# Noise basis determination

Determination of the noise basis for the successful disposal of various types of operations in HFHE.

Last updated

Determination of the noise basis for the successful disposal of various types of operations in HFHE.

Last updated

Defining the active base (computation base) and data structures in the "shifted" space on hypergraphs has its own mechanics built on the map of states. States are redefined by creating a base from the previous map. The new base model must be proven and exhibit characteristics of completeness.

All elements must be placed within the given values and must be integral. The larger the gap between elements, the longer the base for the ciphertext, the more complex the approach to achieving convergence (and vice versa).

Creating a lattice differs from accepted standards and requires redefinition for successful implementation of all necessary functionality. We will not consider the entire process of generating the starting space for managing vectors with encrypted values, but we will outline the basic principles that lie in this area. Such lattice-tables are the basis for any types of operations with ciphers.

This example does not consider the validation system because it requires a separate explanation and additional examples using the mechanics of accounting for all types of keys, such as the bootstrapping key, public key, etc. These mechanisms will be similarly considered in the corresponding articles.

Some numbers that are given are values obtained by approximation. Approximation was achieved by searching through a pre-selected range (approximate ideal values) to achieve the most effective and fastest operations.

Initialization of hash tables with specified parameters (

**obtaining values will be discussed separately**).Computation of factorials and combinations to determine states and construct paths for a future lattice space. The combinatorial combination of two parameters: $xR_{\text{low}}$ and $xW_{\text{low}}$, are positions in the lattice.

The factorial of the sum in the numerator calculates all the options for the permutation in the previously specified space, the product of the factorials of each option reducing this number to the combinations defined in the denominator.

$C(xR_{\text{low}}, xW_{\text{low}}) = \frac{(xR_{\text{low}} + xW_{\text{low}})!}{xR_{\text{low}}! \cdot xW_{\text{low}}!}$

Determination of the noise base, the indices of which are adjusted through recursive checking with the adaptation of $xR\_low$ and $xW\_low$:

Division and rounding down are used to ensure the result is integer.

$\text{new}_{xR_{\text{low}}} = \left\lfloor \frac{xR_{\text{low}} \cdot (xW_{\text{low}} - 1) + \left\lfloor \frac{xW_{\text{low}}}{2} \right\rfloor}{xW_{\text{low}}} \right\rfloor$

Performing logarithmic operations to find criteria for stopping recursive functions:

*The version of the product code responsible for this mechanism. Some parameters (such as the lookup table and sets of approximated values) were excluded from the example due to the large number of preset elements, but will be considered and described in separate articles.*

$\Omega(xR_{\text{low}}) =
\begin{cases}
\left\lfloor \frac{\log_2(xR_{\text{low}})}{6} - 1 \right\rfloor, & \text{if } xR_{\text{low}} > 2 \\
xR_{\text{low}}, & \text{for } xR_{\text{low}} \leq 2
\end{cases}$