MQV

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.

MQV has some weaknesses that were fixed by HMQV in 2005.^[1] A few articles^[2]^[3] offered alternative viewpoint. It is now known that HMQV is vulnerable to a KCI attack when ephemeral public keys are not validated

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

Description

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) in the same way as Alice.
3	Now, Alice calculates $S_a = x + \bar{X} a$ and sends X to Bob.
4	Bob calculates $S_b = y + \bar{Y} b$ and sends Y to Alice.
5	Alice calculates $K = h \cdot S_a (Y + \bar{Y}B)$ and Bob calculates $K = h \cdot S_b (X + \bar{X}A)$ 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.

Correctness

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 $K = h \cdot S_b S_a P$

References

↑ Krawczyk, H. (2005). "HMQV: A High-Performance Secure Diffie-Hellman Protocol". Advances in Cryptology – CRYPTO 2005. Lecture Notes in Computer Science. 3621. pp. 546–566. doi:10.1007/11535218_33. ISBN 978-3-540-28114-6.
↑ Koblitz, Neal (2007). "The Uneasy Relationship Between Mathematics and Cryptography" (PDF). Notices of the AMS. 54 (8): 972–979.
↑ "Letters to the Editor" (PDF). Notices of the AMS. 54 (11): 1454–1456. 2007.

Bibliography

Kaliski, B. S., Jr (2001). "An unknown key-share attack on the MQV key agreement protocol". ACM Trans. Inf. Syst. Secur. 4 (3): 275–288. doi:10.1145/501978.501981.
Law, L.; Menezes, A.; Qu, M.; Solinas, J.; Vanstone, S. (2003). "An Efficient Protocol for Authenticated Key Agreement". Des. Codes Cryptography. 28 (2): 119–134. doi:10.1023/A:1022595222606.
Leadbitter, P. J.; Smart, N. P. (2003). "Analysis of the Insecurity of ECMQV with Partially Known Nonces". Information Security. 6th International Conference, ISC 2003, Bristol, UK, October 1–3, 2003. Proceedings. Lecture Notes in Computer Science. 2851. pp. 240–251. doi:10.1007/10958513_19. ISBN 978-3-540-20176-2.
Menezes, Alfred J.; Qu, Minghua; Vanstone, Scott A. (2005). Some new key agreement protocols providing implicit authentication (PDF). 2nd Workshop on Selected Areas in Cryptography (SAC '95). Ottawa, Canada. pp. 22–32.
Hankerson, D.; Vanstone, S.; Menezes, A. (2004). Guide to Elliptic Curve Cryptography. Springer Professional Computing. New York: Springer. doi:10.1007/b97644. ISBN 0-387-95273-X.

External links

Public-key cryptography

Algorithms

Integer Factorization	Benaloh Blum–Goldwasser Cayley–Purser Damgård–Jurik GMR Goldwasser–Micali Paillier Rabin RSA Okamoto–Uchiyama Schmidt–Samoa

Discrete logarithm	Cramer–Shoup DH DSA ECDH ECDSA EdDSA EKE ElGamal signature scheme MQV Schnorr SPEKE SRP STS

Others	AE CEILIDH EPOC HFE IES Lamport McEliece Merkle–Hellman Naccache–Stern Naccache–Stern knapsack cryptosystem NTRUEncrypt NTRUSign Three-pass protocol XTR

Theory

Standardization

Topics

Cryptography

History of cryptography Cryptanalysis Outline of cryptography

Symmetric-key algorithm Block cipher Stream cipher Public-key cryptography Cryptographic hash function Message authentication code Random numbers Steganography

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MQV

Description

Correctness

See also

References

Bibliography

External links