82 1. ALGEBRAIC THEMES the message. What foils the snoop and makes the RSA method useful for secure communications is that it is at present computationally difﬁcult to factor very large integers. 12 On the other hand, it is computationally easy to ﬁnd large prime num-bers and to do the computations necessary for encryption and decryption. So if my primes p and q are sufﬁciently large, you and I will still be able to do our computations quickly and inexpensively, but the snoop will not be able to factor n to ﬁnd p and q and decrypt our secret message. Example 1.12.3. I ﬁnd two (randomly chosen) 50–digit prime numbers: p D 4588423984 0513596008 9179371668 8547296304 3161712479 and q D 8303066083 0407235737 6288737707 9465758615 4960341401: Their product is n D 3809798755658743385477098607864681010895851155818383 984810724595108122710478296711610558197642043079: I choose a small prime r D 55589 , and I send you n and r by email. I secretly compute m D 19048993778293716927385493039323405054479255779091
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