CRT Application Examples

Cryptography · Axiom Academy

EXAMPLE CRT Application Examples Step-by-step walkthroughs applying the Chinese Remainder Theorem Problem: Classic Three-Congruence System Excellent work! You've completed this CRT example. Here's what we learned: CRT Structure: The Chinese Remainder Theorem solves systems of congruences with pairwise coprime moduli by constructing a unique solution modulo N = n₁ × n₂ × ... × nₖ. Computing Components: For each congruence, compute Nᵢ = N/nᵢ and find its modular inverse Mᵢ such that Nᵢ · Mᵢ ≡ 1 (mod nᵢ) using the Extended Euclidean Algorithm. CRT Formula: The solution is x ≡ Σ(aᵢ · Nᵢ · Mᵢ) (mod N), which combines all congruences into a single value satisfying all conditions simultaneously. Cryptographic Applications: CRT enables RSA speedup (computing signatures/decryptions modulo p and q separately), secret sharing (distributing information across multiple shares), and efficient modular arithmetic in secure systems. Verification: Always verify your solution by checking that x satisfies each original congruence: x ≡ aᵢ (mod nᵢ) for all i. CRT is fundamental to modern cryptography - practice with different moduli sets to master this powerful technique!

This is the written version of the interactive lesson above. See the full Cryptography course.