RSA Encryption Problem: Finding d Using Euclidean Algorithm

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In summary, the problem involved finding the value of d, which is the multiplicative inverse of e mod (p-1)(q-1). The lecturer used the extended Euclidean algorithm to find d=11787, which is the unique positive value that satisfies 3*d = 1 mod 17680. This can also be found using the formula d = (e^-1) mod (p-1)(q-1). More information can be found in "A Concrete Introduction to Higher Algebra" by Lindsay Childs.
  • #1
muso07
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Homework Statement


This is a problem my lecturer did in class but I'm a bit confused.

We have 2 large primes p=137 and q=131, choose e = 3
n = pq = 137.131 = 17947
(p-1)(q-1) = 136.130 = 17680

Compute d = e-1 mod (p-1)(q-1) = 3-1 mod 17680 = 11787.

This is where I'm confused. How did he get d=11787?

Homework Equations

The Attempt at a Solution


I think I'm supposed to use the Euclidean algorithm to find d, but he didn't show how.
 
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  • #2
for d to be the multiplicative inverse of 3 mod 17680 you need 3*d = 1 mod 17680; there will be a unique positive d < 17680 for which this is true, and this d can be found using the extended euclidean algorithm. the following page will be helpful:
http://www-math.cudenver.edu/~wcherowi/courses/m5410/exeucalg.html

for your problem you will start
17680 = 5893(3) + 1
and the euclidean algorithm terminates quickly because
3 = 3(1) + 0
rearranging the line above you get
1 = 17680 - 5893(3)
so you know that d = -5893, which is 11787 mod 17680.
 
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  • #3
Lindsay Childs in "A Concrete Introduction to Higher Algebra" deals with RSA details if you need more information.
 

Related to RSA Encryption Problem: Finding d Using Euclidean Algorithm

1. What is RSA encryption?

RSA encryption is a form of public-key cryptography that allows for secure communication over an insecure network. It uses a pair of keys (public and private) to encrypt and decrypt data, ensuring confidentiality and authenticity.

2. How does RSA encryption work?

RSA encryption works by using the mathematical properties of prime numbers. The public key is generated by multiplying two large prime numbers, while the private key is calculated using these prime numbers. The encryption process involves raising a message to the power of the public key and taking the remainder when divided by a larger number. The decryption process involves raising the encrypted message to the power of the private key and taking the remainder when divided by the same larger number.

3. What makes RSA encryption secure?

RSA encryption is secure because it would take an unfeasibly long time for someone to factor the large numbers used in the key generation process. This means that it is computationally infeasible for someone to determine the private key from the public key, making it virtually impossible to decrypt the message without the private key.

4. Are there any weaknesses in RSA encryption?

One potential weakness of RSA encryption is that if the prime numbers used in key generation are not large enough, it becomes easier for someone to factor them and determine the private key. Additionally, if someone gains access to the private key, they can decrypt all past and future messages encrypted with the corresponding public key.

5. How is RSA encryption used in real-world scenarios?

RSA encryption is commonly used in online communication, such as email and messaging, to ensure the confidentiality and authenticity of the messages. It is also used in digital signatures, secure online transactions, and secure login systems. Many government and financial institutions also use RSA encryption to protect sensitive data.

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