Classical Algebra (Math 326) Written Assignment

due April 6, 2001

Directions. Please submit your assignment typed. If there is more than one page, please staple.

Explain your solutions.

1. Encrypt the string “Sell everything at once!” one character at a time by forming the sequence consisting of the 7th powers of its decimal ASCII codes modulo 11021.

2. The method of the previous problem was used to encrypt a string. The corresponding encrypted sequence is

[5965, 5730, 8607, 5425, 711, 7508, 711, 1237, 6720, 5730, 7508, 327,

7508, 5730, 1237, 9857, 8738, 711, 7508, 5730, 8215, 8607, 711, 8215,

8607, 1332, 8215, 9821, 3953]

What was the original string?

3. Let m be the number

5258168548013597450620892673749432457952803197 .

Encrypt the string ”Two words” by forming the sequence of eth powers of the (decimal) ASCII codes taken modulo m when e = 10001. For ease of transcription you may respond to this exercise by submitting the least non-negative residues mod 128 of the encrypted sequence.

4. Repeat the previous exercise with the following modification. Instead of operating on the decimal ASCII code of each of the nine characters in the string, take the characters 3 at a time, and form the length 3 sequence consisting of the numbers c + b · 128 + a · (128)^{2} for each of the three sequences of numbers [a, b, c] appearing in the sequence of 9 ASCII codes of the string. (Notice that this technique is both more efficient and more “secure” than that of the previous exercise.) For ease of transcription you may respond to this exercise by submitting the least non-negative residues mod 128 of the encrypted sequence.

5. What step in trying to decrypt a string encoded with the known exponent 10001 modulo the number m by the method of the previous exercise makes such decryption difficult? Are you able to find the 3 character string encoded that way to produce the single term sequence

[394070967091950488701702235969152190968955616] ?