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Table of contents
1.
Introduction
2.
The RSA Cryptosystem
3.
Implementing RSA
4.
Frequently Asked Questions
4.1.
What is RSA in cryptography?
4.2.
Is RSA cryptography still used?
4.3.
Is RSA impossible to crack?
4.4.
Is RSA asymmetric or symmetric?
4.5.
What makes RSA weak?
5.
Conclusion
Last Updated: Mar 27, 2024
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Introduction to The RSA Cryptosystem

Author Muskan Sharma
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Ashwin Goyal
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Introduction

Hey Readers!!!

Welcome back to another article on topics related to Cryptography.

What is Cryptography?

Cryptography is the technique for securing communication between the sender and the receiver.

In this article, you'll learn about the RSA Cryptosystem

Let’s begin!!

Introduction to The RSA Cryptosystem

The RSA Cryptosystem

The RSA Cryptosystem can be explained as the cryptosystem working in Zn, where n is the product of two different odd primes, p, and q. 

Note that φ(n) = (p-1)(q-1) for such an integer n.

Let's check to ensure that encryption and decryption are inverse operations. Since

ab ≡ 1 (mod φ(n)),

Also we know that,

ab = tφ(n) + 1

Let for some integer t1. x ∈ Zn*, then we can say,

(xb)a ≡ xtφ(n)+1(mod n)

≡(xφ(n))t x (mod n)

≡ 1t x (mod n)

 ≡ x (mod n)

This shows that (xb)a  ≡ x (mod n) if x ∈ Zn/Zn*. 

Also we can express the RSA Cryptosystem as:

Let n = pq, where p and q are primes. Let P = C = Zn, and define

K = {(n, p, q, a, b) : ab ≡ 1 (mod φ(n))}.

Also, for K we can define it as

eK(x) = xb mod n

and

dK(y) = ya mod n

(x, y ∈Zn).The public key comprises the values n and b, whereas the private key comprises the values p, q, and a.

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Implementing RSA

The RSA Cryptosystem has many facets that need to be discussed, such as the specifics of how the cryptosystem is set up, the effectiveness in encrypting and decrypting, and security concerns. Bob(user) utilizes the RSA PARAMETER GENERATION technique, colloquially known as the algorithm shown below, to configure the system.


RSA PARAMETER GENERATOR

Step 1. Create the two large primes p and q so that p  q 

Step 2. n ← pq and φ(n) ← (p − 1)(q − 1)

Step 3. It choose a random b (1 < b < φ(n)) as gcd(b, φ(n)) = 1

Step 4.a ← b-1 mod φ(n)

Step 5.Private keys - (p, q, a) and Public keys - (n, b).

To the factor, n is one of the most obvious attacks on the RSA Cryptosystem. If this is possible, it is easy to calculate (n) = (p -1)(q -1) and then determine the decryption exponent a from b in the same way as Bob did. (It has been hypothesized, but not yet proven, that breaking the RSA Cryptosystem is polynomially comparable to factoring n).

N = pq must undoubtedly be large enough to be computationally infeasible to factor if the RSA Cryptosystem is to be secure. RSA moduli with up to 768 bits in their binary representation can now be factored by factoring algorithms.

To be on the safe side, it is now advised to choose p and q as 1024-bit primes each; n will then be a 2048-bit modulus. The best factoring algorithms now can't factor in a number this big.

Let's look at the arithmetic processes of encryption and decryption now, setting aside for the time being the issue of where to find 1024-bit primes. One exponentiation modulo n is needed to encrypt (or decrypt) data.

Since n is very large, multiprecision arithmetic must be used to conduct calculations in Zn, and the time needed will depend on how many bits n's binary representation contains.

Assume that x and y are positive numbers with binary representations of k and l bits, respectively.

In other words, k = [log x] + 1 and'= [log y] + 1(Assuming the base of the log is 2). Suppose that k l. It is not difficult to derive big-oh upper bounds on how long it takes to complete various operations on x and y using common "grade-school" arithmetic methods.

Concluding the results :

  1. x + y has time complexity O(k)
  2. x − y has time complexity O(k)
  3. xy has time complexity O(kl)
  4. [x/y]  has time complexity O(l(k − l)).
  5. gcd(x, y) has time complexity O(k3).

With the final item, a gcd can be calculated using the algorithm above. It can be demonstrated that the EUCLIDEAN ALGORITHM's needed number of iterations is O(K). The complexity of a gcd calculation is observed to be O(K3) since each iteration executes a long division that takes time O(K2). A more thorough study can be utilized to demonstrate that the complexity is O(K2) instead.

Frequently Asked Questions

What is RSA in cryptography?

A cryptosystem is a collection of cryptographic algorithms utilized for particular security services or purposes, is built around the RSA algorithm.

Is RSA cryptography still used?

RSA is a secure algorithm, yet IoT makers frequently use insecure implementations of it.

Is RSA impossible to crack?

The common encryption algorithm used on the Internet is RSA. Although well recognized, the technique is very difficult to decipher.

Is RSA asymmetric or symmetric?

RSA is an asymmetric encryption which uses two unique but connected keys.

What makes RSA weak?

Organizations prime numbers are significantly easier to factor in when they have weak random number generators, making it easier for attackers to break the process.

Conclusion

This blog has extensively about the RSA Cryptosystem. We hope this blog has helped you enhance your knowledge about what RSA Cryptosystem is and the implementation of RSA.

If you want to learn more deeply about cryptography, check out the excellent content on the Coding Ninjas Website:

What is the Chinese Remainder theorem?The Partial Information Concerning Plaintext Bits for obtaining the semantic securityOptimality of Deception Probabilities in Cryptography

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