spake2-0.4.2: Implementation of the SPAKE2 Password-Authenticated Key Exchange algorithm

Crypto.Spake2.Math

Description

This module ignores everything about networks, bytes, encoding, hash functions, and so forth. All it does is provide the mathematical building blocks for SPAKE2, as per Simple Password-Based Encrypted Key Exchange Protocols by Michel Abdalla and David Pointcheval.

## How it works

### Preliminaries

Let's say we have two users, user A and user B. They have already agreed on the following public information:

• cyclic group, $$G$$ of prime order, $$p$$
• generating element $$g \in G$$, such that $$g \neq 1$$
• hash algorithm to use, $$H$$

If the connection is asymmetric (e.g. if user A is a client and user B is a server), then they will also have:

• two arbitrary elements in $$M, N \in G$$, where $$M$$ is associated with user A and $$N$$ with user B.

If the connection is symmetric (e.g. if user A and B are arbitrary peers), then they will instead have:

• a single arbitrary element $$S \in G$$

The discrete log of these arbitrary elements must be difficult to guess.

And, they also have a secret password, which in practice will be an arbitrary byte string, but for the purposes of this module is an arbitrary scalar in the group that is a shared secret between both parties (see Crypto.Spake2.Groups for more information on scalars).

### The protocol

This is derived from the paper linked above.

One side, A, initiates the exchange. They draw a random scalar, $$x$$, and matching element, $$X$$, from the group. They then "blind" $$X$$ by adding it to $$M$$ multiplied by the password in scalar form. Call this $$X^{\star}$$.

$X^{\star} \leftarrow X \cdot M^{pw}$

to the other side, side B.

Side B does the same thing, except they use $$N$$ instead of $$M$$ to blind the result, and they call it $$Y$$ instead of $$X$$.

$Y^{\star} \leftarrow Y \cdot N^{pw}$

After side A receives $$Y^{\star}$$, it calculates $$K_A$$, which is the last missing input in calculating the session key.

$K_A \leftarrow (Y^{\star}/N^{pw})^x$

That is, $$K_A$$ is $$Y^{\star}$$ subtracted from $$N$$ scalar multiplied by $$pw$$, all of which is scalar multiplied by $$x$$.

Side B likewise calculates:

$K_B \leftarrow (X^{\star}/M^{pw})^y$

If both parties were honest and knew the password, the keys will be the same on both sides. That is:

$K_A = K_B$

### How to use the keys

The keys $$K_A$$ and $$K_B$$ are not enough to securely encrypt a session. They must be used as input to create a session key.

Constructing a session key is beyond the scope of this module. See createSessionKey for more information.

Synopsis

# Documentation

data Spake2 group Source #

An instance of the SPAKE2 protocol. This represents one side of the protocol.

Constructors

 Spake2 Fieldsparams :: Params group password :: Scalar group

data Params group Source #

The parameters of the SPAKE2 protocol. The other side needs to be using the same values, but with swapped values for ourBlind and theirBlind.

Constructors

 Params Fieldsgroup :: groupThe cyclic group used for encrypting keysourBlind :: Element groupThe "blind" we use when sending out values. Side A refers to this as $$M$$ in the protocol description.theirBlind :: Element groupThe "blind" the other side uses when sending values. Side A refers to this as $$N$$ in the protocol description.

startSpake2 :: (AbelianGroup group, MonadRandom randomly) => Spake2 group -> randomly (Spake2Exchange group) Source #

Initiate the SPAKE2 exchange. Generates a secret (xy) that will be held by this side, and transmitted to the other side in "blinded" form.

data Spake2Exchange group Source #

A SPAKE2 exchange that has been initiated.

computeOutboundMessage :: AbelianGroup group => Spake2Exchange group -> Element group Source #

Determine the element (either $$X^{\star}$$ or $$Y^{\star}$$) to send to the other side.

Arguments

 :: AbelianGroup group => Spake2Exchange group An initiated SPAKE2 exchange -> Element group The outbound message from the other side (i.e. inbound to us) -> Element group The final piece of key material to generate the session key.

Generate key material, $$K$$, given a message from the other side (either $$Y^{\star}$$ or $$X^{\star}$$).

This key material is the last piece of input required to make the session key, $$SK$$, which should be generated as:

$SK \leftarrow H(A, B, X^{\star}, Y^{\star}, K, pw)$

Where:

• $$H$$ is a hash function
• $$A$$ identifies the initiating side
• $$B$$ identifies the receiving side
• $$X^{star}$$ is the outbound message from the initiating side
• $$Y^{star}$$ is the outbound message from the receiving side
• $$K$$ is the result of this function
• $$pw$$ is the password (this is what makes it SPAKE2, not SPAKE1)