Concept:
RSA Algorithm:
Step 1:Calculate value of n = p × q, where p and q are prime no.’s
Step 2:calculate Ø(n) = (p-1) × (q-1)
Step 3:consider d as a private key such that Ø(n) and d have no common factors. i.egreatest common divisor (Ø(n) ,d )= 1
Step 4:consider e as a public key such that (e × d) mod Ø(n) = 1.
Step 5:Ciphertext = message i.e. memod n.
Step 6:message= cipher text i.e. cdmod n.
Calculation:
To find the value of 'd' in the RSA algorithm, we need to calculate the modular multiplicative inverse of 'e' modulo φ(n), where n is the product of the two prime numbers p and q, and φ(n) is the Euler's totient function.
Given:
p = 13
q = 5
e = 7
ciphertext = 6
First, calculate n:
n = p ×q
n = 13 ×5
n = 65
Next, calculate φ(n):
Ø(n) =(p - 1) ×(q - 1)
Ø(n) = (13 - 1) ×(5 - 1)
Ø(n)= 12 ×4
Ø(n) = 48
Now, we need to find the modular multiplicative inverse of 'e' moduloØ(n). In other words, we need to find 'd' such that (e * d) mod Ø(n) = 1.
Using the extended Euclidean algorithm, we can find 'd':
Step 1:
48 = 7 ×6 + 6
7 = 6 ×1 + 1
Step 2:
6 = 1 * 6 + 0
Since the remainder in the last step is 1, we can conclude that the greatest common divisor of 7 and 48 is 1. Therefore, 'd' exists, and it is the coefficient of 7 in the equation:
1 = 7 - 6 ×1
So, 'd' is equal to 7.
Now, to decrypt the ciphertext using the private key (d, n), we can use the formula:
plaintext = (ciphertextd) mod n
Substituting the values:
plaintext = (67) mod 65
Calculating this:
plaintext = 279936 % 65
plaintext = 46
Therefore, the value of 'd' is 7 and the plaintext (decrypted value) of the ciphertext '6' using the private key (d, n) is 46.
So, the correct answer is 7, 46.
FAQs
So, the correct answer is 7, 46.
How to solve RSA algorithm problem? ›
Steps in RSA Algorithm
- Choose two large prime numbers (p and q)
- Calculate n = p*q and z = (p-1)(q-1)
- Choose a number e where 1 < e < z.
- Calculate d = e-1mod(p-1)(q-1)
- You can bundle private key pair as (n,d)
- You can bundle public key pair as (n,e)
RSA Algorithm Example
- Choose p = 3 and q = 11.
- Compute n = p * q = 3 * 11 = 33.
- Compute φ(n) = (p - 1) * (q - 1) = 2 * 10 = 20.
- Choose e such that 1 < e < φ(n) and e and φ (n) are coprime. ...
- Compute a value for d such that (d * e) % φ(n) = 1. ...
- Public key is (e, n) => (7, 33)
- Private key is (d, n) => (3, 33)
Step:
- 1) Calculate value of n = p × q, where p and q are prime no.' ...
- 2) calculate Ø(n) = (p-1) × (q-1)
- 3) consider d as public key such that Ø(n) and d has no common factors.
- 4) consider e as private key such that (e × d) mod Ø(n) = 1.
- p =7, q= 11, e = 13.
- Use step 2 and 4 of RSA algorithm to calculate private key.
The choice of "e" in the RSA algorithm depends on the values of "p" and "q" and the requirement that "e" must be coprime to the totient of n. The value of "e" can be any number that meets this requirement, but commonly used values are 3, 17, and 65537.
What is the formula for RSA algorithm? ›
➢ To create an RSA public/private key pair, here are the basic steps: 1- Choose two prime numbers, p and q such that p ≠ q . 2- Calculate the modulus, n = p × q. 3- Calcuate ϕ( n ) = ( p – 1 ) × ( q – 1 ). 4- Select integer e such that gcd (ϕ( n ), e) = 1 and 1 < e < ϕ( n ).
What is the math behind RSA algorithm? ›
The RSA cryptosystem is composed of three steps: Key generation: Each user u generates two primes p,q, calculates n=pq and φ(n)=(p−1)(q−1), picks a random e (which must be relatively prime to φ(n)) and calculates d=e−1(modφ(n)). The user publishes the public key pubu=(n,e) and keeps for herself the private key priu=d.
What is the trick to find D in RSA algorithm? ›
1 Answer
- All must insure that ed≡1modλ(n), where λ is the Carmichael function. ...
- Instead of ed≡1modλ(n), we can use ed≡1modφ(n), where φ is Euler's totient. ...
- The question uses the later method, and asks how to calculate d=e−1modφ(n); that is, by definition, the integer d∈[0,φ(n)) with φ(n) dividing ed−1.
Since the truth values of ( p ∧ q ) is F and ( p ∧ q ) → q is T, from the table, the truth values of p and q are either T and F respectively or F and T respectively or both F.
How does the RSA algorithm work? ›
RSA utilizes a private and public key pair. The private key is kept secret and known only to the creator of the key pair, while the public key is available to anyone. Either the public or private key can be used for encryption, while the other key can be used for decryption.
Suppose we pick the primes p=3457631 and q=4563413. (In practice we might pick integers 100 or more digits each, numbers which are strong probable primes for several bases.) Suppose we also choose the exponent e=1231239 and calculate d so e d ≡ 1 (mod φ(n)).
What is a real life example of RSA? ›
These are some real-world examples that demonstrate the usage of RSA encryption in practice: Securing email messages in email providers. Encrypting messages in messaging apps and chat rooms. Securing P2P data transfer.
How to choose p and q in RSA? ›
The keys for the RSA algorithm are generated in the following way: Choose two large prime numbers p and q. To make factoring harder, p and q should be chosen at random, be both large and have a large difference.
What is the formula to find the value of E? ›
Euler's number (e) is a mathematical constant such that y = e x is its own derivative. The value of e is approximately 2.71828 (e is an irrational number, so any decimal representation of e will be approximate). Two common ways of calculating Euler's number are e = lim n → ∞ ( 1 + 1 n ) n and e = 1 + 1 1 !
Is an example for RSA? ›
Example 1:
This example uses prime numbers 7 and 11 to generate the public and private keys. Explanation: Step 1: Select two large prime numbers, p, and q. Step 2: Multiply these numbers to find n = p x q, where n is called the modulus for encryption and decryption.
How can I improve my RSA algorithm? ›
A modify RSA algorithm is proposed using “n” distinct prime numbers. A pair of a random number and their modular multiplicative inverse is used to increase the security of the RSA algorithm. Key generation, encryption and decryption time of Modified RSA (MRSA) algorithm to break the system is significantly higher.
How do you solve algorithm problems? ›
Step-by-Step Strategy to Solve Complex Algorithm
- Step 1: Understand The Problem Statement.
- Step 2: Identify the Appropriate Algorithm.
- Step 3: Plan Your Solution.
- Step 4: Implement The Algorithm.
- Step 5: Analyze Time And Space Complexity.
- Step 6: Test And Debug.
- Step 7: Optimize And Refine.
- Step 8: Documentation.
The Mathematics behind RSA. In RSA, we have two large primes p and q, a modulus N = pq, an encryption exponent e and a decryption exponent d that satisfy ed = 1 mod (p - 1)(q - 1). The public key is the pair (N,e) and the private key is d. C = Me mod N.
![[Solved] Using 'RSA' algorithm, if p = 13, q = 5 and e = 7, t (2024)](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAAEAAAABCAQAAAC1HAwCAAAAC0lEQVR42mP8/h8AAvMB+NzmkbcAAAAASUVORK5CYII=)