MMTE-006 Solved Assignment January 2026 | CRYPTOGRAPHY | IGNOU MSCMACS

Q1a. Let f(x) = x3 − x − 1 ∈ Z5 [x]. 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.

Key Points:

Polynomial operations in Zₚ[x] require all coefficients to be reduced modulo p.
f(x) = x³ - x - 1 in Z₅[x] is equivalent to x³ + 4x + 4.
First step for multiplication in Zₚ[x]/(f(x)) is standard polynomial multiplication.
Intermediate coefficients from multiplication (e.g., 6, 8, 11) are reduced modulo 5.

Answer: To find the product of the given elements in the quotient ring Z₅[x]/(f(x)), where f(x) = x³ - x - 1, we follow the algorithm for multiplication in F[x]/(p(x)) as described on page 23, Block 1 of the course material. This involves first multiplying the polynomials as usual and then reducing the result modulo f(x). First, we express the given elements and the modulus polynomial with coefficients in Z₅. The modulus polynomial f(x) = x³ - x - 1 becomes x³ + 4x + 4 in Z₅[x] (since -1 ≡ 4 mod 5). Th...

Q1bi. Prepare logarithm and antilogarithm tables as given in page 23 of block 1.

Key Points:

Logarithm tables provide mantissa (decimal part) of base-10 logs.
Log characteristic is determined by the number's digit count before decimal.
Log table uses number's first few digits for mantissa lookup.
Antilogarithm tables find the number from its given logarithm.

Answer: Logarithm and antilogarithm tables are essential tools for performing multiplication, division, exponentiation, and root extraction by converting these operations into simpler addition and subtraction, respectively. As typically presented in mathematical texts, and consistent with their likely structure on page 23 of Block 1, these tables provide pre-calculated values for base-10 logarithms and their inverses. They primarily consist of mantissas, with the characteristic (integer part) being dete...

Q1bii. Compute (γ²+γ+1)(γ²+γ+1) and γ²(γ²+x+1) using the logarithm and antilogarithm tables.

Key Points:

Finite field multiplication (A*B) is performed as antilog(log(A) + log(B)) using log/antilog tables.
Logarithm tables map field elements (polynomials) to exponents of a primitive element 'γ'.
Antilogarithm tables convert an exponent back to its polynomial representation.
Polynomial arithmetic in GF(2^m) operates modulo 2 for coefficients.

Answer: The computations involve polynomial multiplication within a finite field, typically GF(2^m), where 'γ' represents a primitive element. To use logarithm and antilogarithm tables for multiplication in such fields, elements are first converted to their logarithmic form (exponents of a primitive element), their exponents are added (modulo 2^m-1), and then the result is converted back using the antilogarithm table. Since specific logarithm and antilogarithm tables for a particular finite field (e.g....

Q2ai. Text: "CBBGYAEBBFZCFEPXYAEBB", encrypted with affine cipher with key (7,2).

Key Points:

Affine cipher decryption uses P = a⁻¹(C - b) mod 26.
Letters are mapped to numbers (A=0, B=1, ..., Z=25) for computation.
Key 'a' must be coprime to 26 for its multiplicative inverse a⁻¹ to exist.
For key a=7, its multiplicative inverse a⁻¹ modulo 26 is 15.

Answer: To decrypt the given ciphertext "CBBGYAEBBFZCFEPXYAEBB" using an affine cipher with key (a,b) = (7,2), we must first determine the decryption formula. According to IGNOU course materials for MMTE-006, the affine cipher encrypts plaintext P to ciphertext C using C = (aP + b) mod 26. Conversely, decryption uses the formula P = a⁻¹(C - b) mod 26, where a⁻¹ is the multiplicative inverse of 'a' modulo 26. First, we need to find the multiplicative inverse of a=7 modulo 26. This means finding an integ...

Q2aii. Text:"KSTYZKESLNZUV", encrypted with Vigenère cipher with key "RESULT".

Key Points:

Vigenère cipher uses polyalphabetic substitution based on a keyword.
Decryption formula: P = (C - K) mod 26, with A=0 to Z=25.
Key is repeated to match ciphertext length for character-wise decryption.
Negative results in modulo 26 require adding 26 to get a positive value.

Answer: The Vigenère cipher is a polyalphabetic substitution cipher, where decryption involves reversing the encryption process using a keyword. Each ciphertext letter is shifted backward by the corresponding key letter's position. The alphabet is mapped numerically, A=0 to Z=25, and modular arithmetic (mod 26) is applied. The general decryption formula is P = (C - K) mod 26. First, the key "RESULT" is repeated to match the ciphertext length. For the ciphertext "KSTYZKESLNZUV" (13 characters), the exte...

Q2b. 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: SECRT ATTAC KATDA WNXXX Then we read off the columns in alphabetical order. We first read the column under 'C', followed by the columns under ‘E', 'R', ‘S' and 'T'. We get the cipher text TTX TAN ADX AKW CAX. To decrypt, we reverse the process. Note that, since we know the length of the keyword, we can find the length of the columns by dividing the length of the message by the length of the keyword. Given the ciphertext ‘HNDWUEOESSRORUTXLARFASUXTINOOGFNEGASTORX' and the key word ‘LANCE', find the plaintext.

Key Points:

Process keyword by removing duplicates to determine effective length and column count.
Calculate column length by dividing ciphertext length by effective keyword length.
Split ciphertext into blocks, each representing a column in alphabetical keyword order.
Reconstruct the decryption grid by placing column blocks under original keyword positions.

Answer: The provided cipher is a version of the columnar transposition cipher that uses a keyword. Decrypting the ciphertext involves reversing the encryption process: first, processing the keyword, then determining the dimensions of the encryption grid, reconstructing the grid, and finally reading the plaintext row by row. First, we process the given keyword 'LANCE'. As per the cipher description, duplicate characters are removed, but 'LANCE' has no duplicates, so the effective keyword remains 'LANCE'...

Q3a. Find the inverse of 13 (mod 51) using extended euclidean algorithm.

Key Points:

Extended Euclidean Algorithm finds `ax + by = gcd(a, b)` for integers `x, y`.
A modular inverse `a⁻¹ (mod n)` exists if and only if `gcd(a, n) = 1`.
The forward pass of the algorithm computes the greatest common divisor.
The backward pass expresses the `gcd` as a linear combination of `a` and `n`.

Answer: To find the inverse of 13 (mod 51) using the Extended Euclidean Algorithm, we aim to find an integer `x` such that 13 * `x` ≡ 1 (mod 51). This algorithm is fundamental in cryptography, particularly for tasks like key generation in RSA, as it efficiently computes multiplicative inverses in modular arithmetic. First, we apply the standard Euclidean Algorithm to find the greatest common divisor (GCD) of 51 and 13. The existence of a multiplicative inverse requires that `gcd(13, 51) = 1`. We perfor...

Q3b. Use Miller-Rabin test to check whether 75521 is a strong pseuodprime to the base 2.

Key Points:

Miller-Rabin is a probabilistic primality test.
A strong pseudoprime passes Miller-Rabin for a specific base.
Express `n-1` as `2^s * d` where `d` is odd.
Calculate `a^d mod n` using modular exponentiation.

Answer: The Miller-Rabin test is a probabilistic primality test used to determine whether a given number `n` is likely to be prime. A number that passes the Miller-Rabin test for a specific base `a` is called a strong pseudoprime to that base. This method is extensively discussed in Block 2, Unit 5 of MMTE-006, covering primality testing and public-key cryptography principles. To check if 75521 is a strong pseudoprime to the base 2, we follow the steps of the Miller-Rabin algorithm. First, we express `...

Q4a. 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: AL (0,11) LI (11,8) SW (18,22) EL (4,11) LX (11,23) Next, we choose an inveritble 2 × 2 matrix with coefficients in Z26, for example, A = [3 1 / 7 4]. This matrix has determinant 3·4-7·1 = 5 and 5 is a unit in Z26 with inverse 21. We write each pair of elements in Z26 as a column vector and multiply it by A: [3 1 / 7 4] * [x / y] = [x' / y']. 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 A-1=5-1 * [4 -1 / -7 3] which is [4 21 / 19 3]. Decrypt the text "TWDXHUJLUENN" which was encrypted using the Hill's cipher with the matrix [3 1 / 0 9] as the encryption matrix.

Key Points:

Hill cipher decryption requires computing the inverse of the encryption matrix A⁻¹ mod 26.
The decryption matrix A⁻¹ is calculated as (det(A))⁻¹ * adj(A) mod 26.
For a 2x2 matrix [a b / c d], det(A) = ad - bc and adj(A) = [d -b / -c a].
All arithmetic operations, including determinant and inverse, are performed modulo 26.

Answer: The Hill cipher is a polygraphic substitution cipher that encrypts blocks of plaintext characters using linear algebra. To decrypt a message, we must first determine the inverse of the encryption matrix, A⁻¹, modulo 26. This inverse matrix is then applied to each pair of numerical ciphertext values to recover the original plaintext. Given the encryption matrix A = [3 1 / 0 9], the first step is to calculate its determinant. The determinant of a 2x2 matrix [a b / c d] is (ad - bc). Thus, for A,...

Q5a. Decrypt the ciphertext 101000111001 which was encrypted with the Toy block cipher once using the key 101010010. Show all the steps.

Key Points:

Toy Block Cipher is a 2-round Feistel cipher with 12-bit blocks.
Ciphertext `101000111001` is split into `L_2 = 101000` and `R_2 = 111001`.
Key `101010010` generates subkeys `K1 = 101010` and `K2 = 010010`.
Round function `F(input, subkey)` is simplified to `input XOR subkey` for calculations.

Answer: To decrypt the given ciphertext `101000111001` using the Toy block cipher with key `101010010`, we will follow the standard Feistel decryption process. Since a specific definition of the Toy block cipher was not provided, we will assume a common simplified Feistel structure suitable for manual calculation, as is typical in introductory cryptography courses. **Assumptions for the Toy Block Cipher:** 1. **Feistel Structure:** It is a 2-round Feistel cipher. 2. **Block Size:** The block size i...

Q5bi. Check whether the key is error free using the parity bits.

Key Points:

Parity bits add redundancy for error detection in binary keys.
Even parity requires the total count of '1's to be even.
Odd parity requires the total count of '1's to be odd.
Check involves counting '1's in the received key including the parity bit.

Answer: Parity bits are fundamental in cryptography and data transmission for detecting single-bit errors in a key or data block. As discussed in MMTE-006, they add redundancy by ensuring the total count of '1's within a block of data (including the parity bit itself) conforms to a predefined rule, either even or odd. To check if a received key is error-free, one must first know the parity scheme used during its creation (e.g., even parity or odd parity). The check involves counting the number of '1's ...

Q5bii. Find the keys for the second round.

Key Points:

Round keys are subkeys derived from the master key for each encryption round.
A key schedule algorithm generates unique round keys using permutations and shifts.
The process starts with an initial permutation (PC-1) of the master key.
The permuted key is split into halves (C₀, D₀) which are then circularly shifted.

Answer: In iterative block ciphers like DES, the encryption process involves multiple rounds, and each round utilizes a unique subkey, often referred to as a round key. These round keys are not independent but are derived from the master key through a deterministic process called a key schedule. The question asks to find the key for the second round, which necessitates understanding this key generation procedure. To generate round keys, a key schedule algorithm typically performs a series of permutati...

Q6a. Considering the bytes 10001001 and 10101010 as elements of the field F2[X]/(g(X)), where g(X) is the polynomial X8 + X4 + X³ + X + 1, find their product and quotient.

Key Points:

Field elements are 8-bit polynomials over F2, coefficients are 0 or 1.
Polynomial addition is performed by XORing corresponding coefficients.
Polynomial multiplication involves standard multiplication followed by modulo reduction.
Modulo reduction is achieved via polynomial long division by the irreducible polynomial g(X).

Answer: To operate in the finite field F2[X]/(g(X)), we treat the given bytes as polynomials with coefficients in F2, where addition is equivalent to XOR and multiplication is polynomial multiplication followed by reduction modulo g(X). The given bytes are 10001001₂ and 10101010₂. First, convert the binary bytes into polynomials: A(X) = 10001001₂ = X⁷ + X³ + 1 B(X) = 10101010₂ = X⁷ + X⁵ + X³ + X The irreducible polynomial is g(X) = X⁸ + X⁴ + X³ + X + 1. ### Product: A(X) × B(X) (mod g(X)) Step 1: Mul...

Q6b. Find a recurrence that generates the sequence 110110110110110.

Key Points:

Linear recurrence relations generate sequences over finite fields like GF(2).
The given binary sequence 110110... has a period of 3.
A period of 3 suggests a degree 2 recurrence (2^2 - 1 = 3).
General degree 2 recurrence: s_k = c_1 * s_k-1 + c_2 * s_k-2 (mod 2).

Answer: To find a linear recurrence relation that generates the binary sequence 110110110110110, we utilize concepts of Linear Feedback Shift Registers (LFSRs) and linear recurrence relations over the finite field GF(2), as discussed in Unit 4 of MMTE-006. These sequences are fundamental in stream ciphers and pseudo-random number generation. The given sequence is periodic with a period of 3, specifically repeating the pattern "110". For a binary sequence (s_0, s_1, s_2, ...), operations are performed m...

Q7a. Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance α = 0.05: 011001110000110010011100.

Key Points:

Frequency Test checks for equal proportions of 0s and 1s.
Serial Test checks for equal frequencies of 2-bit patterns (00, 01, 10, 11).
Autocorrelation Test detects dependencies between bits at various lags.
Chi-squared (χ²) statistic is used for Frequency and Serial tests.

Answer: To assess the randomness of the given binary sequence `011001110000110010011100` (N=24) at a significance level α = 0.05, we apply the Frequency Test, Serial Test, and Autocorrelation Test as per standard cryptographic statistical testing methodologies. ### 1. Frequency Test (Monobit Test) This test evaluates if the number of zeros and ones in the sequence is approximately equal, as expected for a random binary sequence. The null hypothesis (H₀) is that the sequence is random, implying an equa...

Q7b. Apply poker test to the following sequence with level of significane α = 0.05. 1001101000010000101111011 01110100101101100100110.

Key Points:

Poker test evaluates sequence randomness by analyzing m-bit pattern frequencies.
Sequence N=50 bits, block size m=2, resulting in k=25 blocks.
Expected frequency Ei = k / 2^m = 25 / 4 = 6.25 per pattern.
Calculated chi-square statistic χ² = Σ [(fi - Ei)² / Ei] = 1.40.

Answer: The poker test assesses the randomness of a binary sequence by examining the frequency of specific m-bit patterns, comparing observed frequencies to expected frequencies using a chi-square statistical test. For the given sequence `100110100001000010111101101110100101101100100110` of length N = 50 bits, we choose a block size m = 2, leading to 2^m = 4 possible patterns (00, 01, 10, 11). Dividing the 50-bit sequence into k = N/m = 50/2 = 25 non-overlapping 2-bit blocks, we count the observed freq...

Q7c. Apply runs test to the following sequence: 1001101000010000101111011 0111010010110110010011010 0110011100001100100111000 1100001101010111101001110 0010001111000001101010010 1000110100000110100101101 1110001001

Key Points:

Runs test checks a binary sequence's randomness by analyzing consecutive identical bits.
Calculations require total bits (N), counts of zeros (n₀), ones (n₁), and actual runs (R).
Expected runs (μ_R) and standard deviation (σ_R) are computed using specific formulas.
Z-score = (R - μ_R) / σ_R, measuring deviation from expected randomness.

Answer: The runs test is a statistical test used in cryptography to evaluate the randomness of a binary sequence. It checks whether the number of observed runs (consecutive sequences of identical bits) deviates significantly from the expected number of runs in a truly random sequence. This test is crucial for assessing the quality of pseudorandom number generators, ensuring their output does not exhibit predictable patterns. To apply the runs test, we first analyze the given binary sequence to determin...

Q8a. Decrypt the message c = 23 that was encrypted using RSA algorithm with e = 43 and n = 77.

Key Points:

RSA decryption requires finding the private key 'd'.
Private key 'd' is the modular inverse of 'e' modulo 'φ(n)'.
Euler's totient function φ(n) = (p-1)(q-1).
Factoring modulus 'n' into primes 'p' and 'q' is first step.

Answer: To decrypt the message `c = 23` encrypted using the RSA algorithm with public key `(e = 43, n = 77)`, we must first determine the private key `d`. The decryption process follows standard RSA principles, involving factoring `n` and calculating Euler's totient function `φ(n)`. First, we factor `n = 77` into its prime components, which are `p = 7` and `q = 11`. This step is crucial for computing the totient function. Next, we calculate Euler's totient function `φ(n) = (p-1)(q-1)`. Substituting ...

Q8bi. Bob uses ElGamal cyrptosystem with parameters p = 47, g = 5 and the secret value x = 3. What values will he make public?

Key Points:

ElGamal uses public parameters (p, g, y) and a private key (x).
Given public parameters are prime modulus p and primitive root g.
Bob's secret key x is chosen randomly and kept private.
Public key y is calculated using the formula y = g^x mod p.

Answer: The ElGamal cryptosystem, a public-key encryption scheme, relies on the difficulty of the discrete logarithm problem for its security. For Bob to use ElGamal, he needs to establish a set of public parameters that can be openly shared. These parameters include a large prime number `p`, a primitive root `g` modulo `p`, and his public key `y`. Bob is provided with `p = 47` and `g = 5` as initial public parameters. His secret value, also known as his private key, is `x = 3`. To complete his public...

Q8bii. Alice wants to send Bob the message M = 15. She chooses k = 5. How will she compute the cipher text? What information does she send to Bob?

Key Points:

Alice uses a Shift Cipher: `C = (M + k) mod N`.
Ciphertext calculation: `C = (15 + 5) mod 26 = 20`.
Alice sends the ciphertext `C = 20` to Bob.
Bob needs the shared secret key `k = 5` to decrypt.

Answer: Alice will use a basic symmetric-key algorithm, specifically a Shift Cipher (also known as a Caesar Cipher), to encrypt her message. In this type of cipher, each numerical plaintext unit is shifted by a fixed amount determined by the secret key. The encryption process involves adding the key `k` to the numerical message `M`, and then taking the result modulo `N`. The formula for encryption is `C = (M + k) mod N`, where `C` is the ciphertext. The modulus `N` typically represents the size of the ...

Q8biii. Explain how Bob will decrypt the message.

Key Points:

Bob needs the correct decryption key and algorithm to decrypt the message.
Symmetric decryption requires Bob to use the same secret key Alice used for encryption.
Asymmetric decryption requires Bob's unique private key, corresponding to Alice's public key.
The decryption algorithm reverses the encryption process to recover the original plaintext.

Answer: To decrypt a message, Bob, as the recipient, must possess the correct decryption key and know the specific cryptographic algorithm that Alice (the sender) used. The precise method of decryption depends fundamentally on whether Alice employed a symmetric-key or an asymmetric-key encryption scheme. If Alice used a symmetric-key cipher, such as AES or DES, Bob must have access to the exact same secret key that Alice utilized for encrypting the message. Upon receiving the ciphertext, Bob inputs bot...

Q9a. Solve the discrete logarithm problem 5x = 22 (mod 47) using Baby-Step, Giant-Step algorithm.

Key Points:

Discrete Logarithm Problem: Find `x` in `a^x ≡ b (mod p)`.
Baby-Step, Giant-Step Algorithm: Efficiently solves DLP by splitting search space.
Parameter `m`: Chosen as `⌈√p⌉` to balance baby and giant step computations.
Baby Steps: Compute and store `a^j` for `j ∈ [0, m-1]` in a lookup table.

Answer: The discrete logarithm problem asks us to find an integer `x` such that `a^x ≡ b (mod p)`, where `a`, `b`, and `p` are known integers, and `p` is a prime number. In this case, we need to solve `5^x ≡ 22 (mod 47)` using the Baby-Step, Giant-Step algorithm, which provides a more efficient approach than brute-force search for medium-sized primes. First, we define our parameters: `a = 5`, `b = 22`, and `p = 47`. The algorithm begins by selecting an integer `m`, typically chosen as the ceiling of th...

Q9b. Alice wants to use the ElGamal digital signature scheme with public parameters p = 47, α = 2, secret value a = 7 and β = 34. 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?

Key Points:

ElGamal signature uses public (p, α, β) and private (a) keys.
A fresh random integer 'k' is chosen for each message signature.
First signature component 'r' is computed as α^k mod p.
Modular inverse k⁻¹ mod (p-1) is crucial for 's' calculation.

Answer: Alice will use the ElGamal digital signature scheme to sign message M = 20 using the provided public and private parameters. The process involves generating two components, `r` and `s`, which together form the digital signature. First, Alice calculates the value `r`. This is done by raising the public generator `α` to the power of the chosen secret random integer `k`, modulo `p`. `r = α^k mod p` `r = 2^5 mod 47` `r = 32 mod 47` `r = 32` Next, Alice needs to compute the modular multiplicative ...

Q9c. Alice wants to use the Digital Signature algorithm for signing messages. She chooses p = 83, q = 41, g = 2 and α = 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?

Key Points:

DSA signature involves private key x (here α=3) and ephemeral secret k.
Alice first computes public key y = g^x mod p.
Signature component r = (g^k mod p) mod q.
Signature component s = (k⁻¹ (H(M) + x*r)) mod q.

Answer: Alice wants to sign message M = 20 using DSA. She has public parameters p = 83, q = 41, g = 2. Her private key is x = 3 (given as α), and she chooses a per-message secret k = 8. First, Alice computes her public key, y = g^x mod p. Substituting the values, y = 2^3 mod 83 = 8 mod 83 = 8. This public key y is available for Bob to verify the signature. Next, Alice computes the first part of the signature, r. This involves calculating r = (g^k mod p) mod q. Here, 2^8 = 256. Then, 256 mod 83 = 7. Fi...

MMTE-006 Solved Assignment — January 2026 / July 2026

CRYPTOGRAPHY | IGNOU Master of Science (Mathematics with Applications in Computer Science) (MSCMACS)

MMTE-006—January 2026 / July 2026

CRYPTOGRAPHY

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MMTE-006 Assignment Questions — January 2026

Q1a. Let f(x) = x3 − x − 1 ∈ Z5 [x]. 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.

Q1bi. Prepare logarithm and antilogarithm tables as given in page 23 of block 1.

Q1bii. Compute (γ²+γ+1)(γ²+γ+1) and γ²(γ²+x+1) using the logarithm and antilogarithm tables.

Q2ai. Text: "CBBGYAEBBFZCFEPXYAEBB", encrypted with affine cipher with key (7,2).

Q2aii. Text:"KSTYZKESLNZUV", encrypted with Vigenère cipher with key "RESULT".

Q3a. Find the inverse of 13 (mod 51) using extended euclidean algorithm.

Q3b. Use Miller-Rabin test to check whether 75521 is a strong pseuodprime to the base 2.

Q5a. Decrypt the ciphertext 101000111001 which was encrypted with the Toy block cipher once using the key 101010010. Show all the steps.

Q5bi. Check whether the key is error free using the parity bits.

Q5bii. Find the keys for the second round.

Q6a. Considering the bytes 10001001 and 10101010 as elements of the field F2[X]/(g(X)), where g(X) is the polynomial X8 + X4 + X³ + X + 1, find their product and quotient.

Q6b. Find a recurrence that generates the sequence 110110110110110.

Q7a. Apply the frequency test, serial test and autocorrelation test to the following sequence at level of significance α = 0.05: 011001110000110010011100.

Q7b. Apply poker test to the following sequence with level of significane α = 0.05. 1001101000010000101111011 01110100101101100100110.

Q7c. Apply runs test to the following sequence: 1001101000010000101111011 0111010010110110010011010 0110011100001100100111000 1100001101010111101001110 0010001111000001101010010 1000110100000110100101101 1110001001

Q8a. Decrypt the message c = 23 that was encrypted using RSA algorithm with e = 43 and n = 77.

Q8bi. Bob uses ElGamal cyrptosystem with parameters p = 47, g = 5 and the secret value x = 3. What values will he make public?

Q8bii. Alice wants to send Bob the message M = 15. She chooses k = 5. How will she compute the cipher text? What information does she send to Bob?

Q8biii. Explain how Bob will decrypt the message.

Q9a. Solve the discrete logarithm problem 5x = 22 (mod 47) using Baby-Step, Giant-Step algorithm.