Why RSA? A Pedagogical Comment

One of the most widely used cryptographic algorithms is the RSA algorithm in which a message m encoded as the remainder c of m modulo n, where n and e are given numbers – forming a public code. A similar transformation c mod n, for an appropriate secret code d, enables us to reconstruct the original message. In this paper, we provide a pedagogical explanation for this algorithm. 1 RSA Algorithm: A Pedagogical Puzzle RSA algorithm: a brief reminder. In many computer transaction, the communicated message is encoded, to avoid eavesdropping. This happens, e.g., every time a credit card information is passed over to some website. In most of such cases, a special RSA algorithm is used to encode the message m; see, e.g., [1]. In this algorithm, two specially selected and publicly available numbers n and e are used to encode the message. The encoded message c has the form of the remainder c = m mod n. The number n is usually at least 100 decimal digits long, and the number e is similarly large. For such large numbers, it is not feasible to compute m simply as m · . . . ·m, by starting with m and e− 1 times multiplying the result by m. Instead, the following much faster algorithm is performed. First, the number e is represented in the binary form, as the sum of powers of two: e = 21 + 22 + . . . + 2p for some k1 > k2 > . . . > kp. For example, 1110 is represented as 10112 = 2 3 + 2 + 2 = 8 + 2 + 1.