[Verse 1]
Computing massive powers, that's the mission we embrace
When base raised to exponent makes your system lose its face
Overflow catastrophe, numbers spiral out of sight
But modular arithmetic keeps calculations tight
Take the base, reduce it first, modulo your target prime
Split exponent binary, process bit by bit in time
Square and multiply technique, ancient wisdom still applies
Fermat's theorem in your pocket, RSA between your eyes
[Chorus]
Mod exp, don't stress, break it down to manageable size
Reduce early, reduce often, keep those numbers crystallized
Binary exponent scanning, left to right or right to left
Modular multiplication, cryptographic theft prevention
[Verse 2]
Start with one accumulator, scan exponent bit array
If the current bit is set, multiply and mod away
Square your base each iteration, modulo operation
Prevents explosion, keeps precision, mathematical salvation
Fast exponentiation, logarithmic time complexity
While naive approach would crumble under exponential density
Montgomery reduction smooths the multiplication flow
Hardware optimization makes the modular magic glow
[Chorus]
Mod exp, don't stress, break it down to manageable size
Reduce early, reduce often, keep those numbers crystallized
Binary exponent scanning, left to right or right to left
Modular multiplication, cryptographic theft prevention
[Bridge]
Public key encryption depends on this foundation
Discrete logarithm hardness, computational frustration
Easy to compute forward, nearly impossible reverse
Modular exponentiation, blessing not a curse
[Verse 3]
Sliding window optimizations, precompute small powers table
Cache the frequently used values when your system's willing able
Side channel attack resistance, constant time implementation
Power analysis protection through careful calculation
From elliptic curve to RSA, Diffie-Hellman key exchange
Modular exponentiation puts security in range
[Outro]
Reduce the base, scan the bits, multiply and square with care
Modular arithmetic magic floating in the digital air
Exponential problems tamed by mathematical precision
Cryptographic backbone, computational decision