uni/year2/semester1/logseq-stuff/pages/DIffie-Hellman Key Exchange.md

• #CT255 - Next Generation Technologies II
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• Groups, Rings, & Fields

  • In mathematics,
    • a group is a set equipped with a binary operation that is associative, has an identity element, and is such that every element has an inverse, e.g., (\mathbb{Z}, +).
    • a ring is a set equipped with two binary operations satisfying properties analogous to those of addition & multiplication of integers, e.g. (\mathbb{Z}, +, *).
    • a field is a set on which addition, subtraction, multiplication, & division are defined and behave as the corresponding operations on rational & real numbers do.

• Diffie-Hellman Key Exchange

  • What is the Diffie-Hellman Key Exchange? #card card-last-interval:: 3.69 card-repeats:: 2 card-ease-factor:: 2.46 card-next-schedule:: 2022-11-22T10:36:42.441Z card-last-reviewed:: 2022-11-18T18:36:42.441Z card-last-score:: 5
    • Diffie-Hellman provides secure key exchange between two partners.
      • The negotiated key is subsequently used for private key encryption / authentication.
      • It uses the multiplicative group of integers modulo n (\mathbb{Z} / n \mathbb{Z})^x.
      • It is based on the difficulty of computing discrete logarithms over such groups, e.g.:
        • 6^3 \text{ mod } 17 = 216 \text{ mod } 17 =12 \text{ (easy) }
        • 12 = 6 ^y \text{ mod } 17 ? \text{ hard }
    • The core equation for the key exchange is
      • K = (A)^B \text{ mod } q

  • Diffie-Hellman: Global Public Elements

    • Select a prime number q and positive and a positive integer a, where a < q and a is a primitive root of q.
    • What is a primitive root? #card card-last-interval:: 2.8 card-repeats:: 2 card-ease-factor:: 2.6 card-next-schedule:: 2022-11-24T08:08:29.696Z card-last-reviewed:: 2022-11-21T13:08:29.697Z card-last-score:: 5
      • a is a primitive root of q, if numbers a \text{ mod } q, a^2 \text{ mod } q, \cdots , a^{q-1} \text{ mod } q are distinct integer values between 1 and (q-1) in some permutation, i.e., elements of (\mathbb{Z} / q \mathbb{Z})^x.
      • Example: a = 3 is a primitive root of (\mathbb{Z} / 5\mathbb{Z})^x, a=4 is not: background-color:: green

  • Generation of Secret-Key

    • Both users share a public prime number q and primitive root a.
    • User A:
      • 1. Select secret number XA with XA < q.
        2. Calculate public value YA = a^{XA} \text{ mod } q (difficult to reverse).
        3. YA is sent to User B.
    • User B:
      • 1. Select secret number XB with XB < q.
        2. Calculate public value YB = a^{XB} \text{ mod } q (difficult to reverse).
        3. YB is sent to User A.
    • User A:
      • User A owns XA and receives YB.
      • Generate secret key: K = (YB)^{XA} \text{ mod } q.
    • User B:
      • User B owns XB and receives YA.
      • Generate secret key: K = (YA)^{XB} \text{ mod } q.
    • Both keys are identical.

  • Diffie-Hellman in Practice

    • The algorithm is used in tandem with a variety of secure network protocols.
      • Provision of secure end-to-end connection.
      • No endpoint authentication - you can't validate who you are talking to.
      • Modulus p typically has a minimum length of 1024 bits.

  • DH & Man-in-the-Middle (MitM) Attacks

    • 
    • Mallory is a MitM attacker and performs message interception & message fabrication.
    • Mallory establishes two individual (secure) connections with Alice & Bob.
    • Neither Alice nor Bob are aware of Mallory's existence (as there is no authentication).