Computer Science & Data Science · Cryptography and security

Diffie–Hellman key exchange (teaching model)

Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.

AdvancedBachelor foundationsSubject specialization

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Formula

A = g^{a} mod p, B = g^{b} mod p, K_{A} = B^{a} mod p = K_{B} = A^{b} mod p = g^{ab} mod p

a, b, g, p, A, B and K are integers or residue classes; no physical unit

Variables & units

Variables & units
Symbol	Meaning	Unit
$p$	Prime modulus defining the finite-field setting	integer
$g$	Public group element of suitable order modulo p	residue class modulo p
$a,b$	Private secret exponents of the two parties	integer
$A,B$	Public values gᵃ mod p and gᵇ mod p	residue class modulo p
$K$	Shared modular group value before key derivation	residue class modulo p

LaTeX

A=g^a\bmod p,\quad B=g^b\bmod p,\quad K_A=B^a\bmod p=K_B=A^b\bmod p=g^{ab}\bmod p

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Derivation & background

The equality follows from (gᵇ)^a≡(gᵃ)^b≡g^(ab) mod p. The teaching model assumes p and g describe a suitable group with sufficiently large order; arithmetic modulo an arbitrary number is not secure parameter selection. The exchange alone does not authenticate either party and is vulnerable to man-in-the-middle attacks. The shared K is not yet an application key: real protocols need authentication, standardized groups, key derivation and secure implementations. Small numbers are only for hand calculation. The RFC supplies the equation; CS2023 lists security as a knowledge area.

Exam blueprint

Validity range

Derivation steps

Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.

1Read the relation from its definition: A=g^a\bmod p,\quad B=g^b\bmod p,\quad K_A=B^a\bmod p=K_B=A^b\bmod p=g^{ab}\bmod p.
2Identify the input, requested output and assumptions of the relation.
3Only as an insecure arithmetic example: p=23, g=5, a=6 and b=15. Then A=5⁶ mod 23=8 and B=5¹⁵ mod 23=19. Alice computes 19⁶ mod 23=2; Bob obtains 8¹⁵ mod 23=2. Check: both equal 5⁹⁰ mod 23=2. These tiny parameters are immediately breakable and must never be used in production.

Rearrangements

Gemeinsames Geheimnis

K=Bᵃ mod p=Aᵇ mod p

Gilt im selben geeigneten endlichen Gruppenmodell.

Task variant

Use the example for Diffie–Hellman key exchange (teaching model) to explain which inputs determine the requested output.

The example connects concrete inputs to its result.
Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.
The relation applies only under its stated assumptions.

For p=23, g=5, a=6 and b=15, calculate both public values and the shared residue.

A=5⁶ mod 23=8 and B=5¹⁵ mod 23=19.
Then B⁶ mod 23=2 and A¹⁵ mod 23=2.
These parameters are an insecure hand-calculation example, not a production key.

Common mistakes

Applying the relation to a different case without checking its stated assumptions.

Check the validity range before substituting values: The equality follows from (gᵇ)^a≡(gᵃ)^b≡g^(ab) mod p. The teaching model assumes p and g describe a suitable group with sufficiently large order; arithmetic modulo an arbitrary number is not secure.

Exam context

Understand modular arithmetic and the idea of key agreement; this is not a guide to production cryptography.

Formula cluster

Cryptography and security

Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.

gcd(a,m)=1 ⇔ a⁻¹ mod m existiert

Euclidean algorithm and modular inverse

c = mᵉ mod n; m = cᵈ mod n

RSA encryption: modular exponentiation

Pcoll ≈ 1−exp(−n(n−1)/(2M))

Hash collision probability (birthday bound)

Worked example

Only as an insecure arithmetic example: p=23, g=5, a=6 and b=15. Then A=5⁶ mod 23=8 and B=5¹⁵ mod 23=19. Alice computes 19⁶ mod 23=2; Bob obtains 8¹⁵ mod 23=2. Check: both equal 5⁹⁰ mod 23=2. These tiny parameters are immediately breakable and must never be used in production.

Applications

Understand modular arithmetic and the idea of key agreement; this is not a guide to production cryptography.

How do you calculate with Diffie–Hellman key exchange (teaching model)?

Here is how to work through a typical Diffie–Hellman key exchange (teaching model) (A=gᵃ mod p; B=gᵇ mod p; K=Bᵃ=Aᵇ mod p) task step by step:

1
Task
Use the example for Diffie–Hellman key exchange (teaching model) to explain which inputs determine the requested output.
Solution path
1. The example connects concrete inputs to its result.
2. Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.
3. The relation applies only under its stated assumptions.
2
Task
For p=23, g=5, a=6 and b=15, calculate both public values and the shared residue.
Solution path
1. A=5⁶ mod 23=8 and B=5¹⁵ mod 23=19.
2. Then B⁶ mod 23=2 and A¹⁵ mod 23=2.
3. These parameters are an insecure hand-calculation example, not a production key.

Frequently asked questions about Diffie–Hellman key exchange (teaching model)

Why do both parties compute the same value?+

Alice publishes A=gᵃ mod p and Bob publishes B=gᵇ mod p while keeping their exponents secret. Alice raises B to a and obtains (gᵇ)ᵃ=gᵃᵇ modulo p. Bob raises A to b and obtains (gᵃ)ᵇ=gᵃᵇ modulo p. Exponent multiplication is commutative, so both residue classes match. An observer sees p, g, A and B but would need to solve a discrete-logarithm problem to recover a private exponent. Teaching model over a suitable finite group; example values are cryptographically insecure and must never be used in production.

What roles do p and g play?+

p defines the modular number system and g is a public group element whose powers form the possible public values. A secure construction needs a suitable subgroup with known, large order; merely saying “p is prime” and choosing any g is not a complete parameter check. This page uses the equation to illustrate modular arithmetic. Production protocols select groups according to current standards and validate public values as required by the protocol. Teaching model over a suitable finite group; example values are cryptographically insecure and must never be used in production.

How does the small-number example work?+

For p=23, g=5, a=6 and b=15, Alice computes A=5⁶ mod 23=8 and Bob computes B=5¹⁵ mod 23=19. Alice obtains B⁶ mod 23=19⁶ mod 23=2; Bob obtains A¹⁵ mod 23=8¹⁵ mod 23=2. Matching values check the equation. The numbers are intentionally small so repeated squaring can be checked by hand. For the same reason, a computer or attacker can recover the private exponents immediately. Teaching model over a suitable finite group; example values are cryptographically insecure and must never be used in production.

Does Diffie–Hellman alone prevent a man-in-the-middle attack?+

No. Without authentication, an attacker can impersonate Bob to Alice and Alice to Bob. The attacker then establishes two separate key agreements and relays messages between them. The bare equation does not prove who shares the key. Real protocols bind the exchange to certificates, signatures or other authenticated mechanisms and need careful key derivation. This page explains the mathematical idea; it is not a protocol specification or security review. Teaching model over a suitable finite group; example values are cryptographically insecure and must never be used in production.

Is K already the key for encrypting a message?+

K is initially only a modular group value and should not be used directly as a symmetric key in real protocols. A standardized key-derivation mechanism processes the shared secret together with context, roles and the protocol transcript to produce appropriately sized key material. An authenticated-encryption mode then protects the message and detects tampering. Randomness, ephemeral-key lifetime, identities and error handling also matter. The equation illustrates agreement on a shared value, not a complete secure communication channel. Teaching model over a suitable finite group; example values are cryptographically insecure and must never be used in production.

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Subject references and further reading

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Diffie–Hellman key exchange (teaching model)

Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.

AdvancedBachelor foundationsSubject specialization

Free to browse · flashcards or a multiple-choice quiz · import from Essential

Variables & units

Symbol

Meaning

Unit

p

Prime modulus defining the finite-field setting

integer

g

Public group element of suitable order modulo p

residue class modulo p

a,b

Private secret exponents of the two parties

integer

A,B

Public values gᵃ mod p and gᵇ mod p

residue class modulo p

K

Shared modular group value before key derivation

residue class modulo p

Derivation & background

Exam blueprint

Validity range

Derivation steps

Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.

1Read the relation from its definition: A=g^a\bmod p,\quad B=g^b\bmod p,\quad K_A=B^a\bmod p=K_B=A^b\bmod p=g^{ab}\bmod p.
2Identify the input, requested output and assumptions of the relation.
3Only as an insecure arithmetic example: p=23, g=5, a=6 and b=15. Then A=5⁶ mod 23=8 and B=5¹⁵ mod 23=19. Alice computes 19⁶ mod 23=2; Bob obtains 8¹⁵ mod 23=2. Check: both equal 5⁹⁰ mod 23=2. These tiny parameters are immediately breakable and must never be used in production.

How do you calculate with Diffie–Hellman key exchange (teaching model)?

Here is how to work through a typical Diffie–Hellman key exchange (teaching model) (A=gᵃ mod p; B=gᵇ mod p; K=Bᵃ=Aᵇ mod p) task step by step:

Task

Use the example for Diffie–Hellman key exchange (teaching model) to explain which inputs determine the requested output.

Solution path

The example connects concrete inputs to its result.
Diffie–Hellman lets two parties derive the same modular group value from public messages without sending their private exponents.
The relation applies only under its stated assumptions.

Task

For p=23, g=5, a=6 and b=15, calculate both public values and the shared residue.

Solution path

A=5⁶ mod 23=8 and B=5¹⁵ mod 23=19.
Then B⁶ mod 23=2 and A¹⁵ mod 23=2.
These parameters are an insecure hand-calculation example, not a production key.

Frequently asked questions about Diffie–Hellman key exchange (teaching model)

Why do both parties compute the same value?+

What roles do p and g play?+

How does the small-number example work?+

Does Diffie–Hellman alone prevent a man-in-the-middle attack?+

Is K already the key for encrypting a message?+

Retain Diffie–Hellman key exchange (teaching model) for exams

Create a curated FSRS exam set for A=gᵃ mod p; B=gᵇ mod p; K=Bᵃ=Aᵇ mod p: formula recall, variables, derivation, rearrangement, worked example, common mistakes and exam context.

Curated formula set · LaTeX · FSRS spaced repetition · import from Essential