Encrypt & Check

Why It Works

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 mod ϕ=1ed\bmod\phi=1
Why RSA decryption worksRSA key derivation recomputed from the toy primes and public exponent.Why RSA decryption works - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

Pin the inverse identity

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

3⋅27=81=2ϕ+13\cdot27=81=2\phi+1
Why RSA decryption worksRSA key derivation recomputed from the toy primes and public exponent.Why RSA decryption works - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

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.

(me)d=mone+kϕ≡m(modn)(m^e)^d=m^{\text{one}+k\phi}\equiv m\pmod n
Why RSA decryption worksRSA key derivation recomputed from the toy primes and public exponent.Why RSA decryption works - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

What this toy example leaves out

These toy values are intentionally checkable, not secure. NOTE: toy-modulus; no-padding; no-side-channel; no-production; never-roll-your-own. Real RSA needs padding and reviewed library code before arithmetic is used in a protocol.

toy arithmetic, not deployed security\text{toy arithmetic, not deployed security}
Why RSA decryption worksRSA key derivation recomputed from the toy primes and public exponent.Why RSA decryption works - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1

Summary

Because e times d is congruent to 1 modulo phi, decryption inverts encryption in this toy setting. NOTE: toy-modulus; no-padding; no-side-channel; no-production; never-roll-your-own. Exact arithmetic on deliberately tiny toy keys; real RSA uses 2048+ bit keys with padding in reviewed libraries.

ed≡1(modϕ)ed\equiv1\pmod{\phi}
Why RSA decryption worksRSA key derivation recomputed from the toy primes and public exponent.Why RSA decryption works - n=55quantityvaluep5q11n=p*q55phi=(p-1)(q-1)40e3d=e^-1 mod phi27e*d mod phi1