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By John C. Bowman

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Decoding is accomplished via the inverse transformation m = Dd (c) = c + d (mod n), where d = −e. That is, encoding is accomplished by addition modulo e and decoding key is accomplished by subtraction modulo e. Caeser adopted the value e = 3 to encrypt the n = 26 symbols of the Roman alphabet, using 0 to represent the letter A and 25 to represent the letter Z. Some fans of the film “2001: A Space Odyssey” even suggest that the computer name HAL is really a shift cipher for IBM, with e = 25! A slight generalization of the shift cipher is the affine cipher, defined by c = Ea,b (m) = am + b (mod n), where a ∈ N is relatively prime to n.

1 RSA Cryptosystem The most well known public-key cipher is the Rivest–Shamir–Aldeman (RSA) Cryptosystem. B. ). The numbers n and e are made publicly available, but d, p, q are kept secret. Anyone who wishes to send a message m, where 0 ≤ m < n, to the receiver encrypts the message using the encoding function c = Ee (m) = me (mod n) and transmits c. Because the receiver has knowledge of d, the receiver can decrypt c using the decoding function M = De (c) = cd (mod n). To show that M = m, we will need the following results.

1 0 1 0 1 0 1 If the vector 1110110 is received, the syndrome is [0, 1, 1]t , which corresponds to the binary number 3, so we know immediately that the a single error must have occurred in the third position, without even looking at H. Thus, the transmitted codeword was 1100110. Remark: For nonbinary Hamming codes, we need to compare the computed syndrome with all nonzero multiples of the columns of the parity-check matrix. • A parity-check matrix for Ham(2, 3) is H= 0 1 1 1 . 1 0 1 2 If the vector 2020, which has syndrome [2, 1]t = 2[1, 2]t , is received and at most a single digit is in error, we see that an error of 2 has occurred in the last position and decode the vector as x = y − e = 2020 − 0002 = 2021.

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