a) Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance :

011001110000110010011100.

 
b) Apply poker test to the following sequence with level of significane 
1001101000010000101111011
01110100101101100100110.

c) Apply runs test to the following sequence:
1001101000010000101111011
0111010010110110010011010
0110011100001100100111000
1100001101010111101001110
0010001111000001101010010
1000110100000110100101101
1110001001

a) Considering the bytes 10001001 and 10101010 as elements of the field  where  is the polynomial  find their product and quotient.

b) Find a recurrence that generates the sequence 110110110110110.

b) Let an activation function be defined as 

Show that  What is the value of  at the origin? Also, find the value of  as v approaches  and 

a) Decrypt the ciphertext 101000111001 which was encrypted with the Toy block cipher once
using the key 101010010. Show all the steps. (5)
b) A 64 bit key for the DES is given below
    11000111 10000101
    11110111 11000001
    11111011 10101011
    10011101 10010001
i) Check whether the key is error free using the parity bits.
ii) Find the keys for the second round.

In this exercise, we introduce you to Hill cipher. In this cipher, we convert our message to numbers, just as in affine cipher. However, instead of encrypting character by character, we encrypt pairs of characters by multiplying them with an invertible matrix with co-efficients in  .

Here is an example: Suppose we want to ENCRYPT "ALLISWELL". Since we require the plaintext to have even number of characters, we pad the message with the character ‘X’. We break up the message into pairs of characters AL, LI, SW, EL and LX. We convert each pair of characters into a pair elements in  as follows:

Next, we choose an inveritble 2×2 matrix with coefficients in , for example,

This matrix has determinant  and 5 is a unit in   with inverse  We write each pair of elements in  as a column vector and multiply it by A:

We then convert each pair of numbers to a pair of characters and write them down. In this example, we get the cipher text "LSPFYGXUEN" corresponding to the plain text "ALLWELL". To decrypt, we convert pairs of characters to pairs of numbers and multiply by    and we have 

Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill’s cipher with the matrix   as the encryption matrix

a) Find the length and order of the following schema:  

i) 

ii)  

iii)  

b) Describe the Function Approximation in MLP. Also, explain Generalization of MLP.

a) Define Kohonen networks with examples.

Consider the set of pattern vectors P. Obtain the connectivity matrix (CM) for the patterns in P (four patterns). 

b) Write the types of Neural Memory Models. Also, give one example of each.

a) Define Error Correction Learning with examples.

Consider a dataset of six points given in the following table, each of which has two features  and  Assuming the values of the parameters c and m as 2 and the initial cluster centers  and  apply FCm algorithm to find the new cluster center after one iteration. 

	3	11
	3	10

	8	12
	10	6
	13	6
	13	5

Let A and B are two fuzzy sets and   and  then find out the following membership values: 

i)                              ii)         iii) 

iv)                              v)         vi)   

b) Apply the “very” hedge on the fuzzy sets defined in Q. 1(b) to get the new modified fuzzy sets. Show the modified fuzzy sets through numeration.

a) Construct the α − cut at α = 4.0 for the fuzzy sets defined in Q. 1(b)

b) Consider a subset of natural numbers from 1 to 30, as the universe of discourse, U. Define the fuzzy sets “small” and “medium” by enumeration.

a) Two sensors based upon their detection levels and gain settings are compared. The following of gain setting and sensor detection levels with a standard item being monitored provides typical membership values to represent the detection levels for each of the sensors.

The universe of discourse is   Find the membership function for the two sensors. Also, verify De-morgon’s laws for these membership functions.

Use Miller-Rabin test to check whether 75521 is a strong pseuodprime to the base 2.

Find the inverse of 13 (mod 51) using extended euclidean algorithm

Another version of the columnar transposition cipher is the cipher using a key word. In this cipher, we encrypt as follows: Given a key word, we remove all the duplicate characters in the key word. For example, if the key word is ‘SECRET’, we remove the second ‘E’ and use ‘SECRT’ as the key word. To encrypt, we form a table as follows: In the first row, we write down the key word. In the following rows, we write the plaintext. Suppose we want to encrypt the text ‘ATTACKATDAWN’. We make a table as follows: 

Then we read off the columns in alphabetical order. We first read the column under ‘C’, followed by the columns under ‘E’, ‘R’, ‘S’ and ‘T’. We get the cipher text TTX TAN ADX AKW CAX. To decrypt, we reverse the process. Note that, since we know the length of the keyword, we can find the length of the columns by dividing the length of the message by the length of the keyword. Given the ciphertext ‘HNDWUEOESSRORUTXLARFASUXTINOOGFNEGASTORX’ and the key word ‘LANCE’, find the plaintext.