Umbra Address Generation

Protocol Description

Each anonymous Umbra address is a standard Solana Ed25519 keypair, derived deterministically from the master seed $S_M$. The derivation uses the NIST-standardized KMAC construction, which acts as a Pseudo-Random Function (PRF).

For the $i$-th Umbra address, a unique 32-byte seed, $\text{seed}_{U_i}$, is generated as follows:

$$\text{seed}_{U_i} \triangleq \text{KMAC}_{S_M}( \text{"Umbra-Address-Generation"} \| i )$$

This output seed is then used to generate a standard Ed25519 keypair, which serves as the user's anonymous Umbra keypair.

$$(sk_{U_i}, pk_{U_i}) \leftarrow \text{Ed25519KeygenFromSeed}(\text{seed}_{U_i})$$

Key Derivation Process

1. Master Seed Input: Uses the 32-byte master seed $S_M$ as the KMAC key
2. Domain Separation: Uses the string $"Umbra-Address-Generation"$ for domain separation
3. Index Uniqueness: Appends the address index $i$ to ensure each address is unique
4. Keypair Generation: Uses the derived seed with standard Ed25519 key generation

Security Analysis

Security Proof: Unlinkability of Umbra Addresses

Goal: Each Umbra keypair $(sk_{U_i}, pk_{U_i})$ must be unlinkable to the master seed $S_M$ and to any other Umbra keypair $(sk_{U_j}, pk_{U_j})$ for $i \neq j$.

Assumption: KMAC is a secure Pseudo-Random Function (PRF).

Argument: The master seed $S_M$ serves as the secret key for the KMAC PRF. By the definition of a secure PRF, the output $\text{seed}_{U_i}$ is computationally indistinguishable from a truly random 32-byte string to any adversary without knowledge of $S_M$. Since the Ed25519 keypair is generated from this pseudorandom seed, the resulting keypair itself, particularly the public key $pk_{U_i}$, appears as a random, unrelated Solana address.

Properties

• Unlinkable: Each Umbra address appears as a random Solana address
• Deterministic: Same master seed always generates the same sequence of addresses
• Indexed: Each address can be generated by incrementing the index parameter
• Standard Compatible: Uses standard Ed25519 keypairs for Solana compatibility
• High Entropy: Each derived seed has full cryptographic entropy