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@sontungmtp.eth

Exploring Fully Homomorphic Encryption Today's digital world faces the challenge of keeping data safe while still being able to use it > FHE allows computations to be performed on encrypted data without ever needing to decrypt it, significantly enhancing data security > This article dive deep into FHE, from theory to real-world applications

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AVX-512 instructions: Advanced Vector Extensions 512 (AVX-512) are specific CPU instructions that support the execution of 512-bit operations. In the context of FHE, leveraging AVX-512 can improve the performance of polynomial arithmetic, which is fundamental in FHE schemes. These instructions enhance the handling of large data sets and complex calculations by providing the means to execute multiple homomorphic operations in parallel, reducing the time required for computations.

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SIMD batching: Single Instruction, Multiple Data (SIMD) batching is a pivotal performance metric in FHE that impacts its efficiency and scalability. By enabling multiple data points to be encrypted into a single ciphertext, SIMD batching allows simultaneous operations on all these data points. This parallel processing capability is especially beneficial for vector operations common in machine learning and statistical analysis, significantly reducing the computational overhead and latency compared to processing data points individually.

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Techniques have addressed many of the computational and efficiency barriers historically associated with FHE:

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FHEW (Fully Homomorphic Encryption from Worse-case assumptions) and TFHE (Fast Fully Homomorphic Encryption over the Torus): These focus on efficient Boolean operations and leverage the Learning With Errors (LWE) framework. Both schemes are optimized for low-latency cryptographic operations crucial for scenarios requiring rapid execution, such as real-time encrypted data processing.

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CKKS (Cheon-Kim-Kim-Song): This scheme is designed for handling encrypted real or complex numbers and performs approximate arithmetic on ciphertexts. CKKS is distinct in its ability to manage the precision of encrypted computations, which is crucial for applications like statistical analysis and machine learning where handling of real numbers is required.

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BGV (Brakerski-Gentry-Vaikuntanathan) and BFV (Brakerski-Fan-Vercauteren): Both schemes are structured around the Ring Learning With Errors (RLWE) paradigm. They are particularly robust in handling integer operations and employ techniques like relinearization and noise management to maintain the integrity and size of ciphertexts during calculations.

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FHE schemes allow secure, arbitrary computation on encrypted data. Their utility is differentiated by the core operations and mathematical foundation:

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Nguyễn Phương Nhi

Given the complexity and breadth of FHE, it's challenging to cover all aspects in a single blog post. For a deeper understanding, we recommend the series TFHE Deep Dive and Homomorphic Encryption 101.

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This means we can take any computable function f, apply it to encrypted data, and then decrypt the result to get the same outcome as if f had been applied to the unencrypted data. Theoretically, FHE changes the way we handle sensitive information by ensuring data remains encrypted throughout its entire processing lifecycle—from analysis to storage, thus protecting it against unauthorized access and potential data breaches.

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FHE extends these capabilities significantly. Unlike basic HE, FHE is not restricted to a single type of operation (addition or multiplication); instead, it supports an unlimited sequence of both, applied in any combination. This allows for the execution of complex algorithms on encrypted data. For example, with FHE, one can compute any function f defined by polynomial expressions, Boolean circuits, or other computational models on encrypted inputs, such that: Dec(Eval(Enc(x), f)) = f(x) for any computable function f, where:

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Multiplication (if the system supports it): Enc(a) x Enc(b) = Enc(a x b) These capabilities ensure that data remains secure and encrypted during computation, making HE valuable for secure computations like aggregating votes or confidential surveys.

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Before the advent of FHE, Homomorphic Encryption (HE) was already advancing secure data computation. HE was designed to allow specific operations on encrypted data, ensuring that information remained protected even during processing. For example, with a basic HE system, one could perform operations like addition and multiplication directly on ciphertexts, which is encrypted data. The mathematical operations in HE are limited to either addition or multiplication but not both simultaneously. This means, if you encrypt two numbers a and b to get Enc(a) and Enc(b), you can perform operations such as:

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In today's digital world, the challenge of keeping data safe while still being able to use it is more critical than ever. With increasing reliance on cloud computing and data sharing, ensuring that sensitive information remains secure throughout its lifecycle is a top priority. This is where Fully Homomorphic Encryption (FHE) comes into play. Introduced by Craig Gentry in 2009, FHE allows computations to be performed on encrypted data without ever needing to decrypt it, significantly enhancing data security. For organizations, this means the ability to fully utilize cloud services without the fear of exposing their most sensitive data. Consider the scenario where confidential financial information or medical records are being processed—FHE ensures that, even if such data were intercepted, it would remain inaccessible to unauthorized individuals. This capability makes FHE not only a powerful tool for security but also essential for maintaining privacy in an increasingly data-driven world.

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Scaling down: The ciphertext modulus is progressively reduced. If the original modulus is q, it is scaled down to a smaller modulus q'. The noise in the ciphertext is scaled down proportionally.

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Modulus switching reduces the size of the modulus and correspondingly scales down the noise in the ciphertext.

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Vectorized FFT kernels: Fast Fourier Transform (FFT) kernels are essential for the efficient implementation of polynomial multiplication in FHE. The use of vectorized FFT kernels enables these operations to be carried out more swiftly and with greater energy efficiency. By optimizing FFT algorithms to utilize SIMD capabilities and AVX-512 instructions, FHE systems can perform necessary convolutions for polynomial multiplication faster, thereby accelerating the overall encryption and decryption processes.

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AVX-512 instructions: Advanced Vector Extensions 512 (AVX-512) are specific CPU instructions that support the execution of 512-bit operations. In the context of FHE, leveraging AVX-512 can improve the performance of polynomial arithmetic, which is fundamental in FHE schemes. These instructions enhance the handling of large data sets and complex calculations by providing the means to execute multiple homomorphic operations in parallel, reducing the time required for computations.

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Limited operations: Suitable for a finite number of operations before noise management becomes necessary again.

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Cumulative errors: Rounding errors can accumulate, affecting the precision of the result.

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