Cryptography

This blog lays the essential mathematical groundwork for understanding modern cryptography. We begin with Group Theory, exploring the definitions of a group, its order, and subgroups—abstract concepts that form the backbone of many public-key cryptosystems. The post then transitions to Vector Spaces and core linear algebra concepts like basis, linear independence, and orthogonality. Finally, we delve into the Gram–Schmidt algorithm, a powerful tool for constructing orthogonal bases, paving the way for more advanced cryptographic topics. [Read More]

This sixth blog in the Cryptography Series explores two fundamental ideas regarding Pseudorandom Generators (PRGs). First, we will learn about the Blum-Micali method, an intelligent sequential construction that allows us to build a PRG capable of extending its output to an arbitrary length from a base PRG that only stretches its input slightly. Next, the post introduces a core security standard: next-bit unpredictability. The highlight of this post is the proof of a classic and surprising result: a PRG is considered secure (indistinguishable from a truly random sequence) if and only if no adversary can efficiently predict its next bit. This provides a solid bridge between statistical theory and practical predictability. [Read More]

This blog delves into two critical aspects of stream ciphers and pseudorandom generators (PRGs). The first part exposes severe vulnerabilities that arise from the misuse of stream ciphers, focusing on two classic attacks: the “two-time pad” flaw, where key reuse leads to complete plaintext disclosure, and “malleability,” which allows an attacker to modify message content undetectably. The second part shifts from attacks to constructions, introducing a method for combining secure PRGs to create a new one with a longer output. The centerpiece of this section is the detailed presentation of the “hybrid argument,” a fundamental and powerful proof technique widely used in modern cryptography to demonstrate the security of protocols. [Read More]

This post explores the stream cipher, a practical solution to the long-key problem of the One-Time Pad. We delve into how a “pseudo-random generator” (PRG) can “stretch” a short key (seed) into a long keystream for encryption. The blog’s core focuses on the formal security definition of a PRG—the concept of “computational indistinguishability”—and presents a detailed proof demonstrating that a secure PRG leads to a semantically secure stream cipher. [Read More]

This blog delves deeper into the meaning and consequences of semantic security, one of the foundational concepts of modern cryptography. We will begin by proving that a semantically secure encryption scheme does not allow an adversary to deduce even the smallest piece of partial information about the original message, such as its parity. Next, through the example of an Internet roulette game, the blog will illustrate how the definition of semantic security is applied to analyze the security of real-world applications. Finally, we will become acquainted with another, equivalent interpretation of semantic security—the bit-guessing game—a powerful and convenient tool commonly used in cryptographic proofs. [Read More]

This blog introduces the notion of message recovery attacks, defines the corresponding security game, and proves that semantic security implies message recovery security via a black-box reduction. A semantic adversary is constructed by wrapping around a message recovery adversary, showing that any success in message recovery would contradict semantic security. This sets the stage for future reductions that interrelate various cryptographic security notions. [Read More]

This blog post explores the concept of Perfect Security in cryptography, which guarantees that observing a ciphertext provides no additional information about the original message. We establish a fundamental theorem stating that Perfect Security is equivalent to the statistical independence of ciphertexts and plaintexts. However, Shannon’s Theorem proves that achieving Perfect Security requires the secret key to be at least as long as the message, making practical implementations infeasible beyond small-scale use cases like the One-Time Pad. [Read More]

In the rapidly evolving world of cryptography, verifying large-degree polynomials efficiently is a challenge. The FRI protocol solves this by enabling the prover to demonstrate that a given polynomial is close to a low-degree one through a series of recursive folding steps. By halving the polynomial’s degree progressively, the FRI protocol reduces computational complexity while preserving the security of the verification process. The prover commits to the polynomial’s evaluations using Merkle trees, ensuring both integrity and consistency, which are verified through collinearity checks. In this blog, we take a deep dive into the mechanics of the FRI protocol, from the commitment phase where the degree is halved to the query phase where the verifier checks consistency. Whether you are a blockchain enthusiast, cryptography expert, or just curious about the future of digital proofs, understanding the FRI protocol will give you a glimpse into the future of secure, scalable cryptographic verification. [Read More]

Dive into the fascinating world of blockchain technology with this engaging exploration of its foundational elements! From understanding the robust structure of Merkle trees that underpins data verification to unraveling the complexities of the Ethereum Virtual Machine (EVM), this blog offers a thorough overview of how decentralized applications and smart contracts operate. Discover how Ethereum revolutionizes trust and security through its unique consensus mechanisms and the role of Ether as its lifeblood. As you embark on this journey, you’ll gain insights that not only enhance your understanding of Ethereum but also prepare you for the cutting-edge innovations of zkEVM. Join me as we unlock the potential of blockchain and envision the future of decentralized technologies! [Read More]

In this blog, we unravel the complex process of verifying circuit constraints within SNARKs, a cornerstone of modern cryptographic protocols. You’ll learn how correctness is ensured across inputs, gate computations, and wiring within a circuit, all without compromising the secrecy of the information involved. By understanding these mechanisms, you gain insights into the robust security that underpins technologies like blockchain. Whether you’re a cryptography enthusiast or a professional in the field, this blog offers a deep dive into the mathematical and logical foundations that make zero-knowledge proofs so powerful. [Read More]