> For the complete documentation index, see [llms.txt](https://docs.zama.org/concrete/llms.txt). Markdown versions of documentation pages are available by appending `.md` to page URLs; this page is available as [Markdown](https://docs.zama.org/concrete/2.4/tutorials/multi_precision.md). # Multi Precision Each integer in the circuit has a certain bit-width, which is determined by the inputset. These bit-widths can be observed when graphs are printed: ``` %0 = x # EncryptedScalar ∈ [0, 7] %1 = y # EncryptedScalar ∈ [0, 15] %2 = add(%0, %1) # EncryptedScalar ∈ [2, 22] return %2 ^ these are ^^^^^^^ the assigned based on bit-widths these bounds ``` However, it's not possible to add 3-bit and 4-bit numbers together because their encoding is different: ``` D: data N: noise 3-bit number ------------ D2 D1 D0 0 0 0 ... 0 0 0 N N N N 4-bit number ------------ D3 D2 D1 D0 0 0 0 ... 0 0 0 N N N N ``` The result of such an addition is a 5-bit number, which also has a different encoding: ``` 5-bit number ------------ D4 D3 D2 D1 D0 0 0 0 ... 0 0 0 N N N N ``` Because of these encoding differences, we perform a graph processing step called bit-width assignment, which takes the graph and updates the bit-widths to be compatible with FHE. After this graph processing step, the graph would look like: ``` %0 = x # EncryptedScalar %1 = y # EncryptedScalar %2 = add(%0, %1) # EncryptedScalar return %2 ``` Most operations cannot change the encoding, which means that the input and output bit-widths need to be the same. However, there is an operation which can change the encoding: the table lookup operation. Let's say you have this graph: ``` %0 = x # EncryptedScalar ∈ [0, 3] %1 = y # EncryptedScalar ∈ [0, 31] %2 = 2 # ClearScalar ∈ [2, 2] %3 = power(%0, %2) # EncryptedScalar ∈ [0, 9] %4 = add(%3, %1) # EncryptedScalar ∈ [1, 39] return %4 ``` This is the graph for `(x**2) + y` where `x` is 2-bits and `y` is 5-bits. If the table lookup operation wasn't able to change the encoding, we'd need to make everything 6-bits. However, since the encoding can be changed, the bit-widths can be assigned like so: ``` %0 = x # EncryptedScalar ∈ [0, 3] %1 = y # EncryptedScalar ∈ [0, 31] %2 = 2 # ClearScalar ∈ [2, 2] %3 = power(%0, %2) # EncryptedScalar ∈ [0, 9] %4 = add(%3, %1) # EncryptedScalar ∈ [1, 39] return %4 ``` In this case, we kept `x` as 2-bits, but set the table lookup result and `y` to be 6-bits, so that the addition can be performed. This style of bit-width assignment is called multi-precision, and it is enabled by default. To disable it and use a single precision across the circuit, you can use the `single_precision=True` configuration option. --- # Agent Instructions This documentation is published with GitBook. GitBook is the documentation platform designed so that both humans and AI agents can read, navigate, and reason over technical content effectively. Learn more at gitbook.com. ## Querying This Documentation If you need additional information that is not directly available in this page, you can query the documentation dynamically by asking a question. Perform an HTTP GET request on the current page URL with the `ask` query parameter, and the optional `goal` query parameter: ``` GET https://docs.zama.org/concrete/2.4/tutorials/multi_precision.md?ask=&goal= ``` `ask` is the immediate question: it should be specific, self-contained, and written in natural language. `goal` is optional and describes the broader end goal you are ultimately trying to accomplish on behalf of the user. GitBook uses it to tailor the answer towards what is most useful for that goal. The response will contain a direct answer to the question and relevant excerpts and sources from the documentation. Use this mechanism when the answer is not explicitly present in the current page, you need clarification or additional context, or you want to retrieve related documentation sections.