Let
We represent the field
by
LEt us write
the table of values is given below:
| i | Vector | i | Vector | ||
| 0 | 1 | (0,0,0,1) | 8 | (0,1,0,1) | |
| 1 | (0,0,1,0) | 9 | (1,0,1,0 | ||
| 2 | (0,0,1,0) | 10 | (0,1,1,1) | ||
| 3 | (1,0,0,0) | 11 | (1,1,1,0) | ||
| 4 | (1,0,0,0) | 12 | (1,1,1,1) | ||
| 5 | (0,1,1,0) | 13 | (1,1,0,1) | ||
| 6 | (1,1,0,0) | 14 | (1,0,0,1) | ||
| 7 | (1,0,1,1) |
i) Prepare logarithm and antilogarithm tables as given in page 23 of block 1.
ii) Compute using the logarithm and antilogarithm tables.
This problem involves computations in the finite field representation \( F_2[y]/(f(y)) \) and requires using logarithm and antilogarithm tables to simplify the given expressions. ### Step 1: Constructing Logarithm and Antilogarithm Tables We will create logarithm and antilogarithm tables based on the given field elements. - Logarithm Table: Each nonzero element in the field is expressed as a power of a primitive element \( y \), and we assign logarithms accordingly. - Antilogarithm Table: This is simply the reverse of the logarithm table, mapping exponents back to field elements. Using the given table of values: | \( i \) | \( y^i \) | Vector | Logarithm | |---------|----------------------|--------------|-----------| | 0 | 1 | (0,0,0,1) | 0 | | 1 | \( y \) | (0,0,1,0) | 1 | | 2 | \( y^2 \) | (0,1,0,0) | 2 | | 3 | \( y^3 \) | (1,0,0,0) | 3 | | 4 | \( y+1 \) | (1,0,0,1) | 4 | _______ ___ _______ ________ __________ _____ _______ ____ __________.
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Alice wants to use the Digital Signature algorithm for signing messages. She chooses p = 83, q = 41, g = 2 and a = 3. Alice wants to sign the message M = 20. She chooses the secret value k = 8. Explain the procedure that Alice will use for computing the signature. What information will she send Bob?
Text:"KSTYZKESLNZUV", encrypted with Vigenère cipher with key "RESULT"
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.
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:
| S | E | C | R | T |
| A | T | T | A | C |
| K | A | T | D | A |
| W | N | X | X | X |
Find the inverse of 13 (mod 51) using extended euclidean algorithm
Decrypt the message c = 23 that was encrypted using RSA algorithm with e = 43 and n = 77.
Solve the discrete logarithm problem (mod 47) using Baby-Step, Giant-Step algorithm.