da1729's Blog

I have been wanting to dive into algebraic cryptanalysis of the modern cryptographic primitives. Almost all of PQC and FHE is based on mathematical lattices, and hard problems based on it. For thos...

In continuation to my MPC blogs and my study of FAEST, here is a blog explaining a method of constructing advanced Zero Knowledge (ZK) proofs by using simpler tools from Multi-Party Computation (MP...

Craig Gentry (yeah the man himself) and Yongwoo Lee proposed a new FHE scheme quite recently (October 2025). The special thing about this new scheme is its algorithmic efficiency towards Homomorphi...

Ok, let’s continue… The major reference for this blog is “A Pragmatic Introduction to Secure Multi-Party Computation” by D. Evans, V. Kolesnikov, M. Rosulek. Yao’s Garbled CircuitsLet’s set the gr...

I am studying the FAEST digital signature protocol (quantum-safe) which requires me to learn concepts like MPC, Zero-Knowledge, and Digital Signatures. So here are my notes and explanation for what...

Gadget Decomposition is one of the key and essential Homomorphic Operations in FHE cryptosystems based on the LWE problem. I have already introduced the LWE problem and how to build a basic cryptos...

Ok this isn’t a technical breakdown like my crypto posts, this is more me just sitting down and letting a thought run loose. Imagine a robot, not running on GPUs or neuromorphic chips, but on some ...

Alright, so I’ve been diving deep into the mathematical foundations behind modern FHE schemes, and honestly, the more I understand the underlying algebra, the more elegant this whole thing becomes. In my previous blog on LWE cryptanalysis, I touched on the basic Learning With Errors problem, but now I want to get into the real mathematical meat of Ring-LWE and the CKKS scheme.

This is gonna be pretty heavy on the math - we’re talking cyclotomic polynomials, Galois theory, Chinese Remainder Theorem, and some serious algebraic number theory. But stick with me, cuz understanding this foundation is crucial for grasping how modern FHE schemes actually work under the hood.

The Learning With Errors (LWE) problem forms the foundation of many post-quantum cryptographic systems. These systems depend on a critical assumption: that no one can distinguish between LWE-generated samples and truly random data. I wanted to test this assumption by building sophisticated machine learning models to see if they could break this fundamental security property.

My goal wasn’t to completely “break” LWE—that would be a monumental achievement. Instead, I aimed to map out where LWE’s security boundaries actually lie in practice. Could a neural network, with its pattern-recognition capabilities, successfully identify LWE samples and violate the core security assumption? Through multiple iterations of model improvements and data refinement, I discovered just how resilient LWE really is. The results highlight why cryptographers rely on standardized libraries with carefully chosen parameters backed by years of analysis.

Matrix Multiplication through MAC operations

Below, I have presented a python code, illustrating matrix multiplication using MAC operation. But, why matrix multiplication only? Because everything is a fking MATRIX!!! (that’s why the film is called Matrix). Physicists, electrical engineers, computer scientists/engineers just love representing everything in matrix, and why not, they make everything more streamlined and easy to represent. Since, we are representing everything in matrices, especially in machine learning and AI, like we have the weights matrices, input vectors, output vectors, etc., we have to do a lot of matrix multiplication and in hardware, using MAC operators, we can easily perform it. Now, carefully look and understand the python code below: