Alice wants to use the ElGamal digital signature scheme with public parameters ,
, secret value
and
. She wants to sign the message
and send it to Bob. She chooses
as the secret value. Explain the procedure that Alice will use for computing the signature of the message. What information will she send Bob?
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) 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
Find the keys for the second round.
6) a) Considering the bytes 10001001 and 10101010 as elements of the field , where g(X) is the polynomial X8 + X4 + X3 + X + 1, find their product and quotient.
Decrypt the message that was encrypted using RSA algorithm with
and
.
Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance :
011001110000110010011100.
Bob uses ElGamal cryptosystem with parameters ,
and the secret value
. What values will he make public?
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.
Find a recurrence that generates the sequence 110110110110110.