Commitment scheme

In cryptography, a commitment scheme allows one to commit to a chosen value (or chosen statement) while keeping it hidden to others, with the ability to reveal the committed value later. Commitment schemes are designed so that a party cannot change the value or statement after they have committed to it: that is, commitment schemes are binding. Commitment schemes have important applications in a number of including secure coin flipping, zero-knowledge proofs, and secure computation.

A way to visualize a commitment scheme is to think of a sender as putting a message in a locked box, and giving the box to a receiver. The message in the box is hidden from the receiver, who cannot open the lock themselves. Since the receiver has the box, the message inside cannot be changed—merely revealed if the sender chooses to give them the key at some later time.

Interactions in a commitment scheme take place in two phases:

In simple protocols, the commit phase consists of a single message from the sender to the receiver. This message is called the commitment. It is essential that the specific value chosen cannot be known by the receiver at that time (this is called the hiding property). A simple reveal phase would consist of a single message, the opening, from the sender to the receiver, followed by a check performed by the receiver. The value chosen during the commit phase must be the only one that the sender can compute and that validates during the reveal phase (this is called the binding property).

The concept of commitment schemes was first formalized by Gilles Brassard, David Chaum, and Claude Crepeau in 1988, but the concept was used without being treated formally prior to that. The notion of commitments appeared earliest in works by Manuel Blum,Shimon Even, and Shamir et al. The terminology seems to have been originated by Blum, although commitment schemes can be interchangeably called bit commitment schemes—sometimes reserved for the special case where the committed value is a bit.

Suppose Alice and Bob want to resolve some dispute via coin flipping. If they are physically in the same place, a typical procedure might be:

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