Polynomial finite fields; ElGamal decryption

In summary: Thank you for the explanation!In summary, the conversation discusses using ElGamal encryption and decryption, specifically in regards to finding r^-a in finite fields. The example from the textbook is used to demonstrate how to solve for r^-a using Fermat's Little Theorem and the properties of finite fields.
  • #1
LANS
24
0

Homework Statement


Given some ElGamal private key, and an encrypted message, decrypt it.


Homework Equations


Public key (F_q, g, b)
Private key a such that b=g^a

Message m encrypted so that r=g^k, t=mb^k

Decrypt: tr^-a = m

The Attempt at a Solution



My problem is finding r^-a. I don't quite get polynomial finite fields, and I can't figure out how to get x such that r^x = r^-a in F_q.



I have an example from the textbook:
Public key (F_23, 7, 4), private key a=6, encrypted message r=21, t=11

The textbook then does 11*21^-6 = 11*21^16 = 11*2^16 = 11*9 = 7

I understand why 11*2^16 = 7 in F_23, but I don't get how 21^-6 = 21^16 and how 21^16 = 2^16.

Any help is appreciated.
 
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  • #2
LANS said:
I understand why 11*2^16 = 7 in F_23, but I don't get how 21^-6 = 21^16 and how 21^16 = 2^16.

Since 23 is a prime number, the field F23 is essentially just the set {0,1,2,...,22} with addition and multiplication carried out modulo 23.

Do you know Fermat's Little Theorem? It says that for any prime p and and any integer a, we have ap-1 ≡ 1 (mod p), which you can use to show 21-6 = 2116 in F23.

Also, since addition in F23 is done modulo 23, you have x = 23 - x for all x ∈ F23, which gives you 2116 = 216.
 
  • #3
Thanks, that helps.

Yes, I do know Fermat's little theorem, I feel silly now for not thinking of it.
 

1. What are polynomial finite fields?

Polynomial finite fields, also known as Galois fields, are mathematical structures used in cryptography. They are finite fields that consist of a fixed number of elements, each represented by a polynomial. These fields are used in many cryptographic algorithms, including the ElGamal encryption scheme.

2. How does ElGamal decryption work?

ElGamal decryption is a process used to decrypt ciphertext encrypted with the ElGamal encryption scheme. It involves using the private key to calculate the inverse of a modular exponentiation, which results in the original plaintext. The decryption process is based on the difficulty of solving the discrete logarithm problem in polynomial finite fields.

3. What is the role of polynomial finite fields in ElGamal decryption?

Polynomial finite fields play a crucial role in ElGamal decryption. The encryption and decryption keys used in ElGamal encryption are elements of a polynomial finite field. The encryption and decryption operations involve modular exponentiations, which are performed in the polynomial finite field. The difficulty of solving the discrete logarithm problem in these fields ensures the security of the ElGamal encryption scheme.

4. How are polynomial finite fields chosen for use in ElGamal decryption?

The choice of a polynomial finite field for use in ElGamal decryption depends on the security requirements of the system. The field must have a large number of elements to make it difficult to solve the discrete logarithm problem. The commonly used fields are based on prime numbers or irreducible polynomials over prime fields.

5. What are some advantages of using polynomial finite fields in ElGamal decryption?

There are several advantages of using polynomial finite fields in ElGamal decryption. These fields allow for efficient computation of modular exponentiations, which are the main operations involved in ElGamal encryption and decryption. Additionally, the use of polynomial finite fields provides a higher level of security, as it is difficult to solve the discrete logarithm problem in these fields. They also allow for the use of smaller key sizes compared to other encryption schemes, making them more efficient for use in resource-constrained environments.

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