Why It Works - RSA with Keys You Can Check

The RSA round trip rests on one modular inverse identity. The key table makes that identity visible before the lesson names the theorem.

highlighted = computed this step

Why the exponents undo each other

The key table already proved that e times d is one modulo phi. That congruence is the arithmetic reason decryption reverses encryption.

ed\bmod\phi=1

Pin the inverse identity

Here e times d is 81, which is 2 times phi plus 1.

3\cdot27=81=2\phi+1

Why that reverses encryption

Euler's theorem says raising by one plus a whole multiple of phi lands back on the same residue for invertible messages. Euler's theorem covers messages coprime to n. RSA in fact recovers every message below n - even 5, 11, or 0, which share a factor with 55 - but the argument for those uses the Chinese Remainder Theorem, beyond this toy.

(m^e)^d=m^{\text{one}+k\phi}\equiv m\pmod n

What this toy example leaves out

These toy values are intentionally checkable, not secure. Real RSA also needs padding such as OAEP before the arithmetic is used safely.

\text{toy arithmetic, not deployed security}

Summary

Because e times d is congruent to 1 modulo phi, decryption inverts encryption in this toy setting. Exact arithmetic on deliberately tiny toy keys; real RSA uses 2048+ bit primes with padding, and timing/side-channels are not modeled here.

ed\equiv1\pmod{\phi}