MQV

Key takeaways

MQV belongs to computer science; place it in that map before memorising details.
Learn the definition first, then one example that makes the definition concrete.
Connect MQV to a quantity you can measure, compute or draw — that is where exam questions come from.
Reproduce the core statement of MQV from memory before moving on to harder problems.

Reference excerpt

MQV (Menezes–Qu–Vanstone) is an authenticated protocol for key agreement based on the Diffie–Hellman scheme. Like other authenticated Diffie–Hellman schemes, MQV provides protection against an active attacker. The protocol can be modified to work in an arbitrary finite group, and, in particular, elliptic curve groups, where it is known as elliptic curve MQV (ECMQV). MQV was initially proposed by Alfred Menezes, Minghua Qu and Scott Vanstone in 1995. It was later modified in joint work with Laurie Law and Jerry Solinas. There are one-, two- and three-pass variants. MQV is incorporated in the public-key standard IEEE P1363 and NIST's SP800-56A standard. Some variants of MQV are claimed in patents assigned to Certicom. ECMQV has been dropped from the National Security Agency's Suite B set of cryptographic standards.

Description Alice has a key pair ( A , a ) {\displaystyle (A,a)} with A {\displaystyle A} her public key and a {\displaystyle a} her private key and Bob has the key pair ( B , b ) {\displaystyle (B,b)} with B {\displaystyle B} his public key and b {\displaystyle b} his private key. In the following R ¯ {\displaystyle {\bar {R}}} has the following meaning. Let R = ( x , y ) {\displaystyle R=(x,y)} be a point on an elliptic curve. Then R ¯ = ( x mod 2 L ) + 2 L {\displaystyle {\bar {R}}=(x\,{\bmod {\,}}2^{L})+2^{L}} where L = ⌈ ⌈ log 2 ⁡ n ⌉ 2 ⌉ {\displaystyle L=\left\lceil {\frac {\lceil \log _{2}n\rceil }{2}}\right\rceil } and n {\displaystyle n} is the order of the used generator point P {\displaystyle P} . So R ¯ {\displaystyle {\bar {R}}} are the first L bits of the first coordinate of R {\displaystyle R} .

Note: for the algorithm to be secure some checks have to be performed. See Hankerson et al.

Correctness Bob calculates:

K = h ⋅ S b ( X + X ¯ A ) = h ⋅ S b ( x P + X ¯ a P ) = h ⋅ S b ( x + X ¯ a ) P = h ⋅ S b S a P {\displaystyle K=h\cdot S_{b}(X+{\bar {X}}A)=h\cdot S_{b}(xP+{\bar {X}}aP)=h\cdot S_{b}(x+{\bar {X}}a)P=h\cdot S_{b}S_{a}P}

Alice calculates:

K = h ⋅ S a ( Y + Y ¯ B ) = h ⋅ S a ( y P + Y ¯ b P ) = h ⋅ S a ( y + Y ¯ b ) P = h ⋅ S b S a P {\displaystyle K=h\cdot S_{a}(Y+{\bar {Y}}B)=h\cdot S_{a}(yP+{\bar {Y}}bP)=h\cdot S_{a}(y+{\bar {Y}}b)P=h\cdot S_{b}S_{a}P}

So the shared secrets K {\displaystyle K} are indeed the same with K = h ⋅ S b S a P {\displaystyle K=h\cdot S_{b}S_{a}P}

… excerpt ends here. Continue reading the full article.

Worked examples

Example 1 — a first encounter with MQV

Start with the simplest possible case. Write down what MQV claims or describes in one sentence, then invent the smallest concrete situation in which that sentence is true. In computer science, the smallest case is usually a single object, a single equation or a single measurement. Check that every symbol or term in your sentence has a meaning in that case.

Example 2 — changing one variable

Take the situation from Example 1 and change exactly one quantity: double it, halve it, or set it to zero. Predict what should happen to MQV before you calculate. Comparing your prediction with the result is the fastest way to find out whether you understand the idea or only the words.

Example 3 — an exam-style question

Typical questions about MQV ask you to (a) state it precisely, (b) apply it to given data, and (c) explain a limitation. Practise writing all three answers in under five minutes; the third part is what separates a full-mark answer from an average one.

Applications of MQV

In research: MQV appears in computer science research whenever the underlying quantities have to be modelled precisely. Papers usually cite it as a starting assumption and then explore where it breaks down.
In technology and industry: Engineering practice reuses MQV in design rules, simulations and safety margins. Knowing the idea lets you read a specification sheet and understand why the numbers look the way they do.
In the classroom: MQV is common in secondary-school and first-year university syllabi. It links to neighbouring topics Elliptic curve cryptography, Key-agreement protocols, so understanding it makes those chapters shorter.
In everyday life: Look for MQV outside the textbook — in sport, cooking, traffic, electronics or the sky above you. An example you found yourself is remembered far longer than one you were given.

How to study MQV in 20 minutes

Read the reference excerpt below once, without taking notes.
Close the page and write down what MQV means in your own words.
Compare your version with the excerpt and mark what you missed.
Work through the three examples above with pen and paper.
Explain MQV out loud to somebody else — or to Teacher Smith in the lgStudy chat.

Frequently asked questions

What is MQV in simple terms?

MQV (Menezes–Qu–Vanstone) is an authenticated protocol for key agreement based on the Diffie–Hellman scheme. Like other authenticated Diffie–Hellman schemes, MQV provides protection against an active attacker.

Why does MQV matter?

Because it connects several computer science ideas at once: it gives you a definition you can apply, a quantity you can calculate, and a way to check whether a result is plausible.

How should I study MQV?

Read the excerpt, restate it from memory, then work through the examples and applications listed on this page. The five-step study plan above takes about twenty minutes.

What does this page cover?

It gives you a compact reference excerpt plus original lgStudy explanations, examples, applications and study material on MQV.