Find a recurrence that generates the sequence 110110110110110.
_____ ____ ________ ____ __________ ________.
________ ____ ________ __________ ________ ________ ___.
__________ _________ _________ ____ ___ ____ ______ _____ __________ _____ ______.
__________ _____ _____ ___ ________ __________ ______ _____ _______ _______ _____.
___ _____ _____ _______ __________ __________ __________ ____.
____ ____ __________ _____ _______ ______ _________ ___ _________ _______ ________.
______ _________ __________ __________ ________ _____ ___.
_______ ___ __________ _____ _________ ______ ____ _______.
_______ ______ _____ _______ ______.
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Find the inverse of 13 (mod 51) using extended euclidean algorithm.
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 Z26.
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 Z26 as follows:
">
Next, we choose an invertible matrix with coefficients in
, for example,
.
This matrix has determinant and
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.
Alice wants to send Bob the message . She chooses
. How will she compute the cipher text? What information does she send to Bob?
Explain how Bob will decrypt the message.
9) a) Solve the discrete logarithm problem using Baby-Step, Giant-Step algorithm.
) Let . We represent the field
by
. Let us write
. The table of values is given below:
|
|
| Vector |
|
| Vector |
| 0 |
|
| 8 |
|
|
| 1 |
|
| 9 |
|
|
| 2 |
|
| 10 |
|
|
| 3 |
|
| 11 |
|
|
| 4 |
|
| 12 |
|
|
| 5 |
|
| 13 |
|
|
| 6 |
|
| 14 |
|
|
| 7 |
|
|
i) Prepare logarithm and antilogarithm tables as given in page 23 of block 1.
ii) Compute and
using the logarithm and antilogarithm tables
Bob uses ElGamal cryptosystem with parameters ,
and the secret value
. What values will he make public?
Decrypt the message that was encrypted using RSA algorithm with
and
.