Find the inverse of 13 (mod 51) using extended euclidean algorithm
To find the modular inverse of \( 13 \) modulo \( 51 \) using the Extended Euclidean Algorithm, we need to find an integer \( x \) such that: \[ 13x \equiv 1 \pmod{51} \] ### Step 1: Apply the Euclidean Algorithm The Euclidean algorithm helps in finding the greatest common divisor (GCD) of 13 and 51. #### Divide _______ __________ ________ ___ ____ ________.
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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?
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.
Alice wants to use the ElGamal digital signature scheme with public parameters p = 47, α = 2, secret value and
. She wants to sign the message M = 20 and send it to Bob. She chooses k = 5 as the secret value. Explain the procedure that Alice will use for computing the signature of the message. What information will she send Bob?
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
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 |
Solve the discrete logarithm problem (mod 47) using Baby-Step, Giant-Step algorithm.
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.