[Verse 1]
When numbers dance in modular space
And greatest common divisors trace
A path to one, the magic starts
Euler's theorem plays its part
If gcd of a and n equals one
Then phi of n becomes the gun
That shoots the power to the top
Where cycles end and patterns stop
[Chorus]
A to the phi of n congruent to one
Modulo n, the cycle's done
Like clocks that tick around the face
Mathematics finds its resting place
When coprime pairs begin to play
Euler's law lights up the way
[Verse 2]
Fermat whispered secrets small
When primes would build their counting wall
But Euler saw a grander scheme
Beyond the prime number dream
Take any base, take any mod
If coprime, then applaud
The power climbs then tumbles down
To one, the mathematical crown
[Chorus]
A to the phi of n congruent to one
Modulo n, the cycle's done
Like clocks that tick around the face
Mathematics finds its resting place
When coprime pairs begin to play
Euler's law lights up the way
[Bridge]
RSA encryption builds its throne
On Euler's work, carved in stone
Choose primes p and q with care
Multiply to get n there
Phi of n equals p minus one
Times q minus one, we're having fun
Pick e coprime to phi's domain
Find d inverse, complete the chain
[Verse 3]
Message m gets raised to e
Ciphertext c is what we see
To decrypt, we use key d
Original message springs free
Why does this mathematical dance
Work with such elegant stance?
Because cd equals one plus k times phi
And Euler's theorem tells us why
[Chorus]
A to the phi of n congruent to one
Modulo n, the cycle's done
Like clocks that tick around the face
Mathematics finds its resting place
When coprime pairs begin to play
Euler's law lights up the way
[Outro]
From Fermat's glimpse to Euler's sight
Cryptography burns bright
In every click and every send
On Euler's theorem we depend