TimeVariantBlake3.tex

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\title{Time-Differential Blake3 Initialization Vector Generation}
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\date{}

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\fancyhead[L]{President Harmon, David R.} % Left side header
\fancyfoot[C]{Copyright 2025 The Mapleseed Inc.} % Centered footer

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\maketitle

\section{Algorithm Definition}

Let $\mathcal{T}$ be the set of system timestamps, and $\mathcal{P}$ be the set of prime numbers. The initialization vector $IV \in \mathbb{F}_{2^{32}}^8$ is generated through the following procedure:

\subsection{Temporal Signature Collection}

For a sequence of timestamps $t_i \in \mathcal{T}$, where $i \in \{1,\ldots,8\}$, define:

\[
\tau(t_i) = t_i \oplus (\nu_{\text{cpu}} \cdot i)
\]

where $\nu_{\text{cpu}}$ represents the CPU frequency in Hz.

The composite temporal signature $\sigma_t$ is defined as:

\[
\sigma_t = \bigoplus_{i=1}^8 \tau(t_i)
\]

\subsection{Prime Distance Calculation}

For each temporal component, calculate the prime distance function $\delta_p$:

\[
\delta_p(x) = \min\{p - x \mid p \in \mathcal{P}, p > x\}
\]

The prime distance vector $\Delta = (\delta_1, \ldots, \delta_8)$ is computed as:

\[
\delta_i = \delta_p(\sigma_t + i\cdot\omega)
\]

where $\omega$ is the word size (32 bits).

\subsection{Entropy Mixing Function}

Define the entropy mixing function $\mathcal{E}: \mathbb{F}_{2^{32}} \rightarrow \mathbb{F}_{2^{32}}$:

\[
\mathcal{E}(x) = x \oplus \text{ROT}_r(x) \oplus \eta
\]

where:
\begin{itemize}
  \item $\text{ROT}_r$ is a right rotation by $r$ bits
  \item $\eta$ is system-specific entropy
  \item $r = \lfloor\log_2(x)\rfloor \bmod 32$
\end{itemize}

\subsection{Initialization Vector Generation}

The final IV components are generated as:

\[
IV_i = \mathcal{E}(\delta_i) \oplus \mathcal{H}(m_i)
\]

where:
\begin{itemize}
  \item $\mathcal{H}$ is an auxiliary hash function
  \item $m_i$ represents system memory statistics
\end{itemize}

\section{Security Properties}

\subsection{Entropy Bounds}

The minimum entropy contribution from each source is bounded by:

\[
H_{\min}(\tau) \geq \log_2(\nu_{\text{cpu}}) + \log_2(t_{\text{precision}})
\]

where $t_{\text{precision}}$ is the system's temporal resolution.

\subsection{Prime Distance Security}

The prime distance function provides a minimum security margin $\sigma$ defined as:

\[
\sigma = \min_{x,y \in \mathbb{F}_{2^{32}}} \{\delta_p(x) - \delta_p(y)\}
\]

This ensures a minimum differential security of $\sigma$ bits.

\section{Implementation Constraints}

The algorithm must satisfy the following constraints:

\begin{enumerate}
  \item Temporal Resolution:
  \[
  t_{\text{precision}} \leq 10^{-9} \text{ seconds}
  \]
  \item Prime Search Boundary:
  \[
  \max_{x \in \mathbb{F}_{2^{32}}} \{\delta_p(x)\} \leq 2^{20}
  \]
  \item Entropy Pool Size:
  \[
  |\eta| \geq 64 \text{ bytes}
  \]
\end{enumerate}

\section{Compression Function Integration}

The modified compression function $G'$ incorporating the dynamic IV is defined as:

\[
G'(h, m, t) = G(h \oplus IV(t), m, t)
\]

where:
\begin{itemize}
  \item $h$ is the input chaining value
  \item $m$ is the message block
  \item $t$ is the current timestamp
  \item $G$ is the original Blake3 compression function
\end{itemize}

The chaining value update function becomes:

\[
h_{i+1} = G'(h_i, m_i, t_i)
\]

\section{Performance Considerations}

The algorithm implements the following optimizations:

\begin{enumerate}
  \item Prime Cache:
  \[
  \mathcal{C}_p = \{(x, \delta_p(x)) \mid x \in \text{recent}(\mathcal{T})\}
  \]
  \item IV Generation Rate Limit:
  \[
  \text{rate}(IV) \leq \min(\nu_{\text{cpu}}/1000, 10^6 \text{ Hz})
  \]
  \item Memory Complexity:
  \[
  \mathcal{O}(\log_2(\max(\mathcal{T})) \cdot |\mathcal{C}_p|)
  \]
\end{enumerate}

\section{Thread Safety Constraints}

For parallel execution, the following invariant must hold:

\[
\forall t_1, t_2 \in \mathcal{T}: t_1 \neq t_2 \implies IV(t_1) \neq IV(t_2)
\]

with probability:

\[
P(IV(t_1) = IV(t_2)) \leq 2^{-256}
\]

\section{Error Bounds}

The algorithm maintains the following error bounds:

\begin{enumerate}
  \item Timing Precision Error:
  \[
  \epsilon_t \leq 10^{-9} \text{ seconds}
  \]
  \item Prime Distance Error:
  \[
  \epsilon_p \leq 2^{-32}
  \]
  \item Entropy Pool Depletion:
  \[
  P(\text{entropy\_depletion}) \leq 2^{-64}
  \]
\end{enumerate}

\end{document}