Verifiable Secret Sharing (VSS)

Verifiable Secret Sharing (VSS) is a cryptographic protocol that enables a secret, such as a private key, to be divided into multiple shares distributed among a group of participants. The protocol ensures that the secret can only be reconstructed when a sufficient number of shares (a threshold) are combined, while any smaller subset reveals nothing. Crucially, VSS adds a verification layer, allowing each participant to cryptographically confirm that their share is consistent with the others and was generated correctly by the dealer, preventing malicious distribution of invalid shares.

The core innovation of VSS is its use of commitment schemes, like Pedersen commitments, or zero-knowledge proofs. When the dealer distributes shares, they also broadcast public commitments to the secret polynomial used in the sharing process. Each participant can use their share and these public commitments to verify that their share is a valid point on the committed polynomial. This process prevents a dishonest dealer from giving different participants inconsistent shares, which could otherwise prevent the secret from being reconstructed later, even by an honest majority.

VSS is a fundamental building block for secure multi-party computation (MPC) and distributed key generation (DKG) protocols. In blockchain contexts, it is essential for creating decentralized and trust-minimized systems, such as threshold signatures for wallets and consensus mechanisms. By ensuring verifiability, VSS moves beyond simple secret splitting to create robust systems where participants do not need to trust the dealer or each other, only the cryptographic guarantees of the protocol itself.

A classic VSS scheme is Feldman's VSS, which uses the hardness of the discrete logarithm problem. The dealer commits to the coefficients of the secret polynomial using public parameters g^{a_i}. Each participant receives a share f(i) and can verify that g^{f(i)} equals the product of the commitments raised to the appropriate powers. If the verification fails, the participant can broadcast a complaint, leading to the dealer's disqualification or a public reconstruction of that specific share, maintaining the protocol's security.

Verifiable Secret Sharing (VSS) is a cryptographic protocol that distributes a secret—such as a private key—among a group of participants, where a minimum threshold of shares is required to reconstruct it, while also providing cryptographic proofs that the shares are consistent and correctly distributed. This prevents a malicious dealer from giving invalid shares to some participants, which would render the secret irrecoverable. The core innovation over basic schemes like Shamir's Secret Sharing is the addition of verifiability, typically achieved using commitments like Pedersen commitments or Feldman's scheme, which bind the dealer to a specific polynomial without revealing it.

The protocol operates in two main phases: distribution and reconstruction. During the distribution phase, the dealer (or a distributed key generation protocol) uses a random polynomial to generate shares, then broadcasts a public commitment to this polynomial. Each participant receives their private share and can verify its validity against this public commitment. This verification ensures that if the dealer is honest, all shares are consistent and will reconstruct to the same secret. If verification fails, participants can broadcast a complaint, leading to the dealer's disqualification or a recovery process.

A critical application of VSS is in Distributed Key Generation (DKG) for threshold cryptography, such as in threshold signatures or secure multi-party computation (MPC). Here, there is no single trusted dealer; instead, each participant acts as a dealer for a sub-share, and the final secret key is never assembled in one place. VSS guarantees that the combined secret is well-defined and that any subset of honest participants can later collaborate to sign or decrypt. This makes it foundational for decentralized systems requiring robust key management, like blockchain validator networks and custody solutions.

The security properties of VSS are formally defined. It must satisfy secrecy (no information about the secret is revealed to unauthorized subsets), correctness (any authorized subset can reconstruct the same secret), and verifiability (players can verify their shares). Modern implementations often leverage elliptic curve cryptography for efficient commitments and zero-knowledge proofs. Protocols like Asynchronous Verifiable Secret Sharing (AVSS) further enhance resilience in unreliable network environments, making VSS a versatile building block for fault-tolerant distributed systems.

Feldman's Verifiable Secret Sharing (VSS) is a cryptographic protocol that extends basic Shamir's Secret Sharing by enabling participants to verify the correctness of their secret shares without learning the original secret. Developed by Paul Feldman in 1987, it prevents a malicious dealer from distributing inconsistent or invalid shares, a critical vulnerability in non-verifiable schemes. The core innovation is the use of cryptographic commitments, where the dealer publishes public values derived from the secret polynomial's coefficients, allowing each participant to mathematically verify that their share is consistent with these public commitments.

The protocol's security relies on the assumed hardness of the Discrete Logarithm Problem. During the distribution phase, the dealer selects a secret s and constructs a random polynomial f(x) of degree t-1 where f(0) = s. For each coefficient a_i of this polynomial, the dealer computes and broadcasts a commitment g^{a_i} within a public cyclic group. When a participant receives their secret share f(i), they can verify it by checking if g^{f(i)} equals the product of the broadcast commitments raised to the appropriate powers. This ensures the share is a valid point on the committed polynomial.

Feldman's VSS is a cornerstone for building secure distributed systems, most notably in threshold cryptography and Byzantine Fault Tolerant (BFT) consensus protocols. It is the verifiable component underpinning Distributed Key Generation (DKG) protocols, which are essential for decentralized custody in applications like threshold signatures for wallets and random beacons. By guaranteeing that all honest participants work from a consistent secret state, it enables protocols to tolerate a bounded number of malicious actors, typically up to t-1 out of n participants, where t is the threshold.