The Quadratic Residue Calculator is a valuable tool used to determine quadratic residues, a significant concept in number theory. By applying a specific formula, this calculator computes the quadratic residue of a given number 'a' under a prime modulus 'p'. Understanding its formula and functionality can aid in various mathematical calculations and cryptographic applications.
Formula of Quadratic Residue Calculator
The formula used by the Quadratic Residue Calculator is as follows:
QR(a, p) = a^((p-1)/2) % p
In this formula:
- 'a' represents the number for which you intend to find the quadratic residue.
- 'p' denotes the prime modulus.
The quadratic residue is calculated by raising 'a' to the power of ((p-1)/2) and then taking the modulus 'p' of the result.
Table of General Terms
To assist, here's a table of general terms related to quadratic residues. These terms are often search and can be helpful without requiring users to perform calculations each time:
| Term | Description |
|---|---|
| Quadratic Residue | The result of the Quadratic Residue Calculator |
| Prime Modulus | The prime number used as the modulus in the calculation |
| Number 'a' | The input number for which the quadratic residue is determined |
| Calculation | Process of computing the quadratic residue |
This table aims to provide quick reference points for users seeking information related to quadratic residues.
Example of Quadratic Residue Calculator
Suppose we want to find the quadratic residue of '5' under the modulus '11'. Using the formula, QR(5, 11) = 5^((11-1)/2) % 11, we perform the calculation as follows:
QR(5, 11) = 5^5 % 11
QR(5, 11) = 3125 % 11 = 4
Hence, the quadratic residue of 5 under the modulus 11 is 4.
Most Common FAQs
A: Quadratic residues have applications in cryptography, number theory, and various mathematical algorithms.
A: 'p' must be a prime number, while 'a' can be any integer relatively prime to 'p'.
A: No, the calculator works specifically for prime moduli.