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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 Menezes, Qu and Vanstone in 1995. It was modified with Law and Solinas in 1998. There are one-, two- and three-pass variants.

MQV is incorporated in the public-key standard IEEE P1363.

Some variants of MQV are claimed in patents assigned to Certicom [1].

MQV has some weaknesses that were fixed by HMQV in 2005 [2]; see [3], [4], [5] for an alternative viewpoint.

ECMQV has been dropped from the National Security Agency's Suite B set of cryptographic standards.

Both MQV and HMQV have weaknesses, that are fixed in the FHMQV protocol (see [6])

## DescriptionEdit

Alice has a key pair (A,a) with A her public key and a her private key and Bob has the key pair (B,b) with B his public key and b his private key.

In the following $\bar{R}$ has the following meaning. Let $R = (x,y)$ be a point on an elliptic curve. Then $\bar{R} = (x\, \bmod\, 2^L) + 2^L$ where $L = \left \lceil \frac{\lfloor \log_{2} n \rfloor + 1}{2} \right \rceil$ and n is the order of the used generator point P. So $\bar{R}$ are the first L bits of the x coordinate of R.

Step Operation
1 Alice generates a key pair (X,x) by generating randomly x and calculating X=xP with P a point on an elliptic curve.
2 Bob generates a key pair (Y,y) by the same way than Alice.
3 Now, Alice calculates $S_a = x + \bar{X} a$ and sends X to Bob.
4 Bob calculates $S_b = y + \bar{Y} b$.
5 Alice calculates $K = h \cdot S_a (Y + \bar{Y}B)$ and Bob calculate $K = h \cdot S_b (X + \bar{X}A)$ where h is the cofactor (see Elliptic_curve_cryptography#Domain_parameters).
6 The communication of secret $K$ was successful. A key for a symmetric-key algorithm can be derived from K.

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

### CorrectnessEdit

Bob calculates: $K = h \cdot S_b (X + \bar{X}A) = h \cdot S_b (xP + \bar{X}aP) = h \cdot S_b (x + \bar{X}a)P = h \cdot S_b S_a P$.

Alice calculates: $K = h \cdot S_a (Y + \bar{Y}B) = h \cdot S_a (yP + \bar{Y}bP) = h \cdot S_a (y + \bar{Y}b)P = h \cdot S_b S_a P$.

So the keys K are indeed the same with $K = h \cdot S_b S_a P$