Welcome back, future privacy architect! Imagine you’re building an AI application that needs to compare sensitive user data—like health metrics or financial patterns—without ever seeing the raw information. How do you find the “distance” or “similarity” between two data points if they’re both encrypted? This chapter tackles exactly that challenge.

In our previous discussions, we explored the fundamentals of Homomorphic Encryption (HE) and Fully Homomorphic Encryption (FHE), understanding HEIR’s crucial role as a compiler for FHE programs. Now, it’s time to bring these powerful concepts to life with a practical example: calculating the Euclidean Distance on encrypted data.

By the end of this chapter, you’ll have a clear conceptual understanding of how to perform this fundamental AI operation privately using HEIR. You’ll grasp not only how the process works but also why HEIR is a vital tool for balancing privacy and utility in AI.

Core Concepts: Euclidean Distance in Private AI

Euclidean Distance is a fundamental metric in countless machine learning algorithms. It quantifies the “straight-line” distance between two points in a multi-dimensional space, essentially telling us how similar or dissimilar two data points are.

What is Euclidean Distance and Why is it Challenging for Privacy?

For two vectors (or data points) A = (a1, a2, ..., an) and B = (b1, b2, ..., bn), the Euclidean Distance d(A, B) is calculated using the formula:

d(A, B) = sqrt( (a1 - b1)^2 + (a2 - b2)^2 + ... + (an - bn)^2 )

This calculation is critical for:

  • Clustering: Grouping similar data points together (e.g., in K-means).
  • Recommendation Systems: Identifying items or users that are “close” to a given query.
  • Classification: Determining which class a new data point belongs to based on its proximity to existing class centroids.

The inherent privacy challenge arises because computing this distance typically requires access to the raw, unencrypted values of A and B. If these vectors contain sensitive personal information, performing this computation on a remote server means exposing that data. This is a non-starter for privacy-preserving applications.

📌 Key Idea: FHE allows us to perform arithmetic operations directly on encrypted data, keeping sensitive information private throughout the computation. HEIR helps translate high-level operations into FHE-compatible instructions.

HEIR’s Role in Private Euclidean Distance

HEIR (Homomorphic Encryption Intermediate Representation) acts as a specialized compiler infrastructure. Instead of developers needing to write intricate, low-level cryptographic code to handle operations on ciphertexts, HEIR allows you to describe your desired computation in a more accessible, higher-level manner.

HEIR then takes this description and translates it into an efficient, FHE-compatible program that can be run using various FHE backends (like SEAL or HElib). This abstraction is critical for making FHE practical for a broader range of developers.

Let’s visualize the high-level flow of computing Euclidean Distance within an FHE context using HEIR:

flowchart TD InputA[Vector A] InputB[Vector B] InputA --> Subtract_Vectors[Subtract Elements] InputB --> Subtract_Vectors Subtract_Vectors --> Square_Differences[Square Elements] Square_Differences --> Sum_Squares[Sum Differences] Sum_Squares --> Compute_Sqrt[Square Root] Compute_Sqrt --> Output_Distance[Encrypted Distance]

This diagram illustrates the logical steps. HEIR’s job is to ensure each of these steps can be executed securely on encrypted data.

Understanding FHE Operations for Euclidean Distance

To compute Euclidean Distance securely with FHE, we need to consider how each mathematical step translates into operations that FHE schemes can perform.

  1. Element-wise Subtraction (a_i - b_i):

    • What it is: Subtracting corresponding elements of two vectors.
    • Why it’s important: This is the first step in finding the difference between points.
    • How FHE handles it: FHE schemes natively support addition and subtraction on ciphertexts. If Enc(a_i) and Enc(b_i) are encrypted values, FHE can compute Enc(a_i - b_i) directly.

  2. Element-wise Squaring ((a_i - b_i)^2):

    • What it is: Multiplying each difference by itself.
    • Why it’s important: Squaring ensures all differences are positive and emphasizes larger differences.
    • How FHE handles it: FHE schemes also natively support multiplication on ciphertexts. If Enc(diff_i) is an encrypted difference, FHE can compute Enc(diff_i * diff_i).

  3. Summation of Squared Differences (sum(...)):

    • What it is: Adding all the squared differences together.
    • Why it’s important: This aggregates the total “distance” before the final square root.
    • How FHE handles it: This is another series of addition operations, which FHE handles efficiently.

  4. Square Root (sqrt(...)):

    • What it is: The final step to get the true Euclidean Distance.
    • Why it’s important: It normalizes the squared sum back to a linear scale.
    • How FHE handles it: This is the trickiest part. FHE schemes primarily operate on polynomials. A true square root function is not a polynomial. Therefore, in FHE, square roots are typically handled in one of these ways:
      • Polynomial Approximation: Using a polynomial (e.g., Taylor series expansion) to approximate the square root over a specific, predefined range. This introduces some precision loss, which must be acceptable for the application.
      • Avoiding the Square Root: If the application only needs to compare distances (e.g., “is point A closer to B than to C?”), then comparing the squared Euclidean distances d(A, B)^2 and d(A, C)^2 is equivalent. This allows you to avoid the complex square root entirely.
      • Post-Decryption: The final result (the sum of squares) is decrypted, and the square root is taken on the plaintext. This requires trusting the party performing the decryption, which might not always align with the privacy goals.

⚡ Real-world insight: Many private AI scenarios, especially those involving clustering or nearest-neighbor searches, only require relative distances. By comparing squared distances, you can often bypass the computationally expensive and precision-losing square root operation in FHE.

🧠 Important: When working with FHE, always consider if an exact computation is strictly necessary or if an approximation or an alternative metric (like squared Euclidean distance) can fulfill your application’s requirements. This often leads to significant performance and precision benefits.

Step-by-Step Implementation (Conceptual)

Since HEIR is a compiler, the “implementation” involves defining the computation graph in a way that HEIR can parse, understand, and translate into FHE-specific operations. As of 2026-08-18, HEIR is in active development, and its end-to-end integration for generating executables is still evolving. The format_assistant/h tool is highlighted for packaging compiled FHE programs.

This means we’ll focus on the logic and structure you would provide to HEIR, rather than writing raw C++ FHE library calls. Assume you have HEIR built and configured (as discussed in earlier chapters, typically involving CMake).

1. Defining the High-Level Operation

Conceptually, you would define a function or program that takes two encrypted vectors and returns their encrypted Euclidean distance. Let’s use a simplified, HEIR-friendly pseudo-language or an IR-like definition to illustrate the logical steps.

// Conceptual definition for HEIR
// This is not runnable code, but illustrates the logical steps HEIR would compile.
// This describes the computation graph, not low-level FHE library calls.

function computeEuclideanDistance(encrypted_vector_A, encrypted_vector_B) returns encrypted_scalar:
    // First, we'll prepare a container for our intermediate results:
    // an encrypted vector to store the squared differences for each element pair.
    encrypted_squared_diffs = new_encrypted_vector_of_same_size_as(encrypted_vector_A)

    // Next, we iterate through each corresponding element of the input vectors.
    // For each pair (a_i, b_i), we'll compute (a_i - b_i)^2.
    for i from 0 to length(encrypted_vector_A) - 1:
        // Retrieve the i-th encrypted element from each input vector.
        // These are still ciphertexts, so their plaintext values remain hidden.
        encrypted_a_i = get_element(encrypted_vector_A, i)
        encrypted_b_i = get_element(encrypted_vector_B, i)

        // Perform element-wise subtraction on the encrypted values.
        // The result, encrypted_diff_i, is also a ciphertext.
        encrypted_diff_i = subtract(encrypted_a_i, encrypted_b_i)

        // Square the encrypted difference. This is an encrypted multiplication
        // of encrypted_diff_i by itself. The result is still encrypted.
        encrypted_squared_diff_i = multiply(encrypted_diff_i, encrypted_diff_i)

        // Store this encrypted squared difference in our intermediate result vector.
        set_element(encrypted_squared_diffs, i, encrypted_squared_diff_i)

    // After iterating through all elements, we have an encrypted vector
    // containing all the squared differences. Now, we sum these up.
    // This sum operation is also performed on ciphertexts, yielding an encrypted scalar.
    encrypted_sum_of_squares = sum_elements(encrypted_squared_diffs)

    // Finally, we need to compute the square root. As discussed, this is often
    // an approximation in FHE using polynomials. HEIR would select the appropriate
    // polynomial approximation based on its configuration and the target FHE backend.
    // We might conceptually call a function like 'approx_sqrt' here.
    encrypted_distance = approx_sqrt(encrypted_sum_of_squares)

    return encrypted_distance
end function

In a real HEIR workflow, you might express this computation using a domain-specific language (DSL) that HEIR can parse, or by directly constructing an Intermediate Representation (IR) program using HEIR’s C++ API. The crucial point is that you’re describing the computation graph or program logic, not writing the intricate low-level FHE operations yourself.

2. Compiling with HEIR (Conceptual Workflow)

Once your computation is defined in an HEIR-compatible format, the next step is to use the HEIR compiler.

  1. Input Program Preparation: You would provide your high-level description (like the pseudo-code above, translated into a formal HEIR-compatible IR format, perhaps an MLIR dialect) to the HEIR compiler. Let’s imagine this is in a file named euclidean_distance.mlir.
  2. Compilation to FHE-specific IR: HEIR processes this program, performs optimizations suitable for homomorphic encryption, and translates it into an FHE-specific intermediate representation. This FHE-specific IR contains instructions tailored for a particular FHE backend (e.g., SEAL, HElib).
  3. Executable Generation: As noted in the HEIR README.md, for an executable, you would use a tool like format_assistant/h. This tool helps in packaging the compiled FHE program into a runnable format that can take encrypted inputs and produce encrypted outputs.

Here’s how a conceptual command-line interaction might look (actual syntax may vary as HEIR evolves):

# Conceptual command-line interaction with HEIR (example, actual syntax may vary).
# These commands illustrate the pipeline, not exact current API.

# Step 1: Assume 'euclidean_distance.mlir' contains your HEIR IR definition.
#         This file conceptually holds the logic described in the pseudo-code.

# Step 2: Compile the HEIR program to an FHE-specific Intermediate Representation (IR).
#         This step translates the generic HEIR IR into instructions optimized for an FHE backend.
#         The '--target-backend SEAL' specifies which FHE library HEIR should target.
heir-compiler --input euclidean_distance.mlir --output-fhe-ir compiled_euclidean.fheir --target-backend SEAL

# Step 3: Generate an executable application from the FHE-specific IR.
#         As per HEIR documentation, 'format_assistant/h' is used for this purpose.
#         This produces a standalone program that can process encrypted data.
format_assistant/h --fhe-ir compiled_euclidean.fheir --output-executable private_euclidean_app

The resulting private_euclidean_app would be an application that, when executed, can accept ciphertexts (encrypted vectors A and B), securely perform the Euclidean Distance calculation on them, and output a ciphertext representing the distance. The client can then decrypt this final ciphertext to get the plaintext distance.

🧠 Important: Current HEIR Status (Checked 2026-08-18) The HEIR GitHub repository explicitly states: “integration between Middle-End and Back-End is not yet well-implemented.” This means while the compiler infrastructure is robust, the end-to-end user experience for generating fully functional FHE executables might still be evolving and require careful attention to the latest documentation. Always refer to the official HEIR GitHub repository for the most current information and usage guidelines.

Mini-Challenge: Private Manhattan Distance

You’ve now walked through the conceptual flow for Euclidean Distance. Let’s test your understanding with a related challenge!

Challenge: Adapt the conceptual HEIR program to calculate the Manhattan Distance (also known as L1 norm) between two encrypted vectors A and B.

The Manhattan Distance is defined as:

d(A, B) = |a1 - b1| + |a2 - b2| + ... + |an - bn|

Hint:

  • How does the core operation sequence differ from Euclidean Distance? Specifically, which mathematical operation is replaced?
  • The absolute value function |x| is not a simple polynomial. How might you conceptually handle |x| in an FHE-friendly way? Consider:
    • Polynomial approximations for |x| over a specific, known range.
    • If your data guarantees that (a_i - b_i) will always be positive within your FHE context, could you simplify? (This is a strong assumption, be careful!)
    • The max(x, -x) trick, and how max itself might be approximated or handled in FHE.

What to observe/learn:

  • How different distance metrics directly translate into different FHE operation sequences.
  • The challenges of implementing non-polynomial functions like abs or max in FHE, and why approximations or careful data design are often necessary.

Common Pitfalls & Troubleshooting

Working with FHE compilers like HEIR, especially during their active development phases, presents unique challenges. Being aware of these can save you significant debugging time.

  1. ⚠️ What can go wrong: High Performance Overheads and Resource Consumption. FHE computations are inherently much slower and require significantly more memory than plaintext operations. Even a seemingly simple Euclidean Distance can take seconds or minutes on encrypted data, depending on chosen FHE parameters, vector dimensions, and the complexity of the FHE scheme.

    • Troubleshooting:
      • Start small: Begin with very small datasets and vector sizes to confirm correctness before scaling up.
      • Optimize FHE Parameters: If you have control over the FHE backend parameters (e.g., polynomial degree, number of slots, security level), experiment with values that meet your security requirements but are optimized for performance.
      • Hybrid Approaches: Consider if FHE is truly needed for all parts of your pipeline. Sometimes, a hybrid approach (e.g., FHE for sensitive parts, trusted execution environments for others) might be more practical.

  2. ⚠️ What can go wrong: Precision Loss with Approximations. Operations like square root and absolute value often require polynomial approximations in FHE. This inherently introduces some loss of precision, which can be critical for applications that demand exact results.

    • Troubleshooting:
      • Understand accuracy needs: Clearly define the acceptable error margin for your application.
      • Test rigorously: Evaluate approximation errors with known inputs and compare against plaintext results.
      • Alternative metrics: Explore if your algorithm can function correctly with squared distances or other FHE-friendly alternatives that avoid problematic operations.

  3. ⚠️ What can go wrong: Data Scaling and Range Issues. FHE schemes operate on integers or fixed-point numbers within a specific, finite range. Input data and all intermediate computation results must be carefully scaled to fit these constraints to avoid overflow, underflow, or incorrect results.

    • Troubleshooting:
      • Normalize data: Always normalize or scale your data before encryption.
      • Track value ranges: Be meticulous about estimating the maximum possible value any intermediate computation (e.g., (a_i - b_i)^2) can reach and ensure it stays within the FHE scheme’s limits.

  4. ⚠️ What can go wrong: Difficulty in Debugging Encrypted Computations. When issues arise, debugging FHE programs is extremely challenging because you cannot inspect intermediate encrypted values. All you see are ciphertexts.

    • Troubleshooting:
      • Extensive plaintext testing: Thoroughly test your logic with plaintext data before attempting FHE compilation.
      • Small, controlled examples: Use minimal examples with known inputs and expected outputs to isolate issues.
      • Leverage compiler messages: Pay close attention to HEIR’s output and any error or warning messages, as these are your primary clues.

  5. ⚠️ What can go wrong: HEIR’s Evolving Development State. As highlighted, HEIR is under active development. Features, APIs, and documentation might change rapidly, and some functionalities might not be fully mature.

    • Troubleshooting:
      • Stay updated: Regularly refer to the latest HEIR GitHub repository for the most current information, examples, and best practices.
      • Be prepared for experimentation: Expect that you might need to experiment or adapt your approach as the project evolves.

Summary

You’ve taken a significant step forward in building practical privacy-preserving AI applications! This chapter guided you through the conceptual implementation of Euclidean Distance on encrypted data using HEIR.

Here are the key takeaways from our journey:

  • Euclidean Distance is vital for many AI tasks, but poses a direct privacy challenge with sensitive data.
  • HEIR acts as a powerful compiler, abstracting away the low-level FHE complexities, allowing developers to define computations at a higher level.
  • We broke down the Euclidean Distance formula into FHE-compatible operations: element-wise subtraction, squaring, and summation.
  • The square root operation in FHE often requires polynomial approximations or can be avoided if only relative distances are needed.
  • We walked through a conceptual HEIR workflow, demonstrating how a high-level description of the computation can be translated into an FHE-ready program.
  • You tackled a mini-challenge to consider Manhattan Distance, highlighting the translation of different mathematical functions into FHE and the specific challenges of non-polynomial operations.
  • We discussed crucial common pitfalls in FHE development, including performance overheads, precision loss, data scaling, debugging difficulties, and HEIR’s evolving nature, along with practical troubleshooting strategies.

This chapter has equipped you with a deeper understanding of how to bridge the gap between AI algorithms and privacy preservation. In the next chapter, we’ll continue to explore more advanced use cases and dive into other essential FHE operations, building on the strong foundation you’ve established here.

References

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