1.1. Secret Sharing Problem Motivation

So, let us start with the problem of secret sharing what exactly it is motivation real world application. So, for that imagine a banking application and the way a locker is operated. So, I do not know whether you have a locker account in a bank or not, but I do have and the way locker account is maintained or operated in the bank is as follows. Whenever you want to open or get access to your locker, you have to go along with your key. And apart from your key there is another key which is held by the manager.

Secret sharing is a cryptographic concept that is vital for secure data management. It can be understood using the analogy of a bank locker system. In this system, a customer needs two keys to access their locker: one key that they possess and another that is held by the bank manager. Both keys must be used together, meaning neither party can access the locker alone. This setup illustrates the essence of secret sharing where information is divided among multiple parties to ensure security and prevent unauthorized access.

Consider a bank locker like a digital password. You hold one part of the password (your key), while a bank manager holds another. Together, they unlock the locker (or access the information). If someone wanted to steal your secrets, they would need both parts to get past the security.

Another interesting motivation for secret sharing problem is the following. So it is believed that in the 1990’s, the access to Russia's nuclear weapon was done in the following fashion. So, the password or the credential for launching the nuclear weapon was shared among 3 top entities of the country, namely the President, Prime Minister and the Defence minister.

This example highlights the critical use of secret sharing in national security. The authorization for launching nuclear weapons was secured through a system where three high-ranking officials (the President, Prime Minister, and Defense Minister) each held a part of the launch code. To activate the weapon, at least two of these three officials would need to cooperate, significantly reducing the risk of unilateral action that could endanger national security.

Imagine the launch of a nuclear weapon as an extremely sensitive project at work. If only one person had access to the information, it could lead to a disaster. Instead, a team needs to come together to make a decision, ensuring that a consensus is reached before any critical action is taken.

So now, let us abstract both the examples that we had seen by this general problem of what we call as (n, t) secret sharing and this problem or this primitive was independently introduced by Turing award winner, Adi Shamir in 1979.

The (n, t) secret sharing problem formalizes the approaches discussed previously. It states that among 'n' parties, a secret can be reconstructed only if at least 't' of these parties cooperate. For instance, in our earlier example with the nuclear launch codes, if n = 3 (the three officials) and t = 2, then at least two officials must agree to unlock the secret.

Think of a group project where three members need to submit their pieces to complete the project (n=3). However, they decide that at least two members must agree on changes before anything is submitted (t=2). This prevents one member from making a hasty decision that could affect everyone.

So, that is why to make the problem interesting and to model what exactly we had seen in the previous examples, we need the following requirements from this sharing mechanism.

In a secret sharing scheme, two essential requirements must be satisfied: First, no group of fewer than 't' shareholders should be able to reconstruct the secret—this ensures security. Second, if 't + 1' or more shareholders cooperate, they must be able to reconstruct the secret unambiguously—this ensures usability.

Consider a safe that can only be opened with multiple keys. If you have one key, you can't access the safe (security), but if you have more than one, you can open it without confusion (usability).

In some sense, you can imagine that this is the kind of system gives you more security more robustness in the sense that if one of the 3 entities say either the President or the prime minister or the Defence minister gets compromised and leaks the password, then an enemy country can launch the nuclear weapon.

This feature of requiring multiple parties to share the secret enhances the overall security of the system. If only one member is compromised, the system remains secure as long as the second member is not compromised. This layered security concept reduces risk.

Imagine a safe with a combination lock that requires three numbers. If you only know one number and cannot access the others, it becomes challenging to unlock. This means if one combination is leaked, the entire safe isn't compromised unless the other numbers are known.

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