Math Problem Statement

מהו המפתח הסודי המשותף שחושב על ידי שני הצדדים בעזרת תהליך ההחלפה של דיפי-הלמן אם: n = 23, g = 5, ואחד הצדדים בוחר את החלק הסודי שלו להיות 3, בעוד שהצד השני בוחר את החלק הסודי שלו להיות 9?

Solution

The image contains a cryptographic problem about the Diffie-Hellman key exchange. Below is a detailed analysis of the question:


Question (translated and analyzed):

What is the shared secret key, computed using the Diffie-Hellman process? If:

  • n=23n = 23
  • g=5g = 5
  • One party chooses their private key to be a=3a = 3
  • The other party selects their private key to be b=9b = 9

Solution:

The Diffie-Hellman key exchange process involves two parties (often called Alice and Bob) who agree on:

  1. A prime number n=23n = 23 (the modulus).
  2. A generator g=5g = 5 (base value used for exponentiation).

Each participant chooses a private key:

  • Alice chooses a=3a = 3.
  • Bob chooses b=9b = 9.

Then:

  • Alice sends Bob the value A=ga(modn)A = g^a \pmod{n}.
  • Bob sends Alice the value B=gb(modn)B = g^b \pmod{n}.

Finally, both compute the shared secret key independently:

  • Alice computes K=Ba(modn)K = B^a \pmod{n}.
  • Bob computes K=Ab(modn)K = A^b \pmod{n}. Since both computations result in the same value, they both end up with the same shared secret.

Step-by-Step Calculations:

  1. Alice sends Bob: A=ga(modn)=53(mod23)=125(mod23)=10A = g^a \pmod{n} = 5^3 \pmod{23} = 125 \pmod{23} = 10

  2. Bob sends Alice: B=gb(modn)=59(mod23)B = g^b \pmod{n} = 5^9 \pmod{23}

    • First, calculate 59=19531255^9 = 1953125.
    • Now, 1953125(mod23)=111953125 \pmod{23} = 11.
  3. Alice computes the shared secret: K=Ba(modn)=113(mod23)K = B^a \pmod{n} = 11^3 \pmod{23}

    • First, calculate 113=133111^3 = 1331.
    • Now, 1331(mod23)=191331 \pmod{23} = 19.
  4. Bob computes the shared secret: K=Ab(modn)=109(mod23)K = A^b \pmod{n} = 10^9 \pmod{23}

    • First, calculate 109=100000000010^9 = 1000000000.
    • Now, 1000000000(mod23)=191000000000 \pmod{23} = 19.

Thus, the shared secret key is: 19\boxed{19}


Do you want more details on any step or further explanation on Diffie-Hellman exchange?

Related Questions:

  1. What is the purpose of the generator gg in Diffie-Hellman?
  2. How can you enhance the security of Diffie-Hellman with larger primes?
  3. What makes the shared secret in Diffie-Hellman secure against eavesdropping?
  4. Can the Diffie-Hellman method be used for symmetric encryption key sharing?
  5. What vulnerabilities does Diffie-Hellman have in modern cryptography?

Tip:

In cryptography, it’s important to choose sufficiently large prime numbers to prevent brute force attacks or discrete logarithm attacks.

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Math Problem Analysis

Mathematical Concepts

Cryptography
Modular Arithmetic
Number Theory

Formulas

A = g^a mod n
B = g^b mod n
K = B^a mod n
K = A^b mod n

Theorems

Diffie-Hellman Key Exchange

Suitable Grade Level

Undergraduate - Computer Science or Cryptography course