Introduction to HEIR’s Compiler Architecture

Welcome back! In our previous chapters, we explored the fascinating world of Homomorphic Encryption (HE) and Fully Homomorphic Encryption (FHE), understanding how they empower us to perform computations directly on encrypted data. We also introduced HEIR (Homomorphic Encryption Intermediate Representation), Google’s ambitious open-source project designed to be an end-to-end FHE compiler.

But how does HEIR actually do this magic? How does it take a standard program and transform it into one that can run securely on encrypted data, preserving privacy? This chapter will pull back the curtain, taking you on a journey into the heart of HEIR: its compiler architecture and the crucial role of its Intermediate Representation (IR).

Understanding HEIR’s architecture is key to grasping its power and potential. It helps us see how complex FHE operations are managed, optimized, and eventually executed. By the end of this chapter, you’ll have a clear mental model of how HEIR translates your privacy-preserving AI ideas into reality.

The Compiler’s Lens: Deconstructing HEIR

At its core, HEIR is a compiler. Just like a C++ compiler translates your human-readable code into machine instructions, HEIR translates your high-level computation logic into a form executable by a homomorphic encryption scheme. This translation isn’t simple; it involves several intricate stages.

What is a Compiler Architecture?

Think of a traditional compiler as a factory with specialized departments. Each department takes the output from the previous one, performs a specific task, and then passes it along.

Generally, compilers are divided into three main phases:

  1. Frontend: This department understands your original source code (e.g., Python, C++). It checks for syntax errors, builds an abstract syntax tree (AST), and often converts the code into a generic, high-level Intermediate Representation (IR).
  2. Middle-End: This is the optimization hub. It takes the IR from the frontend and performs various transformations to make the code faster, more efficient, or smaller, without changing its core logic.
  3. Backend: This final department takes the optimized IR and generates the actual machine code or instructions for a specific target platform (e.g., x86, ARM).

HEIR’s Specialized Compiler Stages

HEIR follows a similar, yet specialized, three-stage pipeline, tailored specifically for the unique challenges of Fully Homomorphic Encryption.

Frontend: Bridging to FHE

The HEIR frontend’s job is to take a program written in a standard language or a more abstract computation graph and convert it into HEIR’s FHE-aware Intermediate Representation. This initial step is where the compiler begins to understand your program’s intent and identify operations that can be performed homomorphically.

Middle-End: FHE-Specific Optimizations

This is where the real FHE magic happens in terms of optimization. Homomorphic encryption schemes have specific computational costs and limitations (e.g., noise growth, bootstrapping). The HEIR middle-end is designed to apply FHE-specific transformations to the IR. This might include:

  • Operation Rewriting: Replacing standard arithmetic operations with their FHE-compatible equivalents (e.g., turning a regular add into an fhe.add).
  • Noise Management: Optimizing the order of operations to minimize noise accumulation, which is critical for FHE to maintain data integrity over many computations.
  • Bootstrapping Strategy: Deciding when and how to perform bootstrapping (a noise-reduction technique) to enable deeper computations without losing precision.
  • Circuit Optimization: Restructuring the computation to be more efficient for encrypted execution, potentially by reordering operations or combining them.

Backend: Targeting Cryptosystems

The HEIR backend is responsible for translating the optimized FHE-IR into concrete operations for a specific homomorphic encryption library or cryptosystem. This could involve generating code for libraries like SEAL, TFHE, or OpenFHE, ensuring that the operations are correctly implemented according to the chosen scheme’s parameters and API.

📌 Key Idea: HEIR acts as an abstraction layer, allowing developers to write high-level code while handling the complex FHE-specific transformations and optimizations behind the scenes.

flowchart TD A[High-Level Program] --> B[HEIR Frontend] B --> C[HEIR IR High-Level] C --> D[HEIR Middle-End] D --> E[HEIR IR Optimized FHE] E --> F[HEIR Backend] F --> G[Target FHE Operations] subgraph Compiler["HEIR Compiler"] B C D E F end

Figure: Simplified HEIR Compiler Pipeline

⚡ Quick Note: As of 2026-08-18, the HEIR project is actively under development. The GitHub repository notes that the “integration between Middle-End and Back-End is not yet well-implemented.” This means that while the conceptual architecture is clear, practical end-to-end compilation to an executable FHE program might require using specific tools like format_assistant/h for certain stages, rather than a fully integrated pipeline. This is a common characteristic of cutting-edge research projects.

Intermediate Representation (IR): The Universal Translator

The concept of an Intermediate Representation (IR) is central to any sophisticated compiler, and especially for HEIR.

What is an IR?

An IR is an abstract form of your program that sits between the original source code and the final machine code. It’s like a blueprint that describes the computation without being tied to a specific programming language or a specific hardware architecture.

Why is this useful?

  • Abstraction: It frees the middle-end from worrying about the quirks of many different source languages and the backend from dealing with many different hardware architectures.
  • Optimization Target: All optimizations can be applied to this single, unified representation, rather than having to write separate optimizers for each source language or target platform.
  • Modularity: Different frontends can target the same IR, and different backends can consume the same IR.

HEIR’s FHE-Aware IR

HEIR’s Intermediate Representation is particularly interesting because it’s designed to represent computations in a way that is aware of homomorphic encryption constraints. It’s built upon the MLIR (Multi-Level Intermediate Representation) framework, which provides a flexible way to define different levels of abstraction for an IR.

This allows HEIR’s IR to:

  • Represent FHE Primitives: It can express operations like add_encrypted, multiply_encrypted, rotate_encrypted_vector, and bootstrap as first-class citizens.
  • Track FHE-Specific Properties: The IR can carry metadata about encrypted values, such as their noise level, the encryption context, or the underlying plaintext type. This is crucial for the middle-end to make informed optimization decisions.
  • Enable Progressive Lowering: MLIR’s multi-level nature means the IR can start at a high, abstract level (e.g., “perform matrix multiplication”) and gradually be “lowered” through successive transformations into more concrete, FHE-specific operations (e.g., “perform dot_product operations on encrypted vectors, followed by a bootstrap”).

Conceptual Example: From High-Level to FHE-IR

Let’s imagine a simple function that adds two numbers. In a high-level language, it’s straightforward:

def add_numbers(a, b):
    return a + b

When this goes through HEIR, the IR might evolve:

Stage 1: High-Level IR (Conceptual)

Initially, the HEIR frontend might convert this into an IR that looks very similar to the original operation, but within the HEIR framework:

// Represents the initial operation
func @add_numbers(%arg0: i32, %arg1: i32) -> i32 {
  %0 = "std.add"(%arg0, %arg1) : (i32, i32) -> i32
  "std.return"(%0) : (i32) -> ()
}

This is a simplified, non-FHE specific representation, similar to what a general compiler might produce. It uses i32 for 32-bit integers.

Stage 2: FHE-Aware IR (Conceptual)

Now, the HEIR middle-end steps in. If arg0 and arg1 are determined to be encrypted inputs, the middle-end would transform the std.add operation into an FHE-specific addition. It might also add annotations about the encryption context.

// After FHE lowering, assuming inputs are encrypted
func @add_numbers(%arg0: !fhe.ciphertext<i32>, %arg1: !fhe.ciphertext<i32>) -> !fhe.ciphertext<i32> {
  // Use an FHE-specific add operation
  %0 = "fhe.add"(%arg0, %arg1) : (!fhe.ciphertext<i32>, !fhe.ciphertext<i32>) -> !fhe.ciphertext<i32>
  // Potentially insert bootstrapping if noise is too high, or other optimizations
  // %1 = "fhe.bootstrap"(%0) : (!fhe.ciphertext<i32>) -> !fhe.ciphertext<i32>
  "fhe.return"(%0) : (!fhe.ciphertext<i32>) -> ()
}

Notice how the type signature changes to !fhe.ciphertext<i32> (indicating an encrypted 32-bit integer) and the operation itself becomes fhe.add. This is a conceptual example of how HEIR’s IR can represent FHE operations and track encrypted types.

🧠 Important: The actual syntax of HEIR’s MLIR-based IR is much more detailed and complex, involving specific dialects for various FHE operations and parameters. This example is highly simplified to illustrate the concept of transformation.

Step-by-Step: Defining a Computation for HEIR’s IR

While HEIR is a compiler, for this chapter focused on architecture and IR, we’ll explore how you would define a simple FHE computation using its conceptual Intermediate Representation. This is how you’d express your privacy-preserving logic in a way that HEIR can understand and process.

Let’s define a simple computation: multiplying an encrypted number by a plaintext constant, then adding an encrypted number. This illustrates both FHE-specific operations and interactions with unencrypted data.

Step 1: Declare the Function and Input Types

First, we need to declare a function within the HEIR IR. We’ll specify its name, arguments, and return type. For encrypted values, we use the !fhe.ciphertext type. For plaintext constants, we’ll use a standard integer type like i32.

Imagine a function compute_private_value that takes two encrypted integers (%encrypted_x, %encrypted_y) and a plaintext integer (%constant). It will return an encrypted integer.

// Define a function that takes two encrypted inputs and one plaintext constant
func @compute_private_value(
    %encrypted_x: !fhe.ciphertext<i32>,
    %encrypted_y: !fhe.ciphertext<i32>,
    %constant: i32
) -> !fhe.ciphertext<i32> {
    // Operations will go here
}

Here, func declares a function, @compute_private_value is its name, and the arguments are explicitly typed. The -> !fhe.ciphertext<i32> indicates the return type.

Step 2: Perform Encrypted Multiplication with a Plaintext Constant

One common FHE operation is multiplying an encrypted value by a known, unencrypted constant. This operation is typically much faster and doesn’t increase noise as much as encrypted-encrypted multiplication. HEIR’s IR would represent this with a specific FHE operation, perhaps fhe.mul_scalar.

Let’s multiply %encrypted_x by %constant.

// ... (previous function declaration) ...
func @compute_private_value(
    %encrypted_x: !fhe.ciphertext<i32>,
    %encrypted_y: !fhe.ciphertext<i32>,
    %constant: i32
) -> !fhe.ciphertext<i32> {
    // Multiply encrypted_x by the plaintext constant
    %multiplied_val = "fhe.mul_scalar"(%encrypted_x, %constant) : (!fhe.ciphertext<i32>, i32) -> !fhe.ciphertext<i32>
    // ... next operation ...
}

Here, %multiplied_val is a temporary variable holding the result. The "fhe.mul_scalar" operation explicitly states that it takes an encrypted integer and a plaintext integer, and produces an encrypted integer.

Step 3: Perform Encrypted Addition

Next, we want to add the result of our multiplication (%multiplied_val) to our second encrypted input (%encrypted_y). This is a standard encrypted-encrypted addition.

// ... (previous function declaration and multiplication) ...
func @compute_private_value(
    %encrypted_x: !fhe.ciphertext<i32>,
    %encrypted_y: !fhe.ciphertext<i32>,
    %constant: i32
) -> !fhe.ciphertext<i32> {
    %multiplied_val = "fhe.mul_scalar"(%encrypted_x, %constant) : (!fhe.ciphertext<i32>, i32) -> !fhe.ciphertext<i32>

    // Add the multiplied value to the second encrypted input
    %final_result = "fhe.add"(%multiplied_val, %encrypted_y) : (!fhe.ciphertext<i32>, !fhe.ciphertext<i32>) -> !fhe.ciphertext<i32>
    // ... return value ...
}

The "fhe.add" operation takes two encrypted integers and returns an encrypted integer. This is a common FHE primitive.

Step 4: Return the Final Encrypted Value

Finally, the function needs to return the computed encrypted value.

// ... (previous operations) ...
func @compute_private_value(
    %encrypted_x: !fhe.ciphertext<i32>,
    %encrypted_y: !fhe.ciphertext<i32>,
    %constant: i32
) -> !fhe.ciphertext<i32> {
    %multiplied_val = "fhe.mul_scalar"(%encrypted_x, %constant) : (!fhe.ciphertext<i32>, i32) -> !fhe.ciphertext<i32>
    %final_result = "fhe.add"(%multiplied_val, %encrypted_y) : (!fhe.ciphertext<i32>, !fhe.ciphertext<i32>) -> !fhe.ciphertext<i32>

    // Return the final encrypted result
    "fhe.return"(%final_result) : (!fhe.ciphertext<i32>) -> ()
}

The "fhe.return" operation signifies the end of the FHE computation within this function.

⚡ Real-world insight: This textual IR is what HEIR’s middle-end would consume and optimize. While you wouldn’t typically write this by hand for complex programs, understanding its structure is crucial for debugging and understanding how HEIR processes your FHE logic. Tools like mlir-opt (part of the MLIR ecosystem) can be used to apply passes and observe IR transformations.

Mini-Challenge: Tracing an Encrypted Multiplication

Let’s test your understanding of how an operation might evolve within HEIR.

Challenge: Imagine you have a high-level function multiply_numbers(x, y) that returns x * y. If both x and y are encrypted inputs, describe conceptually how the multiply operation might be represented in:

  1. The initial, high-level IR stage (similar to std.add).
  2. The FHE-aware IR stage, considering it’s an encrypted multiplication.

Think about the types involved and the operation name. You don’t need to write perfect MLIR syntax, just explain the conceptual changes.

Hint: Focus on the type annotations and the operation name in your conceptual IR snippets. How would they reflect the encrypted nature of the data? Remember the !fhe.ciphertext type and the FHE-specific operation prefix.

Common Pitfalls & Troubleshooting in Early FHE Compilers

Working with cutting-edge tools like HEIR, especially in its active development phase, comes with its own set of challenges.

  1. Navigating Development Stage Limitations: As mentioned, HEIR is evolving. You might encounter features that are not fully integrated or documented. Expect to consult the source code and GitHub issues more frequently than with mature tools. The current integration between the middle-end and backend is still being built out (as of 2026-08-18).
    • Troubleshooting: Always check the latest README.md on the HEIR GitHub for updates on feature completeness and recommended workflows. Join any community forums or mailing lists if available.
  2. Complexity of FHE Operations: Even with a compiler, understanding the underlying FHE operations and their constraints (like noise growth and bootstrapping) is crucial for designing efficient private AI applications. If your program doesn’t compile or performs poorly, it might be due to FHE limitations rather than a compiler bug.
    • Troubleshooting: Review the basics of FHE scheme limitations (e.g., multiplicative depth, bootstrapping cost). Simplify your computation to isolate the problematic part. Consider if your algorithm is inherently FHE-friendly.
  3. Debugging Encrypted Computations: Debugging a program that runs on encrypted data is inherently difficult because you cannot directly inspect intermediate values. Errors might manifest as incorrect final results or performance bottlenecks.
    • Troubleshooting: Start with small, unencrypted test cases to verify logic. Gradually introduce encryption, comparing results. HEIR’s IR can be inspected to understand the transformations, which is a powerful debugging tool for compiler-level issues. Look for tools or passes within HEIR that allow for “plaintext evaluation” during development.

Summary

In this chapter, we’ve taken a deep dive into the internal architecture of HEIR, Google’s FHE compiler, and demystified the concept of Intermediate Representation.

Here are the key takeaways:

  • HEIR functions as a specialized compiler with a Frontend, Middle-End, and Backend, each tailored for FHE.
  • The Frontend translates high-level programs into an initial IR, understanding the program’s intent.
  • The Middle-End applies crucial FHE-specific optimizations, such as noise management, bootstrapping strategy, and circuit restructuring.
  • The Backend targets specific FHE cryptosystems, generating executable operations for libraries like SEAL or OpenFHE.
  • Intermediate Representation (IR) is the core abstraction, acting as a universal blueprint for the program.
  • HEIR’s IR, built on MLIR, is FHE-aware, representing encrypted types (!fhe.ciphertext) and FHE-specific operations (e.g., fhe.add, fhe.mul_scalar), and enabling progressive lowering.
  • We conceptually walked through defining a simple FHE computation directly in HEIR’s IR, illustrating how operations like encrypted addition and scalar multiplication are expressed.
  • Working with HEIR currently requires acknowledging its active development status and potential limitations in full end-to-end integration.

Understanding HEIR’s architecture and IR is fundamental to leveraging its power for private AI. In the next chapter, we’ll start getting our hands dirty with setting up the HEIR environment and exploring how to define simple computations for FHE in a more practical context.

References

This page is AI-assisted and reviewed. It references official documentation and recognized resources where relevant.