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MAT 1113
Tutorial 6

1. Suppose that we choose p = 23, q = 31, and n = 29, encrypt 572 using the public keys z and n.

2. Consider the (2, 7) encoding function e:
e(00) = 0000000
e(01) = 1001001
e(10) = 1100111
e(11) = 1000110
(i) Find the minimum distance of e.
(ii) How many errors will e detect?

3. Find the minimum distance of the (3, 8) encoding function e:
E(000) = 00000000 e(100) = 01100101 e(001) = 01110010
E(101) = 10110000 e(010) = 10011100 e(110) = 11110000
E(011) = 01110001 e(111) = 00001111

4. Consider the (2, 6) encoding function e:
e(00) = 000000
e(01) = 011110
e(10) = 101010
e(11) = 111000
(a) Find the minimum distance of e.
(b) How many errors will be detected?

5. Consider the (3, 9) encoding function e:
e(000) = 000000000 e(100) = 010011010
e(001) = 011100101 e(101) = 111101011
e(010) = 010101000 e(110) = 001011000
e(011) = 110010001 e(111) = 110000111
(a ) Find the minimum distance of e.
(b) How many errors will e detected?

6. Consider the (3, 7) encoding function e:
e(000) = 0000000 e(100) = 0101110
e(001) = 1001001 e(101) = 1110001
e(010) = 0011001 e(110) = 0101110
e(011) = 0011010 e(111) = 1100110
(a ) Find the minimum distance of e.
(b) How many errors will e detected?




7. Consider the (2, 6) encoding function e:
e(00) = 000000
e(01) = 010110
e(10) = 101101
e(11) = 111001
(a) Find the minimum distance of e.
(b) How many errors will e detect?

8. Find the weight of each of the following words in B5:
(a) x = 01000 (b) x = 11100 (c) x = 00000 (d) x = 11111

9. Consider the (6, 7) parity check code. For each of the received words, determine whether an error will be detected.
(a) 1010011 (b) 1011101

10. A word is encoded using the parity check code and it is transmitted. For the following received word, decode the word using a single-error correcting code procedure.
0101111

11. Consider the following encoding function e: B3 B5.
e(000) = 00000 e(001) = 11101 e(010) = 01110 e(100) = 10101
e(101) = 11010 e(110) = 00001 e(011) = 10110 e(111) = 11111

(a) Determine the minimum distance of the encoding function.
(b) Determine how many errors the encoding function can detect.

12. Consider the (7, 8) parity check code. For each of the received words, determine whether an error will be detected.
(i) 10101101.
(ii) 01111000.

13. In the RSA public-key cryptosystem, to encrypt a, compute c = an mod z and send c to the holder of the public keys z and n, where z is chosen as the product of two primes p and q. Assume that we choose our primes to be p = 17 and q = 29, and n = 13, encrypt 398 using the public keys z and n.

14. Find the weight of the word 110110001.

15. Consider the (7, 8) parity check code. For each of the received words, determine whether an error will be detected.
(i) 11100100.
(ii) 01010111.




16. Decode the received word using single-error correcting code procedure.
(a) 1110011
(b) 0110010
(c) 0001101

17. In the RSA public-key cryptosystem, to encrypt a, compute c = an mod z and send c to the holder of public keys z and n, where z is chosen as the product of two primes p and q. Assume that we choose our primes to be p = 13 and q = 17, and n = 5, encrypt 144 using the public keys z and n.

18. Find the minimum distance of the (2, 4) encoding function e:
e(00) = 0000
e(10) = 0110
e(01) = 1011
e(11) = 1111
Hence, determine how many errors the encoding function can detect.

19. Consider (6, 7) parity check code. For each of the received words, determine whether an error will be detected.
(a) 1101010
(b) 0011111

20. A word is encoded using the parity check code and it is transmitted. For each of the following received words, decode the words using single-error correcting code procedure.
(i) 0101001.
(ii) 1011011.

21. In the RSA public-key cryptosystem, to encrypt a, compute c = an mod z and send c to the holder of public keys z and n, where z is chosen as the product of two primes p and q. Assume that we choose our primes to be p = 11 and q = 19, and n = 7, encrypt 124 using the public keys z and n.

22. Consider the (2, 6) encoding function e:
e(00) = 000000
e(01) = 010010
e(10) = 101001
e(11) = 110110

(i) Find the minimum distance of e.
(ii) How many errors will e detect?

23. Find the weight of each of the following words in B6:
(i) 110010.
(ii) 011011.


24. Find the weight of each of the following words in B6:
(i) 011010.
(ii) 100111

25. Consider the (6, 7) parity check code. For each of the received words, determine whether an error will be detected.
(i) 1001010.
(ii) 0001111.

26. A word is encoded using the parity check code and it is transmitted. For each of the following received words, decode the words using single-error correcting code procedure.
(i) 0110110.
(ii) 1110110.

27. Consider the (2, 5) encoding function e:
e(00) = 00000
e(01) = 01101
e(10) = 00110
e(11) = 11011
(i) Find the minimum distance of e.
(ii) How many errors will e detect?

28. A word is encoded using the parity check code and it is transmitted. For each of the following received words, decode the words using single-error correcting code procedure.
(i) 1101011.
(ii) 0110011.

29. In the RSA public-key cryptosystem, to encrypt a, compute c = an mod z and send c to the holder of the public keys z and n, where z is chosen as the product of two primes p and q. Assume that we choose our primes to be p = 13 and q = 23, and n = 10, encrypt 238 using the public keys z and n.

30. Consider the (2, 6) encoding function e:
e(00) = 000000
e(01) = 011101
e(10) = 110011
e(11) = 111000
(i) Find the minimum distance of e.
(ii) How many errors will e detect?





ANSWER:
1. c = 113
2. (i) minimum distance = 2 (ii) k 1
4. (a) minimum distance = 2 (b) k 1
7. (a) minimum distance = 2 (b) k 1
8. (a) 1 (b) 3 (c) 0 (d) 5
9. (a) error will not be detected (b) error will be detected
10. 0100
11. (a) minimum distance = 1 (b) k 0
12. (i) error will be detected (ii) error will not be detected
13. c = 40
14. 5
15. (i) error will not be detected (ii) error will be detected
16. (a) 1010 (b) 0111 (c) 0011
17. c = 196
18. minimum distance = 1 , k 0
19. (a) error will not be detected (ii) error will be detected
20. (i) 1101 (ii) 1010
21. c = 53
22. (i) minimum distance = 2 (ii) k 1
23. (i) |x| = 3 (ii) |x| = 4
30. (i) minimum distance = 3 (ii) k 2