Extended Euclidean Algorithm

The Extended Euclidean Algorithm is an extension of the Euclidean Algorithm for finding the greatest common divisor (GCD) of two integers. In addition to computing the GCD, it also finds the coefficients

x and
$$

y such that

ax+by=gcd(a,b) for given integers

a and b.

Algorithm Steps:

1. Initialization: Initialize

   ${𝑥}_1=1$

   y1​=0, 

   
   ${}_{}$

   x2​=0, and
   ${𝑦}_2=1$

2. Loop: Repeat the following steps until the remainder becomes zero:
   • Divide
     

     $a$ by $b$ to get quotient 

     
     $$

     $q$ and remainder
     $\mathrm{<br>}$

     $r$: 𝑎
     $=𝑏𝑞+𝑟$

     
     $<span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;">Update</span>$

     
     $𝑎=\mathrm{𝑏 <span\; class="base"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="mord\; mathnormal"></span></span>}$

     
     $\mathrm{<span\; class="base"\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><span\; class="mord\; mathnormal">b</span><span\; class="mspace"></span><span\; class="mrel">=</span>r</span><span\; style="font-family:\; Georgia,\; \text{'}Times\; New\; Roman\text{'},\; \text{'}Bitstream\; Charter\text{'},\; Times,\; serif;"><br></span>}$

   • Update

     ${\mathrm{<br>}}_{}$

   • xi+1​=xi−1​−qxi​ and 

     
     ${}_{}$

     yi+1​=yi−1​−qyi​.

3. Termination: When the remainder becomes zero, the GCD is the last nonzero remainder

   $$

   r. The coefficients
   $𝑥$

   y provide the Bézout’s identity.

Use Cases:

• Modular Inverse: The Extended Euclidean Algorithm can be used to find the multiplicative inverse of
  

  $a$ modulo $m$ when 

  $a$ and $m$ are coprime.

• Diophantine Equations: It can be used to solve Diophantine equations of the form

  $$

  ax+by=c.

Advanced Encryption Standard (AES)

The Advanced Encryption Standard (AES) is a symmetric-key block cipher used for encryption and decryption of electronic data. It replaced the older Data Encryption Standard (DES) and became the de facto encryption standard. AES operates on fixed-size blocks of data and supports key lengths of 128, 192, and 256 bits.

Key Features:

1. Block Size: AES operates on 128-bit blocks of data, regardless of the key size.
2. Key Lengths: AES supports three key lengths: 128 bits, 192 bits, and 256 bits.
3. Substitution-Permutation Network (SPN): AES uses a round-based structure known as an SPN, which involves multiple rounds of substitution and permutation operations.
4. Key Expansion: AES employs a key expansion algorithm to generate round keys from the original key.
5. Rounds: The number of rounds in AES depends on the key size: 10 rounds for a 128-bit key, 12 rounds for a 192-bit key, and 14 rounds for a 256-bit key.
6. Round Operations: Each round of AES consists of four main operations: SubBytes (byte substitution), ShiftRows (row shifting), MixColumns (column mixing), and AddRoundKey (round key addition).
7. Security: AES is designed to be secure against known cryptographic attacks, including differential and linear cryptanalysis.

Use Cases:

• Data Encryption: AES is widely used to encrypt sensitive data in various applications, including communication protocols (e.g., HTTPS), disk encryption, and file encryption.
• Cryptography Standards: AES is included in cryptographic standards such as SSL/TLS, IPsec, and SSH.
• Secure Communication: It ensures confidentiality and integrity of data transmitted over insecure channels.

The Extended Euclidean Algorithm is a fundamental algorithm for computing the GCD of two integers and finding Bézout’s coefficients. It has applications in modular arithmetic, cryptography, and solving Diophantine equations. The Advanced Encryption Standard (AES) is a widely adopted symmetric-key block cipher used for secure data encryption. It provides strong security and is used in various applications where data confidentiality is critical.