Introduction to Cryptography - Part A

In a (3, 7) Shamir secret sharing scheme with modulus 59, three of the shares are (5, 11), (10, 43) and (20, 34). Another share is (7, k), but the value of k is unreadable. Find the correct value of k

1. Here, the Shamir scheme is (3,7). Here, three shares will be used for finding M as S(x)= M+ si (x).

M + 5.s = 11 (mod 59)

M + 10.s = 43 (mod 59)

M + 20.s = 34 (mod 59)

S(x)= 84- 42(k) (mod 59)

Now, for (7, k), we have s (7) = 84-42(7) (mod 59) = 39.

That is, k= 39.

Introduction to Cryptography - Part B

3. Block cipher has been considered as encryption algorithm. Block cipher is an algorithm where input size is b which has been produced b bits ciphertext. However, it should be divided when the input range of b bits will be larger. AES stands for the encryption standard which is in advanced level.

There will be various hash functions that are in use but will not be secure that way. The secure hash function will be considered as a cryptographic function. The working policy of hash function is it will be used for data transformation. The algorithm that has been used in this function has been included with bitwise operations, compression functions and modular additions.

The main purpose of using SHA is encrypting passwords while the server is responsible for keeping records of specific hash value. This process has been playing an important role as the cyber hackers will

Only be able to find the hash function. The actual password cannot be shown there. Here, the secure hash design has been showing that at first the password was a simple text and after using the hash function the password totally changed and a type of hexadecimal format has been shown. This is the significance of hash function.

4. Let the probability has been taken such a way that no person will have the same birthday among the 40 people.

The steps here:

First student was born on a specific day where the probability was 365/365.

The next person of the class has 364 days while the probability of this student was 364/365.

The third person was born in between the limited 363 days while the probability was 363/365.

This similar pattern will be continued till the last person has the probability of 326/365 and the student will have 39 days.

Now the step is to multiply 40 probabilities below:

(365/365) *(364/365) *(363/365) …. (327/365) * (326/365) =

Here, the calculation can be a little bit difficult where the calculation can be done in another way:


Here, factorial function needs to be used for better calculation: 365! =

(365! / 325!) where the calculation can be {365! / (325! *36530) = 2.5104/ (7.439*7.3924) = 0.53

From the above calculation it can be seen that there will be a 53% probability that two people will share the same day as their birthday.

6. There will be two keys in cryptosystem such as public key for encryption and private key for decryption. Here, the public key that has been provided for Alice (p,g,h). Here, p has been used as a big prime while g has been used as mod generator for p. The value has been given where h= ga. Here, a has been mentioned as the secret key for Alice.

Plaintext Ciphertext Ciphertext Plaintext

Here, the diagram has been shown the communication process of Bob and Alice. First bob needs to create a plain text and after that the text will be converted into ciphertext and when the message will enter into Alice’s device Alice must have a public key that will help her to open the message.

There will be rainbow table attack that has been done by Eve. Here, the part of the plaintext he has received will need to be matched with other parts. If the part has been matched with other parts in the specific chain then it will continue the process and will be able to get the full plaintext from the start end. The plain text that has been already converted with hash functions. Then the process will check the hash text either it matches with the original hash passwords then the plain text which has been received will be the original password. If the process has not matched with the hash functions then it will be reduced again for the match. If the part of the plain text has not been matched with other part in the chain then the person will be able to reduce the specific hash function. This is the way Eve will be able to access the messages.

The random number generator can be broken also while the output of b has been decided as 2 which has the probability is 90%. Now through rainbow cable attack, Eve can not be able to encrypt the c3 and c4 as the random number generator was in bad sense. Hence, here the hash function can be manipulated through E1Gamal cryptosystem. The rainbow table will allow the attacker to pass the string by converting md5 hash function. Here, the hash function can be,

HashMD5 (12345678) = 45d15ad35bb700hf364d78h61708bd.

Here, the first 8 hash function can be changed through the 8 characters. Then the function will be re-hashed as:

HashMD5(45d15ad3) = 67fh24ff54680hf453ad98acf56426gd.

This need to be repeated till a number of hashes have been present in the chain. However, this has been included with one output chain where the encryption has been started from the plain text with the help of ciphertext and then it has been ended with the newly generated hash function. The c3 and c4 has been generated through this way where the MD5 function has been changed.

Bibliography for Introduction to Cryptography

Gul, E., & Ozturk, S. (2019). A novel hash function based fragile watermarking method for image integrity. Multimedia Tools and Applications78(13), 17701-17718.

Indrayani, R., Nugroho, H. A., Hidayat, R., & Pratama, I. (2016, October). Increasing the security of MP3 steganography using AES Encryption and MD5 hash function. In 2016 2nd International Conference on Science and Technology-Computer (ICST) (pp. 129-132). IEEE.

Liu, H., Kadir, A., & Liu, J. (2019). Keyed hash function using hyper chaotic system with time-varying parameters perturbation. IEEE Access7, 37211-37219.

Mouha, N., Raunak, M. S., Kuhn, D. R., & Kacker, R. (2018). Finding bugs in cryptographic hash function implementations. IEEE transactions on reliability67(3), 870-884.

Teh, J. S., Tan, K., & Alawida, M. (2019). A chaos-based keyed hash function based on fixed point representation. Cluster Computing22(2), 649-660.

Remember, at the center of any academic work, lies clarity and evidence. Should you need further assistance, do look up to our Computer Science Assignment Help

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