divided into n shares, and any t of these shares can collaboratively perform cryptographic operations like decryption or signing. This approach enhances security by ensuring that no single party holds the entire private key, and a minimum number of parties must cooperate to perform critical operations.
2. Attribute-Based Encryption (ABE): ABE allows data to be encrypted under a public key associated with a set of attributes. Users possess private keys corresponding to specific attributes, and they can decrypt the data if their attributes satisfy the encryption policy. This method enables fine-grained access control without assigning multiple private keys to a single public key.

Real-World Use Cases

While the direct assignment of multiple private keys to a single public key is uncommon due to security concerns, the alternative approaches mentioned above have practical applications:

• Threshold Cryptography: Used in securing cryptographic keys in distributed systems, ensuring that sensitive operations require collaboration among multiple parties, thereby reducing the risk of key compromise.

• Attribute-Based Encryption: Applied in scenarios like secure data sharing in cloud environments, where access control is based on user attributes rather than individual keys.