$$ \newcommand \pk {\mathrm{pk}} \newcommand \sk {\mathrm{sk}} \newcommand \fv {\text{first}} \newcommand \lv {\text{last}} \newcommand \Sign {\mathrm{Sign}} \newcommand \Verify {\mathrm{Verify}} \newcommand \Rand {\mathrm{Rand}} $$

Identity, Authorization, and Authentication

A player is uniquely identified by a 256-bit string \( I \) called an address.

Each player owns exactly one participation keypair. A participation keypair consists of a public key \( \pk \) and a secret key \( \sk \).

A keypair is defined in the specification of participation keys in Algorand. Each participation keypair is valid for a range of protocol rounds \( [r_\fv, r_\lv] \).

Let \( m \), \( m’ \) be arbitrary sequences of bits.

Let \( B_k \), \( \bar{B} \) be 64-bit integers representing balances in μALGO with rewards applied.

Let \( \tau \), \( \bar{\tau} \) be 32-bit integers, and \( Q \) be a 256-bit string.

Let \( (\pk_k, \sk_k) \) be some valid keypair.

A secret key supports a signing procedure

$$ y := \Sign(m, m’, sk_k, B_k, \bar{B}, Q, \tau, \bar{\tau}) $$

Where \( y \) is opaque and cryptographically resistant to tampering, where defined.

Signing is not defined on many inputs: for any given input, signing may fail to produce an output.

The following functions are defined on \( y \):

• Verifying: \( \Verify(y, m, m’, \pk_k, B_k, \bar{B}, Q, \tau, \bar{\tau}) = w \) where \( w \) is a 64-bit integer called the weight of \( y \). \( w \neq 0 \) if and only if \( y \) was produced by signing by \( \sk_k \) (up to cryptographic security). \( w \) is uniquely determined given fixed values of \( m’, \pk_k, B_k, \bar{B}, Q, \tau, \bar{\tau} \).

• Comparing: Fixing the inputs \( m’, \bar{B}, Q, \tau, \bar{\tau} \) to a signing operation, there exists a total ordering on the outputs \( y \). In other words, if \( f(\sk, B) = \Sign(m, m’, \sk, B, \bar{B}, Q, \tau, \bar{\tau}) = y \), and \( S = \{(\sk_0, B_0), (\sk_1, B_1), \ldots, (\sk_n, B_n)\} \), then \( \{f(x) | x \in S\} \) is a totally ordered set. We write that \( y_1 < y_2 \) if \( y_1 \) comes before \( y_2 \) in this ordering.

• Generating Randomness: Let \( y \) be a valid output of a signing operation with \( \sk_k \). Then \( R = \Rand(y, \pk_k) \) is defined to be a pseudorandom 256-bit integer (up to cryptographic security). \( R \) is uniquely determined given fixed values of \( m’, \pk_k, B_k, \bar{B}, Q, \tau, \bar{\tau} \).

The signing procedure is allowed to produce a nondeterministic output, but the functions above must be well-defined with respect to a given input to the signing procedure (e.g., a procedure that implements \( \Verify(\Sign(\ldots)) \) always returns the same value).