1.12. AN APPLICATION TO CRYPTOGRAPHY81Similarly,pdividesaha. But thenahais divisible by bothpandq, and hence bynDpqDl:c:m:.p; q/.nLetrbe any natural number relatively prime tom, and letsbe aninverse ofrmodulom,rs1.modm/. We can encrypt and decryptnatural numbersathat are relatively prime tonby first raising them to therthpower modulon, and then raising the result to thesthpower modulon.Lemma 1.12.2.For all integersa, ifbar.modn/;thenbsa .modn/:Proof.WritersD1CtmThenbsarsDa .modn/, by Lemma1.12.1.nHere is how these observations can be used to encrypt and decryptinformation: I pick two very large primespandq, and I compute thequantitiesn,m,r, ands. I publishnandr(or I send them to you privately,but I don’t worry very much if some snoop intercepts them). I keepp,q,m, andssecret.Any message that you might like to send me is first encoded as anatural number by a standard procedure; for example, a text message is
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