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Table of contents
What are Threshold Schemes
Types of Threshold Schemes
Shamir Scheme
Visual Threshold Schemes
Frequently Asked Questions
How do Threshold signatures work?
What is the access structure of threshold schemes?
What are some benefits of cryptography?
What are the different kinds of encryption schemes?
Which encryption is faster?
Last Updated: Mar 27, 2024

What are Threshold Schemes

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Ashwin Goyal
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Cryptography is a branch of computer science focusing on securely accessing and sending info between two or more parties without using a third party.  In practice, these techniques keep info or secrets between parties safe from enemies. 

In the big picture, cryptography attempts to construct secure ecosystems that can function with or without the participation of trusted entities. Threshold schemes are one of the most secure approaches for carrying out cryptographic operations. 

what are threshold schemes

This article will discuss what are threshold schemes in detail. We will discuss its methodology along with its application. We will also discuss the types of threshold schemes. So without further ado, let’s get started!

What are Threshold Schemes

Threshold cryptography gets its name from the idea that an individual keyholder cannot open a lock on their own. Instead, there must be a certain number of keyholders – a "threshold" number working together to open the lock. 

Let’s have a look at a real-life example. In the later 20th century, when nuclear weapons were introduced, the control of nuclear weapons depended upon a similar “two-out-of-three” presence. The three members involved were the Defense Minister, the Defense Ministry, and the President. It means to control nuclear weapons, two out of three members among them should be present.

In threshold schemes, the key is distributed among users such that no single one can deduce the secret without the consent of others. Threshold schemes are complex because keyholders must be able to collaborate without knowing each other's key. 

The threshold schemes overcome the flaws in traditional cryptography because we can access the info by using a threshold number of keys, not all of them. We can still access data even if some of the keys are affected or destroyed.

threshold schemes diagram


Following is the methodology of threshold schemes:

  • Let t, w be positive integers, while t ≤ w. Here, t and w are the numbers of key holders.
  • (t, w)-threshold scheme shares a key K among a set of w participants.
  • Any t participants can use their keys and find the value of K, but it won't be possible with t − 1 participant.


Threshold schemes provide higher levels of security than many other cryptographic algorithms. As a result, it was adopted by many cryptographic protocols, processes, and applications.

Their applications can be found in various industries, including the Internet of Things, cloud computing, authentication, ad-hoc and sensor networks, etc.

A generalised application can be said as:

  • It implements a digital sign technique in which no single component can ever access the signing key.
  • Even if one component is compromised. The output, encryption and decryption must be implemented correctly.
  • Even though some (but not all) randomness sources are biased or unavailable, they generate unbiased randomness.
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Types of Threshold Schemes

There are the following types of techniques where we use threshold schemes

  • Shamir Threshold Scheme
  • Visual Threshold Schemes

Shamir Scheme

Shamir threshold scheme is a cryptographic algo. It is an algo in which a key is divided into parts, with each participant receiving a unique portion, and some or all of the parts are required to rebuild the key.

Counting on all participants to combine the key may take time and effort. Thus, we utilise the threshold approach, in which any k of the components is sufficient to reconstruct the original secret.

shamir schemes

Suppose that our key is k= 1234.

We will divide the key into 6 parts (n=6), where any subset of 3 parts (s=3)is sufficient to reconstruct the key. Let’s suppose two nums at random, 166 and 94.

a1,a2 value

We will now make a polynomial to produce divided keys,

polynomial fxn

We will then construct 6 parts for the polynomial: 


We give each participant a different single point (both and f(x)).


To reconstruct the key, any 3 points will be enough.

Let us take these three points,


Now we will use Lagrange algo to compute the polynomial:




The key is the constant value of the polynomial, i.e, k=1234.

Visual Threshold Schemes

So far, we've discussed threshold schemes as finite groups. But isn’t there any possibility of having visual or infinite field cryptography? Naor and Shamir proposed that the rectangular image, made up of black and white pixels, can be used for visual cryptography. They studied a threshold scheme scenario. They said images must be made up of black and white pixels for visual threshold schemes. 

When superimposing many black-and-white pixels of images. The share of images is printed on transparencies. The reconstruction is done "visually" by stacking the shares. As a result, a scheme of this type is referred described as a visual threshold scheme.

visual threshold scheme

Frequently Asked Questions

How do Threshold signatures work?

Threshold signatures are digital signs in which signers can create groups in which only specified group subsets can sign on the group's behalf.

What is the access structure of threshold schemes?

The access structure of a threshold scheme is a collection of subsets that are authorised to generate signs.

What are some benefits of cryptography?

Some benefits of cryptography include Confidentiality, Authentication, Data Integrity and Non-repudiation.

What are the different kinds of encryption schemes?

There are two types of encryption today: symmetric and asymmetric encryption. The name is derived from using the same key for encryption and decryption.

Which encryption is faster?

Symmetric encryption is faster than asymmetric encryption.


This article briefly discussed what are threshold schemes. We discussed its methodology along with its application. We will also discuss the types of threshold schemes. We hope this blog has helped you understand what are threshold schemes. If you like to learn more, you can check out our articles: 


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