Time-Differential Blake 3 Initialization Vector Generation

President Harmon, David R.

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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:

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:

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

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).

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:

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

Initialization Vector Generation

The final IV components are generated as:

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

where:

• $\mathcal{H}$ is an auxiliary hash function
• $m_i$ represents system memory statistics

Security Properties

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.

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.

Implementation Constraints

The algorithm must satisfy the following constraints:

1. Temporal Resolution:

   $$ t_{\text{precision}} \leq 10^{-9} \text{ seconds} $$

2. Prime Search Boundary:

   $$ \max_{x \in \mathbb{F}_{2^{32}}} {\delta_p(x)} \leq 2^{20} $$

3. Entropy Pool Size:

   $$ |\eta| \geq 64 \text{ bytes} $$

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:

• $h$ is the input chaining value
• $m$ is the message block
• $t$ is the current timestamp
• $G$ is the original Blake3 compression function

The chaining value update function becomes:

$$ h_{i+1} = G'(h_i, m_i, t_i) $$

Performance Considerations

The algorithm implements the following optimizations:

1. Prime Cache:

   $$ \mathcal{C}_p = {(x, \delta_p(x)) \mid x \in \text{recent}(\mathcal{T})} $$

2. IV Generation Rate Limit:

   $$ \text{rate}(IV) \leq \min(\nu_{\text{cpu}}/1000, 10^6 \text{ Hz}) $$

3. Memory Complexity:

   $$ \mathcal{O}(\log_2(\max(\mathcal{T})) \cdot |\mathcal{C}_p|) $$

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:

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

Error Bounds

The algorithm maintains the following error bounds:

1. Timing Precision Error:

   $$ \epsilon_t \leq 10^{-9} \text{ seconds} $$

2. Prime Distance Error:

   $$ \epsilon_p \leq 2^{-32} $$

3. Entropy Pool Depletion:

   $$ P(\text{entropy_depletion}) \leq 2^{-64} $$

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