2. Mapping components to numbers

The strength of cryptography keys is defined in terms of entropy, roughly "how random" the keys are. Think about picking a colored ball randomly from a bucket full of balls. There are a few factors at play that determine how random picking a ball of a specific color is. If the bucket size increases and more balls are added, then the randomness also increases: so the volume of the bucket is one factor. However, another factor is how different the colors in the bucket are. No matter how large the bucket is, if the balls are all the same color, our randomness does not increase. These factors are called "dimensions". The size of each factor (e.g., 1 color vs 100 different colors) is called "cardinality". We can multiply them to find how big our random "space" is. Because computers use binary numbers, in cryptography, numbers are usually picked from a space that has a size equal to a power of 2. For that reason, we need to be able to map cross-stitch patterns to spaces that have the proper, computer-friendly size.

In the case of cross-stitch, we can think of the components outlined in step 1 as our dimensions. The cardinality is how many options of each component is available. For example, suppose our patterns had 2 flowers with 5 options for type of flower, to yield 2*5=10 different options. Alternatively, we could design our patterns to have 3 flowers and 4 flower designs, for 3*4=12 options. Here, each flower is a dimension, the cardinality is the number of flower types, and the space is the number of options. We can choose higher/lower dimensions and cardinality to yield different spaces. The tricky thing is, again, that our space needs to be a power of two. So this quickly becomes an interesting combinatorics problem to figure out how to pick cross-stitch shapes, colors, and layouts that multiply to yield the correct space. For example, suppose our cross-stitch design was k horizontal lines of specific colors. If we have n colors total, we need to solve how to find n and k such that they produce the correct computer-friendly size. The equation below formalizes somewhat the process to do so. 

...So this step mostly involves designing equations and then begging WolframAlpha to do the math for me.