In: Computer Science

Explain the mathematical proof that One time pad (OTP) is perfectly secure under Ciphertext Only attack (COH).

The OTP is an encryption technique that cannot be cracked, but
it requires the use of a one-time pre-shared key the same size as,
or longer than, the message being sent. In this technique, a plain
text is paired with a random secret key. Then, each bit or
character of the plain text is encrypted by combining it with the
corresponding bit from the pad using modular addition.

The resulting ciphertext are going to be impossible to decrypt or
break if the subsequent given below four conditions are met:

- The Key must be truly random.
- The Key must be a minimum of as long because the plaintext.
- The Key must not be ever reused in whole or partially
- The key must be kept completely secret.

It has also proven that any cipher with the property of perfect secrecy must use keys with effectively the same requirements as OTP keys. Digital version of one-time pad cipher are employed by nation for critical diplomatic and military communication, but the issues of secure key distribution have made them impractical for many applications.

**Example:**

Suppose Joy wishes to send the message "HELLO" to Roy. Assume two pad of paper containing identical random sequences of the letters were previously produced and securely issued to both of them. Joy chooses the appropriate unused page from the pads. The way to do this, normally arranged for in advance, as for instance "use the next available sheet for the next message".

The material that is on the selected sheet is the key for the message. Each letter from the pad will be combined in a predetermined way. (It is commonly used way, but not required, to assign each letter a numerical value, "A" is 0, "B" is 1, "C" is 3 and so on.)

The technique to combine the key and the message using modular addition. The numerical values of corresponding message and key letters are added together, modulo 26. So, if key material begins with "XMCKL" and the message is "HELLO", then the coding be done as follows:

```
H E L L O → (message)
7 (H) 4 (E) 11 (L) 11 (L) 14 (O) → (message)
+ 23 (X) 12 (M) 2 (C) 10 (K) 11 (L) → (key)
= 30 16 13 21 25 → (message + key)
= 4 (E) 16 (Q) 13 (N) 21 (V) 25 (Z) → (message + key) mod 26
E Q N V Z → (ciphertext)
```

If number is larger than 25, then the remainder after subtraction of 26 is taken in modular arithmetic fashion. This means that if the computations "go past" of Z, the sequence starts again from A.

The ciphertext to be sent to Roy is thus "EQNVZ". Roy use the matching key page and the same process, but in reverse, to obtain plain text(message sent by joy will be decrypted). Here the key is subtracted from the ciphertext, again using modular arithmetic fashion:

```
E Q N V Z → (ciphertext)
4 (E) 16 (Q) 13 (N) 21 (V) 25 (Z) → (ciphertext)
- 23 (X) 12 (M) 2 (C) 10 (K) 11 (L) → (key)
= -19 4 11 11 14 → (ciphertext – key)
= 7 (H) 4 (E) 11 (L) 11 (L) 14 (O) → (ciphertext – key) mod 26
H E L L O → (message)
```

from the above, if a number is negative, then 26 is added to make the number zero or higher.

Thus Roy recovers Joy's plain text, the message "HELLO". Both Joy and Roy destroy the key sheet immediately after use, thus preventing reuse and attack against the cipher.

**Mathematical Proof:**

Claude Shannon proved, that the one-time pad has a property he termed perfect secrecy; that is, the ciphertext C gives absolutely no additional information about the plain text. This is because, given a truly random key that is used only once for the message and than destroyed, a ciphertext can be translated into any plain text of the same length, and all are equal. Thus, the priori probability of a plaintext message M is the same as the a posteriori probability of a plaintext message M given the ciphertext.

**Mathematically, this is expressed as
H(M)=H(M|C)** , where **H(M)** is the
information entropy of the plaintext and **H(M|C)** is
the conditional entropy of the plaintext for the given ciphertext
C. **Η** is the capital Greek letter eta.
This implies that for every message M and its corresponding
ciphertext C, there must be at least one key K that will bind them
as a one-time pad. Mathematically, this means
, where **K,C,M** are the distinct quantity of key,
cipher and message

Another way of saying that perfect secrecy is based on the idea that for all messages m1,m2 in message space M, and for all ciphers c in cipher space C, we have

where **Pr** represents the probabilities, taken
over a choice **k** in key space **K**
over the coin tosses of a probabilistic algorithm,
**E.** Perfect secrecy is a strong notion of the
cryptanalytic difficulty.

what is MAC address flooding attack ? briefly explain any one
meathod to prevent this attack?

When there is only one buyer in the market
a. then the market will be perfectly competitive
b. a monopsony exists
c. a closed shop exists.
d. the supply curve for the good will be perfectly
elastic.
Any practice that forces employers to use more labor than they
would otherwise use is
a. monopolistic exploitation
b. a craft union
c. closed shop
d. featherbedding.
IN MICROECONOMICS

Consider the one-time pad with messages and key-streams both
being binary sequences. Suppose that the system is used
erroneously, so that two messages have been encrypted using the
same key. What information can an adversary that hears the two
ciphertexts deduce about the plaintexts? Show this mathematically
using EXOR logic

1. Draw equilibrium of a perfectly competitive firm under the
following conditions (one diagram for each):
a) Short run profits
b) Short run losses
c) Long run equilibrium with zero economic profits
2. State the condition (in terms of costs and revenues) that
describes allocative efficiency that occurs in LR equilibrium and
explain what happens to Producer Surplus in this case.
3. A firm accepts a price of $22 in a market consisting of many
producers all making essentially the...

Write the Faraday’s Law. mathematical expressions (integral form
only). Explain each symbol, each detail in each of these equations
– by words. Draw sketches when they are necessary for your
explanation.

1.
Adjusting entries are needed
Select one:
a. only under the cash basis of accounting.
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c. only under IFRS
d. to produce relevant financial information for
users.
e. to update accounts at the beginning of the accounting
period.
2.
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and collects $3,250 of this. It incurs $4,200 in expenses on
account during the current year and pays $2,600 of them. Walmart
would report what...

Give an example of a perfectly competitive market (or one that
closely approximates one). Explain why you believe this market is
competitive. If you can’t think of such a market, explain why it is
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Use the thermodynamic arguement to explain why the perfectly
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form.
Use the same equation to explain why higher temperatures
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Which of the weak forces is the most important in the overall
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(a) Under what condition, does the angular momentum become a good quantum number? (Explain your answer with the mathematical description.)
(b) In that case, what do Hamiltonian and angular momentum operators share? (Briefly explain why you came up with the answer)
(c) In general, boundary conditions give discrete eigenvalues. What is the 'boundary condition' for a rotation problem? (What is the physical reason for discrete angular momentum values?)
(d) Can you explain why the orbital angular momentum operator cannot...

Choose one type of proof from the list below and explain how you
might apply the same principles to help ensure that a piece of
software meets desired specification and has fewer bugs. Be sure to
reference software testing standards as well as the mathematical
proof principles. proof by example or counter-example proof based
on if-then and if and only if contingencies proof by contradiction
proof using cases proof by induction

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