Converting a Decimal Number to Binary Methods

1. Method 1: Divide by 2 Shortcut (Most Common and Easiest)

   1. This is the method most people use mentally or on paper.
   2. Steps:

      1. Divide the number by 2.

         1. Write down the remainder (it will be 0 or 1).
         2. Use the quotient (the result of the division) and repeat the steps until the quotient becomes 0.
         3. The binary is the remainders read from bottom to top.

      2. Example: Convert 13 to binary

         Division	Quotient	Remainder
         13 ÷ 2	6	1
         6 ÷ 2	3	0
         3 ÷ 2	1	1
         1 ÷ 2	0	1

      3. When the quotient is 0, stop.

      4. Now read remainders from bottom to top:

         1. → 1101

      5. So, 13 (decimal) = 1101 (binary)

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2. Method 2: Tip for Small Numbers (Head Method):

   1. You can memorize powers of 2:

      Power of 2	Value
      2⁷	128
      2⁶	64
      2⁵	32
      2⁴	16
      2³	8
      2²	4
      2¹	2
      2⁰	1

   2. Example: Convert 19 to binary using powers of 2

      1. 19 ≥ 16 → 1
      2. 19 - 16 = 3
      3. 8 > 3 → 0
      4. 4 > 3 → 0
      5. 2 ≤ 3 → 1 → 3 - 2 = 1
      6. 1 = 1 → 1

   3. Write it:

      1. 16 8 4 2 1
      2. 1 0 0 1 1 → 10011

Vernam Cipher in Cryptography

1. What is Vernam Cipher?

   1. The Vernam Cipher is a symmetric key cryptographic technique invented in 1917 by Gilbert Vernam.
   2. It’s also known as the One-Time Pad Cipher when the key is truly random and as long as the message.
   3. In simple words:
      1. It hides a message by combining it with a secret random key using the XOR operation (Exclusive OR).
   4. Key Principles
      1. Plaintext (P): The original message we want to protect.
         1. Example: HELLO
      2. Key (K): A random set of letters/numbers (same length as plaintext).
         1. Example: XMCKL
      3. Ciphertext (C): The encrypted message.
         1. Example: Something unreadable like EQNVZ.

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2. Vernam Cipher (Using XOR)

   Bit P	Bit K	Result (C = P ⊕ K)
   0	0	0
   0	1	1
   1	0	1
   1	1	0

   So, XOR flips bits when the key bit = 1, and keeps bits the same when the key bit = 0.

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3. How it Works (Step by Step)

   1. Assign numbers to alphabets (A=0, B=1, …, Z=25).
   2. Convert both the Plaintext and the Key to numbers.
   3. Combine them using XOR (⊕).
   4. Convert the numbers back to alphabets → that gives you the Ciphertext.
   5. To decrypt, apply the same key again → you get back the original plaintext.
   6. Alphabet to Number Mapping

      1. Here’s the full mapping table:

         A = 0 (00000)   B = 1 (00001)   C = 2 (00010)   D = 3 (00011)
         E = 4 (00100)   F = 5 (00101)   G = 6 (00110)   H = 7 (00111)
         I = 8 (01000)   J = 9 (01001)   K = 10 (01010)  L = 11 (01011)
         M = 12(01100)   N = 13(01101)   O = 14(01110)   P = 15(01111)
         Q = 16(10000)   R = 17(10001)   S = 18(10010)   T = 19(10011)
         U = 20(10100)   V = 21(10101)   W = 22(10110)   X = 23(10111)
         Y = 24(11000)   Z = 25(11001)
         

      2. Each letter is represented in 5 bits.

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4. Example 1

5. Example 2:

6. Summary