# Keys to Understanding WPA3 – SAE : Diffie-Hellman Key Exchange, Elliptic Curve Cryptography and Dragonfly Key Exchange

WPA3 certification is introduced by Wi-Fi Alliance in 2018 as a successor to WPA2. It aims to alleviate the vulnerabilities in WPA2 and provide more secure wireless networks.  It introduces new concepts like Simultaneous Authentication of Equals (SAE), dragonfly key exchange, NIST elliptical curve cryptography etc. To make it easier to understand WPA3 as a whole, I will be discussing each component individually in detail. WPA3 replaces Pre-Shared Key with Simultaneous Authentication of Equals (SAE) to derive the Pairwise Master Key (PMK) which enables secure communication even when the password is compromised. To understand how this is achieved, we need to understand how Diffie-Hellman key exchange and elliptical curve cryptography work in conjunction with Dragon fly key exchange.

Diffie-Hellman Key Exchange establishes session key between two entities without actually having to exchange any key information over a public insecure channel. Let’s get into the security terms of Alice and Bob being the two entities. Alice and Bob agree on two numbers g and p where p is a prime number. Alice chooses her private key to be a and Bob chooses b.

Alice calculates gamod p and sends it to Bob. Bob calculates gbmod p and sends it to Alice. This exchange happens over an insecure channel. Alice and Bob will perform the same multiplicative operation with modulo p against the values received.

```Alice             <--agree on g and p-->           Bob
gamod p            <----Exchange---->           gbmod p
(gbmod p)amod p      --Derive key--     (gamod p)bmod p```

For example, consider a=4 b=3 p=23 and g=5.

```Alice             <--agree on g=5 and p=23-->   Bob
gamod p = 4          <----Exchange---->      gbmod p = 10
(gbmod p)amod p = 18   --Derive key--   (gamod p)bmod p = 18```

The strength of the algorithm lies in the fact that (gbmod p)amod p is same as gbamod p and with large values of a,b and p it will be computationally close to impossible to obtain gbamod p without knowing the private keys a and b. This is an example of a trapdoor function which is nothing but a one-way function that states for a given x it is easy to calculate y = f(x) but very difficult to find x = f-1(y).  The basic concept of DH Exchange cannot be explained better without the paint analogy.

In this analogy g and p are common paint, a and b are secret colors and gabmod p is the common secret derived. This was one of the earliest implementations of Diffie Hellman algorithm. CWSP-206 study guide explains the same concept with different trapdoor function.

Here George and Billy agree on using 3 and 5 as their commonly agreed numbers and the operation they use is raised to the power.

```George (35=243)           ------------         Billy (35=243)
secret 4, 2434           <------------>        secret 7,  2437
(2434)7                   ------------         (2437)4```

Now that we have a good idea of what DH key exchange means, let’s take a look at Elliptic Curve Cryptography (ECC).

Elliptic curves like the one shown in the picture are set of points bound by the equation y2 = x3 + ax +b. Different curves use variations of this equation. To derive PMK, WPA2 uses a well-known hash function on the password whereas in WPA3, the password is indexed onto a point on the curve which is then used as generator to hash and derive the PMK. Hashing a password directly can be susceptible to dictionary attack. But it becomes very difficult doing it on generator points on an elliptic curve because change in a single character in the password can lead to a different generator point; hashing of which can result in a totally different PMK.

WPA3 also makes it impossible to derive PMK of individual sessions even when the password is compromised. Knowing the password can help the hacker identify the generator point on elliptic curve but due to the integration of Diffie-Hellman with ECC into Dragonfly key exchange makes it impossible to derive individual session PMK. The trap door function in this case could be scalar multiplication. According to discrete logarithmic problem, for two points Q and P on the elliptic curve where Q = n.P (n times P), it is impossible to determine ‘n’ based on only Q and P.

Let’s take a deeper dive into Dragonfly Key Exchange

The client device and access point in this diagram are both configured with a password for authentication. Client device chooses a secret A and access point chooses secret B. At this point let’s assume the password is already compromised and the hacker knows the generator point for PMK. Client hashes the secret A with generator point and transmits DH Hash A. Access point does similar process with secret B to create DH Hash B and transmits it to client. Having received DH Hash B, client hashes it with secret A to derive the PMK and access point hashes its secret B to derive the same PMK following the DH exchange as described earlier. Without knowing secret A and secret B, the hacker will not be able to derive PMK just from the password.

I hope this helped in understanding the WPA3 – SAE fundamentals. If you are interested in learning more I recommend the video playlist from Mojo networks on youtube which provides a simplified yet informative explanation on WPA3 concepts. I will be writing another blog post on frame exchanges during WPA3 – SAE authentication in the future.

References:

1. CWSP-206 Study and Reference Guide from Certitrek
2. Wikipedia
3. Youtube playlist on WPA3 Enhancements by Mojo Networks