Decrypting RSA: Using d_K, d_p, d_q, M_p, M_q and x_p,x_q to Prove y^d=x mod n

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SUMMARY

The discussion focuses on the mathematical proof of the RSA decryption process using the parameters d_K, d_p, d_q, M_p, M_q, x_p, and x_q. It establishes that y^d is congruent to x mod n, where n is the product of two primes p and q. The proof involves applying Fermat's theorem to demonstrate y^d = y^(d_p) mod p and y^d = y^(d_q) mod q, followed by the use of the Chinese Remainder Theorem (CRT) to derive the final result for x.

PREREQUISITES
  • Understanding of RSA encryption and decryption principles
  • Familiarity with modular arithmetic
  • Knowledge of Fermat's Little Theorem
  • Basic concepts of the Chinese Remainder Theorem (CRT)
NEXT STEPS
  • Study the application of Fermat's Little Theorem in cryptography
  • Learn about the Chinese Remainder Theorem and its applications in number theory
  • Explore the RSA algorithm in detail, including key generation and encryption
  • Investigate modular exponentiation techniques for efficient computation
USEFUL FOR

This discussion is beneficial for cryptographers, computer scientists, and students studying cryptography or number theory, particularly those interested in RSA encryption and decryption methods.

hope2009
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In RSA: d_K (y)=y^d mod n and n=pq. Define

d_p=d mod(p-1)

d_q=d mod(q-1)
Let

M_p=q^(-1) mod p
M_q=p^(-1) mod q
And

x_p=y^(d_p ) mod p
x_q=y^(d_q ) mod q
x=M_p qx_p+M_q px_q mod n

Show that y^d=x mod n
any help would be appraciated, thanks
 
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homework eh?

use fermat's thm to prove y^d = y^(d_p) mod p (same for q)
show x = x_p mod p (same for q)
then use CRT to solve for x
 

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