Is There a Solution to the Equation A=x^b mod p?

  • Thread starter Chu
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In summary, The conversation discusses the possibility of solving for x in the equation a=x^b mod p, with known values for p, a, and b. The speaker believes there is a solution, but it may be difficult to find. They mention that raising every remainder to the power of b may solve the equation, but there are certain circumstances in which this may not be the best solution. It is also noted that finding generators of the group of units mod a prime is a well-known and difficult problem.
  • #1
Chu
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Say you are given a=x^b mod p, where p, a, and b are known. Is there a way to solve this? I am pretty sure there is . . . but it is driving me nuts.

-Chu
 
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  • #2
Try solving x^2 = 2 (mod 3).
 
  • #3
Muzza said:
Try solving x^2 = 2 (mod 3).

Sorry, I'll rephrase. I know a solution must exist from the choices of p,b,and a (this is part of a crypto algorithm where they know x, I do not, and I am wondering if I have sufficent info to solve for it).
 
  • #4
You can of course solve it - all one needs is to raise every remainder to the b'th power. However, I don't think there is a better solution except in certain circumstances.

obviously b must divide [tex]\varphi(x)[/tex], but that doesn't give a sufficient condition for a solution, or even tell you what it is.

The reason for my scepticism is that it is a well known and difficult problem to find generators of the group of units mod a prime (which is a cyclic group).
 

1. What does the equation A=x^b mod p mean?

The equation A=x^b mod p is a mathematical expression used in modular arithmetic. It represents finding the remainder when the number A is divided by p, with a base of x and an exponent of b.

2. How do you solve A=x^b mod p?

To solve A=x^b mod p, you can use the exponentiation by squaring method. First, calculate x^b, then take the remainder when divided by p. This will give you the value of A.

3. What makes the equation A=x^b mod p solvable?

The equation A=x^b mod p is solvable if the base x and the modulus p are coprime, meaning they do not share any common factors except 1. If this condition is met, then a unique solution for A exists.

4. Can the equation A=x^b mod p have multiple solutions?

Yes, the equation A=x^b mod p can have multiple solutions if the base x and the modulus p are not coprime. In this case, there are multiple values of A that satisfy the equation.

5. What are the practical applications of A=x^b mod p?

The equation A=x^b mod p has various practical applications, such as in cryptography, number theory, and computer science. It is used in encryption algorithms, generating random numbers, and checking for primality of large numbers.

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