A Method for Securely Comparing Integers using Binary Trees

Question

1. What is multi-party computation (MPC)?

2. What is Homomorphic Encryption (HE) used for?

3. What protocols use garbled circuits for secure integer comparison?

4. What is the ideal functionality FGT in the DEFINITIONS section?

Accepted Answer

Multi-party computation (MPC) is a cryptographic technique that allows several parties to compute a function on their private inputs without revealing any information other than the function's output. It is used in various scenarios such as comparing private integers securely in machine learning, private auctions, and secure enterprise benchmarking. MPC ensures privacy and security while enabling collaborative computations among multiple parties.

Accepted Answer

Homomorphic Encryption (HE) allows computations on ciphertexts by generating an encrypted result whose decryption matches the result of a function on the plaintexts. It enables secure data processing without revealing the underlying data. HE algorithms include key generation, encryption, and decryption processes. HE supports addition, multiplication, and constant operations on encrypted data, making it useful for secure data analysis, privacy-preserving computations, and secure cloud computing. Fully Homomorphic Encryption (FHE) enables arbitrary computations on encrypted data, while Leveled FHE allows computations up to a fixed circuit depth. HE is a crucial technology for enhancing data security and privacy in various applications.

Accepted Answer

Kolesnikov et al. proposed schemes that use garbled circuits for secure integer comparison. These schemes include Lin and Tzeng [37, 48], Fischlin [22], Blake and Kolesnikov [6], and Garay et al. [24]. They rely on the arithmetic black-box model and require access to the bit representation of the integers. Some schemes have a constant number of rounds, while others have log-logarithmic rounds. Recent work by Bourse et al. [11] and Iliashenko and Zucca [30] also contribute to this field, but they use specific AHE constructions and FHE schemes, respectively.

Accepted Answer

The ideal functionality FGT takes inputs from both the server and client, computes a bit output, and outputs a value of 1 if the input is greater than or equal to a certain threshold, and 0 otherwise. It is illustrated in Figure 1 and can be extended to a symmetric scenario. The protocol involves homomorphic computations and encryption to ensure secure sharing of intermediate results between parties. This functionality is crucial in scenarios where only the client receives the output, and the server computes the result before sending it. The protocol can be adapted to prevent parties from learning intermediate results, enhancing security in larger settings.

Accepted Answer

The ideal/real world paradigm in security and correctness is a formalization where for each party, there must exist a simulator that generates a distribution computationally indistinguishable to the party's view. This paradigm ensures that parties learn only what they are allowed to, and it is crucial in multi-party protocols. The simulator takes the party's input and the sequence of messages received during the protocol execution to generate a distribution that is indistinguishable from the party's view. This concept is essential in ensuring the security and correctness of protocols, especially in the semi-honest security model where parties follow the protocol but may try to learn more information from its execution.

Accepted Answer

The comparison problem can be seen as a classification problem using a decision tree by building the tree on server input and evaluating it using client input. Given a server's input, a client's input classifies as 1 if >= or 0 otherwise. The tree edges are labeled with bits, and the evaluation is done by traversing the tree along the path of. However, since the path representing - is encrypted, all paths need to be evaluated and aggregated. Evaluating a path on encrypted inputs involves computing bit equality between edge labels and the bits of the inputs, and adding or multiplying the results along the path. This process allows for efficient classification of inputs based on the comparison problem.

Accepted Answer

The root node in a binary tree data structure serves as the starting point of the tree. It is the node with no parent node, and all other nodes in the tree are descendants of the root node. The root node is essential for traversing the tree and accessing its elements. In the given data structure, the root node's parent pointer is initially null, indicating that it has no parent. The root node's left and right child pointers are also initially null, indicating that it does not have any child nodes. As the tree is constructed, the root node's child pointers will point to the respective child nodes, and the parent pointer will point to the root node itself. The root node plays a crucial role in defining the structure and hierarchy of the binary tree, allowing for efficient traversal and manipulation of the tree's elements.

Accepted Answer

The binary tree is created by labeling the leaves with 0 or 1 based on the input bit. For FHE, if the bit is 1, a leaf node with cLabel = 0 is inserted on the left and a new node on the right. If the bit is 0, a leaf node with cLabel = 1 is inserted on the right and a new node on the left. For AHE, the traversal works similarly, but no leaf node is inserted on the left from the traversed path. The tree contains only paths that can be evaluated to zero, i.e., paths labelled with integers that are larger or equal to .

Accepted Answer

The randomization and permutation of ciphertexts in Algorithm 3 EvalPaths serve to protect the server's privacy and prevent leakage of information. Randomization ensures that the encrypted costs at the leaves are not predictable, while permutation of ciphertexts prevents leakage of the tree structure and potential information about it. By excluding the case of randomly generating a ciphertext that decrypts to zero, the algorithm maintains privacy and security. These operations are crucial in guaranteeing that only intended information is disclosed during the evaluation process.

Accepted Answer

In the constant case, AHE creates a tree with paths labeled with integers larger or equal to zero. If the comparison value is zero, the tree has one additional leaf. For values smaller than 2-1, the tree traverses to the left. To handle the case of zero, the server replaces the first encrypted bit with a ciphertext of 0 and omits the rightmost path of the tree during evaluation. This ensures that only the comparison bit is revealed, maintaining security during the computation process.

Accepted Answer

In the Shared Output Bit scheme, the server randomly chooses between computing GT (e.g., [ >= ]) or LT (e.g., [ <= ]) functionality. The server then computes GT if the chosen bit equals 0, and computes LT otherwise. This choice does not leak which operation was evaluated, as the tree structure remains the same for both GT and LT computations. After computing the desired functionality, the server returns ciphertexts to the client, with at most one encrypting 0 and the rest encrypting random plaintexts. The client can extract its share of the comparison bit and send back the necessary ciphertext for further computation. This approach is not specific to the Shared Output Bit scheme but also works with other AHE-based comparison protocols.

Accepted Answer

To handle encrypted inputs for AHE, we need to simulate the Not and Xor operations under AHE. The Not-operation is computed as ! = 1 - Add(1, MulCons(-1, -1)). The Xor-operation is computed as = -. However, the Xor-operation may result in 0 1 = -1, so we handle this before aggregating the paths. We build the normal cmp-tree using the encrypted input and apply the Not-operation. Then, we evaluate the bits on the tree and apply the Xor-operation on the bits along the paths. To ensure that the sum along the paths is not equal to 0, we multiply the Xor-result at level by 2 before aggregating the results. This multiplication can be applied on an AHE ciphertext. Lemma 6.9 ensures that the sum on such a path is always different to 0. The server creates the cmp-tree, evaluates the encrypted input, and sends a permuted vector to the client, allowing them to deduce the comparison bit.

Accepted Answer

An inverse normal cmp-tree is a binary tree with a right-oriented structure. It consists of a rightmost path of length, labelled with the bits of an integer. The deepest inner node of this path has a left leaf node labelled with the integer. Each inner node of the path has a left child leaf node, and the label on the right edge is 1 - (). The label on the left edge is 1 - (), and the label on the left child leaf node is 1 - (). This structure is different from the left-oriented structure of Definition 6.4, but it can also be represented with a left-oriented structure by labeling all leaves except the leftmost one with 1 - (). This definition is based on the Less Than (LT) function and its computation.

Accepted Answer

In node evaluation, the algorithm performs exactly one Not gate due to the fact that the left and right edges of an inner node are always labelled with opposite bits. For the encrypted case (Section 6.3), we need one Not and one Xor operation. Hence, we have in total Not-operations. This means that the number of operations in node evaluation depends on the specific algorithm and its implementation, but generally, it involves a single Not gate and potentially an Xor gate for encrypted cases.

Accepted Answer

In the FHE Binary Circuit, for binary encoding, there are 2 additions in node evaluations, 0 additions in path evaluation, and 2 additions during leaves aggregation, resulting in a total of 2 additions. For the encrypted case, an additional Xor operation is required at each node, leading to 2 additions and a total of 3 additions. Path aggregation requires 3 - 2 multiplications, resulting in a total of 3 - 2 multiplications. The optimized implementation reduces the depth to log, but increases the number of multiplications. For arithmetic encoding, each Xor operation requires 1 addition and 1 multiplication, while the Not operation requires 1 - . The total number of additions is 2, and the total number of multiplications is 3 - 2. Comparing to previous work, Cheon et al.'s scheme requires 2 - 2 additions and 2 - 3 + (-1) multiplications, while our optimized implementation uses precomputation for improved running time.

Accepted Answer

In the encrypted case, AHE requires 2 homomorphic operations for node evaluation and 3 -1 operations for paths evaluation, considering leaves. Additionally, 2 constant multiplications are needed to prevent issues explained in Section 6.4. In the constant case, AHE requires 2 -2 operations for paths evaluation, as leaves can be omitted. Compared to previous work, Veugen's scheme requires 4 moves between parties and 2 rounds, with a complexity lower bound of 2/3 2 + 3 + 2/3. Our scheme, however, only requires 3 -2 operations, making it a significant improvement over the optimized DGK protocol and Veugen's scheme.

Accepted Answer

The optimized implementation uses a two-dimensional array [(1, . . ., + 1), (1, 2, 3)] with + 1 rows and three columns. The array is initialized with the cmp-tree of = 3, where the first column stores the labels on the leftmost path. Column 2 and 3 store the right-oriented paths from the first row to the last one. The last row and the last column store leaf labels, where on the last row, only the first cell is filled. The evaluation process involves storing [] == [] in cell 1, its negation in cell 2, and ( []) in cell 3. This corresponds to the computation of decision bits in cell 1 and 2, and the leaf node labels in cell 3. The optimized implementation also uses a direct acyclic graph described in [45] for efficient evaluation of inner nodes. The implementation consists of computing a dependency list (DL) table for each element of the matrix, which contains cells' numbers along a multiplication path. This optimized implementation is more efficient than the binary tree representation and is suitable for FHE schemes with fixed parameters.

Accepted Answer

Veugen's scheme, DGK, and Joye and Salehi's scheme have different communication and computation efforts. Veugen's scheme requires the same communication as DGK and our scheme, but it also requires additional Paillier encryption for large randomized plaintexts. Joye and Salehi's scheme sends fewer ciphertexts and requires two rounds instead of one in the encrypted case. In terms of computation effort, Veugen's scheme and Joye and Salehi's scheme perform better than the original DGK scheme. However, Joye and Salehi's scheme performs slightly better for small bit-lengths (<= 32), while Veugen's scheme performs better for large bit-lengths (>= 64). Our optimized implementation is always better than the naive implementation. In the encrypted case, both Veugen's scheme and Joye and Salehi's scheme require additional Paillier ciphertext operations, which increase the running time. However, our encrypted case still performs better than the original DGK scheme. The network cost for the extra protocol round is not included in the evaluation. Our optimized implementation only needs memory for the two-dimensional array, making it easier to implement and evaluate compared to Cheon et al.'s scheme, which requires additional memory for computing products and storing log ciphertexts during evaluation.

Accepted Answer

Integer comparison plays a crucial role in machine learning (ML) applications. ML classifiers, such as decision trees (DTs), require integer comparison to classify inputs. Privacy-preserving techniques are essential in ML to protect the privacy of the model and user's data. For instance, private DT schemes rely on Distributed Garbled Key (DGK) [42, 51] or Fully Homomorphic Encryption (FHE) [45]. In the context of range queries, Tueno et al. [46] proposed an application using search tree structure and order-preserving encryption (OPE) to overcome the limitations of private-key OPE. Additionally, integer comparison is used in benchmarking and auctions, where the goal is to securely compute the ranked element among private integers. The DGK protocol was proposed for online auctions, enabling privacy-preserving market clearing price computation. In biometrics, integer comparison is required during the authentication or identification phase, where biometric features are scanned and compared with stored ones. Privacy-preserving biometric matching protocols, such as the one proposed by Blanton and Gasti [7] using DGK, ensure the protection of sensitive biometric data.

Accepted Answer

The proposed protocol for secure integer comparison is based on Homomorphic Encryption (HE) and uses binary trees for evaluation. It is a non-interactive solution suitable for various applications or as a subroutine in larger protocols. The protocol includes FHE and AHE modes with extensions and optimizations. Improved data representations and evaluations are implemented to reduce computational overhead. The protocol ensures secure comparison by concluding that strictly dominates, making the sum never equal to zero.

A Method for Securely Comparing Integers using Binary Trees

Chat with Paper

AI Agents for this Paper

Most frequently asked questions

1. What is multi-party computation (MPC)?

2. What is Homomorphic Encryption (HE) used for?

3. What protocols use garbled circuits for secure integer comparison?

4. What is the ideal functionality FGT in the DEFINITIONS section?

5. What is the ideal/real world paradigm in security and correctness?

6. How can the comparison problem be viewed as a classification problem using a decision tree?

7. What is the role of the root node in a binary tree data structure?

8. How is the binary tree created for AHE and FHE schemes?

9. What is the purpose of randomization and permutation of ciphertexts in Algorithm 3 EvalPaths?

10. How does AHE handle comparison to zero in the constant case?

11. How does the server choose between GT and LT functionality in the Shared Output Bit scheme?

12. How can we handle encrypted inputs for AHE?

13. What is the structure of an inverse normal cmp-tree?

14. How many operations occur in node evaluation?

15. How does FHE Binary Circuit perform in terms of additions and multiplications?

16. How does the complexity of AHE differ in encrypted and constant cases?

17. What is the optimized implementation data structure?

18. How does Veugen's scheme compare to DGK and Joye and Salehi's scheme in terms of communication and computation effort?

19. What are the applications of integer comparison in machine learning?

20. What is the proposed protocol for secure integer comparison?

Citations

Efficient and Accurate Homomorphic Comparisons

Secure Branching Program Evaluation

Private Computation On Set Intersection With Sublinear Communication

References

Data Mining: Practical Machine Learning Tools and Techniques

Data Mining Practical Machine Learning Tools and Techniques

A public key cryptosystem and a signature scheme based on discrete logarithms

Public-key cryptosystems based on composite degree residuosity classes

Fully homomorphic encryption using ideal lattices

Related Papers (5)

How to build a faster private information retrieval protocol

FINE: A Fully Informed aNd Efficient communication-induced checkpointing protocol for distributed systems

Controlling Concurrent Accesses in Multimedia Database Systems

Fully homomorphic encryption using ideal lattices

A Protocol for Privately Reporting Ad Impressions at Scale