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Table of contents
1.
Introduction
2.
ElGamal Cryptosystem
3.
Index Calculus
4.
Frequently Asked Questions
4.1.
Why do we use index calculus algorithms?
4.2.
What is one-way function notation?
4.3.
Which cryptosystem is based on a discrete logarithm?
4.4.
Can we use algorithms based on a discrete logarithmic problem for a small prime modulo?
5.
Conclusion
Last Updated: Mar 27, 2024
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Calculus but Index Calculus Method

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Introduction

Calculus is a mathematical concept that studies the continuous change or rate of change in a particular problem. In this blog, we will discuss calculus with the help of the index calculus method. The primary motive of this blog is to make you understand how we can use calculus in cryptography.

Cryptography is a field that studies how to develop efficient algorithms for the encryption of data. Algorithms mainly developed are based on mathematical expressions, so it will be difficult to breach the algorithms using computational power.

Index Calculus Method

The index calculus method is also one of the mathematical concepts that are used to implement cryptography. But before we discuss the relationship between cryptography and index calculus, we need to understand the ElGamal Cryptosystem and discrete logarithm problem.

ElGamal Cryptosystem

The ElGamal Cryptosystem is a public key cryptography system based on the discrete logarithm problem. A discrete logarithm problem is finding an exponent in a particular computational time or time complexity. A discrete Logarithm problem is defined as: 

In a multiplicative group (G, ·), an element α ∈ G has order n, and an element β ∈ (α).

Find the unique integer x, 0 ≤ x ≤ n − 1, such that x= β. We will denote integer x by logx= β; it is called the discrete logarithm of β.
 

A discrete logarithm problem follows the concept of a one-way function.  A one-way function is a function that is easier to implement in one way and hard to revert or inverse the implementation. As mentioned, ElGamal Cryptosystem is based on discrete logarithm problems, so let’s understand the one-way function using the ElGamal Cryptosystem.
 

The ElGamal Cryptosystem is defined as, suppose p is a prime in the Discrete Logarithm problem in (Zp ∗ , ·) is infeasible, and α ∈ Zp ∗ be a primitive root of p. and define K = {(p, α, a, β):  = a mod p}.

The values p, α, and β are the public keys, and a is the private key. For K = (p, α, a, β), and for a (secret) random number k ∈ Zp−1, define enc(x, k) = (y1, y2), where y1 =  k mod p and y2 = x k mod p.

For y1, y2 ∈ Zp ∗ , define dec(y1, y2) = y2(y1 a ) −1 mod p.


Example

Let's assume Person A wants to send the message X = 911 to person B. 

Suppose prime number p = 19 and primitive root α = 3. Now let's consider a = 5.

ElGamal example

Now, let's consider k = 3 as the random integer chosen by person A.

ElGamal example

And

ElGamal example

We will send y1 and y2 as ciphertext, and Person B will decipher this using the decryption formula.
 

According to the definition 

ElGamal example

By solving the above equation, we get X = 911 which is the message sent from person A to person B.
 

Points to remember

  • Always remember that a value should be secret and only shared with the receiver on a secured channel to implement the decryption. 
     
  • In ElGamal Cryptosystem, the prime number p and the value will be public.
     
  • Try to use a huge prime number because a small prime number can be easily computed, and an eavesdropper can guess the secret key very quickly.
     
  • The above problem is an example of a one-way function because, as you can observe, it is not easy to think or compute the value of exponent a for a huge integer.
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Index Calculus

There are many algorithms to solve the discrete logarithmic problem, such as the shanks algorithm, The Pohlig-Hellman algorithm, The Pollard Rho algorithm, and many more.

The index calculus algorithm is also one of the methods to solve the discrete logarithmic problem. 

Index calculus algorithm is used to solve a specialized discrete logarithmic in ZP* where p is the prime number and is a primitive root element of modulo p. The index calculus method is divided into two steps. But before we discuss those steps, you need to know about the factor base. The factored base is a set of B, including small prime integers.
 

Step 1

It is a pre-computation step in which you need to find out the logarithms of the set B primes in factor base. With the following expression, we will build congruences modulo p. With the following expression, we will build C congruences modulo p. Here C is bigger than B, and C = B+10.

index calculus expression

For 1 ≤ j ≤ C, we can write these congruences in the following expression.

index calculus expression

In order to generate the C congruences in the desired form, take a random value of x and compute expresssion it to determine all the factor bases in B.
 

Step 2

In this step, we will compute the discrete logarithm of a particular element with the help of discrete logarithms of set B and denote it as.

index calculus expression

Here, s is a random integer((1 ≤ s ≤ p − 2).

When we factor the on factor base B, we will obtain the following congruence form.

index calculus expression

Frequently Asked Questions

Why do we use index calculus algorithms?

We use the index calculus algorithm to solve discrete logarithmic problems.

What is one-way function notation?

A one-way function is a function that is easier to implement in one way and hard to revert or inverse the implementation.

Which cryptosystem is based on a discrete logarithm?

ElGamal Cryptosystem is based on the DLP or discrete logarithmic problem.

Can we use algorithms based on a discrete logarithmic problem for a small prime modulo?

You can use it, but it is not recommended because, for small prime modulo, the attacker can easily compute the plaintext.

Conclusion

In this blog, we discussed the ElGamal cryptosystem and discrete logarithmic problems. We have also discussed the index calculus method to solve discrete logarithmic problems.

To learn more about cryptography, check out the following articles.

 

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