Chinese Remainder Theorem on large exponents

In summary, to use the Chinese Remainder Theorem to shrink down the massive a^27,654,321 term, you will need to split the modulus into coprime elements, find the inverse of a for each modulus, and use the Chinese Remainder Theorem to combine the solutions and find the final solution for x. This solution will be unique modulo 100,000,000.
  • #1
SeventhSigma
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Say I have a^27,654,321 modulo 100,000,000 (where Euler's Theorem no longer helps because totient(100,000,000) = 40,000,000 which is larger than my exponent). How do I use the Chinese Remainder Theorem here to shrink down my massive a^b term I have here?

It seems like I need to first split up my modulus into coprime elements, so 2^8 *5^8. But from here I'm just not sure what I need to be doing other than trying to solve:

x = a^27,654,321 modulo 2^8
x = a^27,654,321 modulo 5^8

1. How do I correctly solve this?
2. Is x technically equivalent to a^27,654,321 modulo 100,000,000 ?
 
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  • #2


1. To solve this using the Chinese Remainder Theorem, you will need to find the inverse of a modulo both 2^8 and 5^8. This can be done using the extended Euclidean algorithm. Once you have the inverses, you can use the Chinese Remainder Theorem to combine the solutions for each modulus and find the final solution for x.

2. Yes, x is technically equivalent to a^27,654,321 modulo 100,000,000. This is because the Chinese Remainder Theorem states that if two numbers are coprime, then there exists a unique solution modulo their product. In this case, 2^8 and 5^8 are coprime, so the solution for x will be unique modulo 2^8 * 5^8, which is equivalent to 100,000,000.
 

1. What is the Chinese Remainder Theorem on large exponents?

The Chinese Remainder Theorem is a mathematical theorem that helps to solve systems of congruence equations, where the modulus (or divisor) of each equation is relatively prime. When dealing with large exponents, the theorem can be used to efficiently compute the remainder of a number when divided by different prime numbers.

2. How is the Chinese Remainder Theorem on large exponents used in cryptography?

In cryptography, the Chinese Remainder Theorem is used as part of the RSA algorithm, which is commonly used for secure data transmission. The theorem allows for efficient decryption of large numbers by breaking them into smaller parts and then combining the results to obtain the original number. This makes it a crucial tool in modern cryptography.

3. Can the Chinese Remainder Theorem be used for non-prime moduli?

Yes, the Chinese Remainder Theorem can be used for non-prime moduli as long as the moduli are relatively prime. However, using non-prime moduli may not always lead to a unique solution, which may make the theorem less useful in certain situations.

4. What are the limitations of the Chinese Remainder Theorem on large exponents?

One limitation of the Chinese Remainder Theorem is that it can only be used when the moduli are relatively prime. Additionally, the theorem may not always provide a unique solution, especially when dealing with non-prime moduli. It also requires a significant amount of computation when dealing with large exponents.

5. How does the Chinese Remainder Theorem on large exponents compare to other methods for solving systems of congruence equations?

The Chinese Remainder Theorem is a more efficient method for solving systems of congruence equations compared to other brute force methods. However, it may not always provide a unique solution and requires the moduli to be relatively prime. Other methods, such as the Extended Euclidean Algorithm, may be more useful in certain situations but can be more computationally intensive.

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