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Table of contents
Pairing-based cryptography
Identity-based encryption (IBE)
Bilinear Diffie-Hellman Assumption
Boneh-Franklin Identity-based Cryptosystem 
Frequently Asked Questions
How does identity-based encryption function?
What are cryptographic identifiers?
What is the motivation for proposing identity-based encryption?
What exactly is identity-based cryptosystem authentication?
Will an identity-based cryptosystem eventually replace PKI?
Last Updated: Mar 27, 2024

Boneh-Franklin Identity-Based Cryptosystem

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Privacy is a significant worry for everyone in today's environment. In today’s world, messages you transmit are encoded into ciphertext before being transferred across the communication channel. When the other person receives it, it is decrypted to reveal the message you wrote. A cryptosystem is a system that performs this process. In this blog, we will be discussing the Boneh-Franklin identity-based cryptosystem. We will also be discussing pairing-based cryptography. Let's get started.

Boneh-Franklin Identity-Based Cryptosystem


cryptosystem is also known as a cipher system. A cryptosystem can be implemented using human methods, machine methods, or software. It consists of the following: 

  • Encryption method, 
  • Decryption algorithm
  • A well-defined triad of text spaces: plaintexts, ciphertexts, and key texts.
Cryptosystem vector image

The encryption algorithm converts plaintext to ciphertext for any given key text

The decryption method transfers the ciphertext to the plaintext for the appropriate key text. 

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Pairing-based cryptography

Pairing-based cryptography uses a mapping to design or analyze cryptographic systems by pairing components from two cryptographic groups into a third group

Digital signature

Some examples of pairing-based cryptography systems are:

  • Digital signature 
  • Creation of crypto-keys 
  • Functional encryption 
  • Attribute-based encryption 
  • production of anonymous credentials 

Identity-based encryption (IBE)

Pairing-based cryptography includes a section called identity-based encryption (IBE). It generates a public key, such as an email address, using a person's identity. IBE enables the sender to encrypt the data without requiring the recipient's public key to be sent or validated. This is useful if exchanging public keys beforehand is impossible or inconvenient.

The Boneh-Franklin Identity cryptosystem is appropriate for encrypting huge volumes of plaintext. Pairing is done through an elliptic curve to build many cryptographic systems. The difficulty of the bilinear Diffie-Hellman assumption is used to secure the Boneh-Franklin Identity-based cryptosystem.

Bilinear Diffie-Hellman Assumption

The Bilinear Diffie-Hellman (BDH) assumption is necessary to achieve cryptosystem security. Most discrete-log and Diffie Hellman issues are reducible to BDH, despite the lack of proven decline from any of those problems to BDH. 

  • Let e: G1G1! G2 is the bilinear mapping, where G1 and G2 are cyclic groups
  • Many-to-one reduction from any BDH must involve an efficient mapping': G2! G1, where '(gx) = f(x)P
  • Use the function before or after the call to Oracle BDH in the reduction.
  • If f(x) = axn +b for any constants a,b,n, then it could be utilized as an oracle for a probabilistic polynomial time solution to the Diffie-Hellman problem in G2.

Boneh-Franklin Identity-based Cryptosystem 

Consider the conditions:

  • q: prime number
  • G1, G2: groups of order q 
  • Pairing equation: G1 × G2 → G3
  • P: generator of G2, and let 
  • n: positive integer
  • M: master key

M = (Mpub , Mpriv)       

Here we can denote Mpriv = s and Mpub = sP

We know that h1 : {0, 1}* → G1 \ {0} and h2 : G2 → {0, 1}^n are public hash functions. 

For user key generation, we can see that: 

For any user U: 


 eq2and eq3

For encryption

Here the plaintext is of set {0, 1}^n

For every x ∈ {0, 1}^n

  • Take a random value r ∈ Zq*.
  • Calculate  
    • y1 = rP
    • eq4

Therefore, ciphertext y = (y1, y2)

For decryption:

Now we know the ciphertext y = (y1, y2), hence calculate eq5 

The value received for variable x is the decrypted plaintext.

Frequently Asked Questions

How does identity-based encryption function?

Identity-based encryption is a public-key encryption in which users generate a public key from a known unique identifier. An authenticated third-party server creates the secret key from the public key.

What are cryptographic identifiers?

Identity-based cryptography is public-key cryptography. An email address, domain name, or actual IP address can all be a part of the public string identifiers.

What is the motivation for proposing identity-based encryption?

A public key encryption method in 1984, in which the public key may be any string. Identity-Based Encryption (IBE) refers to encryption systems of this sort. The initial motivation for identity-based encryption was to make certificate administration in email systems easier.

What exactly is identity-based cryptosystem authentication?

Identity-based cryptosystem authentication is a procedure that ensures an entity's identification by using an authentication method that confirms the entity's identity.

Will an identity-based cryptosystem eventually replace PKI?

ID-based PKCs will replace traditional public key infrastructures.


This article incorporates information about the Boneh-Franklin identity-based cryptosystem and explains pairing-based cryptography.

If you think this blog has helped you enhance your knowledge, or if you want to learn more, check out our articles:


  1. Cryptosystem
  2. Cryptography
  3. What Do You Mean by Cryptosystem?
  4. Public Key Cryptography
  5. Monte Carlo Simulation


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