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?
The ElGamal digital signature scheme works as follows: we are given the public parameters ( p = 47 ), ( \alpha = 2 ), Alice's secret key ( a = 7 ), and her public key ( \beta = 34 ). Alice wants to sign the message ( \mathcal{M} = 20 ) using a random secret ( k = 5 ) that is coprime to ( p-1 = 46 ). Step 1: Compute ( r ) The signature component ( r ) is calculated as: [ r = \alpha^k \mod p ] [ r = 2^5 \mod ______ ____ ____ ______ _________ ________ _____ _________ ______ ________ ___.
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______ _______ __________ _____ __________ _____ _____.
__________ ________ ___ _________ _________ ____ __________ ______ _______.
____ ____ _____ ____ ___ ___.
________ _____ ___ ________ ____ ___ _______ __________ _____ ___ ______ _________.
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_____ _______ _____ ____ __________ _____ _____ ____ ______ ________.
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_____ _______ ___ __________ ___ ___ ______ ___ ___.
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_______ ________ _____ _______ _______ ________.
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_______ ________ ______ ____ ____ _______ _________ __________ ___ __________ _______ ___.
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___ _______ ___ _____ __________.
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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
Bob uses ElGamal cryptosystem with parameters ,
and the secret value
. What values will he make public?
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?
Alice wants to send Bob the message . She chooses
. How will she compute the cipher text? What information does she send to Bob?
Let . Find the product of x2 + 2x + 1 + (f(x)) and x2 + 3x - 1 + (f(x)) using the algorithm in page 23, block 1. You should show all the steps as in example 11, page 22, block 1.
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.
Decrypt the message that was encrypted using RSA algorithm with
and
.