[Verse 1] When numbers get massive and powers grow wild Computing base to the exponent styled But memory crashes and time runs away Modular math saves the computational day Take your base, your exponent, your modulus too Break it down to pieces, that's what we do Instead of computing the giant result Use properties that make big numbers tumble and bolt [Chorus] Mod ex, mod ex, break it down in steps Square and multiply, no computational debts Mod ex, mod ex, keep the numbers small Binary digits guide you through it all Power mod n, that's the key we hold Efficient algorithms worth their weight in gold [Verse 2] Start with one, that's your result so far Read the exponent bits, each binary star If the bit is zero, just square what you got If the bit is one, multiply on the spot But here's the trick that makes it all work clean Take modulo n after every routine Keep those numbers bounded, never let them grow Overflow protection in the modular flow [Chorus] Mod ex, mod ex, break it down in steps Square and multiply, no computational debts Mod ex, mod ex, keep the numbers small Binary digits guide you through it all Power mod n, that's the key we hold Efficient algorithms worth their weight in gold [Bridge] RSA encryption depends on this game Diffie-Hellman too, they use the same frame Cryptography relies on powers so vast But modular methods make the computation fast From right to left, read each binary bit Square the result, then conditionally hit With multiplication when the bit is set Efficient and secure, place your bet [Verse 3] Time complexity linear in the bit count No more exponential amounts to surmount Memory stays constant, space efficiency tight Modular exponentiation done right Applications everywhere in the digital age From secure communications to the crypto stage Master this algorithm and you'll understand How to tame the giants with mathematics grand [Chorus] Mod ex, mod ex, break it down in steps Square and multiply, no computational debts Mod ex, mod ex, keep the numbers small Binary digits guide you through it all Power mod n, that's the key we hold Efficient algorithms worth their weight in gold [Outro] When powers grow massive beyond all control Modular methods achieve the goal Break it down, keep it tight Mod ex makes the computation right
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