1.
Introduction
2.
3.
Simplified (t, t)-threshold Scheme
4.
4.1.
What is the basic concept behind threshold cryptography, and how does this help in encryption?
4.2.
Why is TTS used in cryptography?
4.3.
How do threshold cryptosystems help in protecting data?
5.
Conclusion
Last Updated: Mar 27, 2024
Medium

# A Simplified (t, t)-threshold Scheme

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## Introduction

In a world full of diversity, we all have our very own languages that we speak. We know how to hide things from people we don't want to show and make them available only to those we want to show them. In a similar way, cryptography has its own family. Every system has a different style and set of ways. Today we will take a look at the (t, t)-threshold Scheme. Let us find out more below.

The foundation of threshold cryptography is a cryptosystem which secures information. This gets done by encrypting it and spreading it among a cluster of fault-tolerant computers. The message is encrypted with a public key, and the associated private key gets distributed among the persons who are involved. To decrypt an encrypted communication or sign a message using a threshold cryptosystem, more than one participant must join in the decryption or signing process. This number has to be greater than some threshold number.

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## Simplified (t, t)-threshold Scheme

The main topic of discussion here is that we want to see a simplified construction for threshold schemes.

This is for the following special case below:

This construction is applicable to any key set with

and

We should note that it is not necessary for 'm' to be prime in this scheme, nor is it necessary for

So now let us imagine that there is a person D who wants to share the following key:

D will carry out the following steps in Cryptosystem. We can see it below.

1. This is done independently at random.

D secretly chooses t-1 elements of Zm, y1, . . . , yt-1.

2. D computes the following equation.

3. For 1 â‰¤ i â‰¤ t, D will give the share yi to Pi.

It is worth noting that the t participants can compute K using the following formula.

Now the question arises if t-1 participants can compute K.

Clearly, the first "t - 1" participants cannot do so. This is because their shares are "t -1" independent random numbers.

Consider we have the t-1 participants in the following set:

where

The shares are possessed by these t-1 participants.

and

By summing their shares, they can compute

They do not, however, know the random value :

Therefore they do not know the value of K. As a result, we have a scheme with (t, t)-thresholds.

### What is the basic concept behind threshold cryptography, and how does this help in encryption?

Threshold cryptography enables each keyholder to lock a secret such that one other keyholder is not able to unlock the lock on its own. The secret must be unlocked by a certain minimum number(threshold number) of all keyholders.

### Why is TTS used in cryptography?

TTS(Threshold Signature Scheme) technique is used for producing a single digital signature from many people who signed(different signers).

### How do threshold cryptosystems help in protecting data?

Threshold cryptosystems encrypt and distribute secrets among a group of independent, fault-tolerant computers to secure information. The ability of a system to function even after errors or malfunctions is known as fault tolerance.

## Conclusion

We read about a Simplified (t, t)-threshold scheme in the article. We tried to get a simple idea of it and answered a few relevant questions. Refer to our courses and explore Coding Ninjas Studio to find more exciting stuff. You can also look into the interview experiences and solve different problems. Look into our Guided paths, test series, libraries and resources to know more.

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