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what is the benefit of lcm in cryptography? and what is the effect of lcm on public key?

note * lcm = Least common multiple


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closed as unclear what you're asking by CodesInChaos, DrLecter, Gilles, e-sushi, poncho Dec 10 at 16:06

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Lcm or least common multiple is simply a concept in math, it does not have a special meaning (or benefit) in cryptography. The least common multiple of two numbers $x$ and $y$ is simply the smallest number $z$ so that both $x$ and $y$ divide $z$. For example $lcm(9, 30) = 90$ because $90 = 9\cdot10$ and $90 = 3\cdot30$ and there is no smaller number that both $9$ and $30$ divides. –  Guut Boy Dec 9 at 13:48

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Regarding public key: In two prime RSA you often compute the secret key from the public as $d=e^{-1} \pmod {(p-1)(q-1)},\;$ but using $d=e^{-1} \pmod {\mathrm{lcm}(p-1,q-1)}$ may have some advantages.

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Note that it has fewer advantages than you'd hope: if you use the CRT optimization (which most people do), then both $e^{-1} \pmod {(p-1)(q-1)}$ and $e^{-1} \pmod {\mathrm{lcm}(p-1,q-1)}$ lead to the same value of $dp$ and $dq$, and so they operate exactly the same. –  poncho Dec 9 at 15:28

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