XOR: A Reversible Transformation in Cryptography
J. N. Pfeiffer · Zenodo (CERN European Organization for Nuclear Research) · 2026
Author’s Note — On Determinism and Randomness This book grew out of an exploration undertaken largely for the pleasure of following a simple question: how do deterministic transformations and randomness meet inside cryptography? Cryptographic algorithms are deterministic mathematical objects, yet many of their security guarantees depend critically on randomness, independence, unpredictability, and non-reuse. XOR became an unexpectedly rich point of entry into that contrast. Its behavior is algebraically exact, deterministic, reversible, and easy to compute; nevertheless, the security of constructions built around it can change completely according to the statistical properties of a key or keystream. Following that tension led naturally from Boolean algebra and finite fields to entropy, perfect secrecy, pseudorandomness, stream ciphers, AES, differential analysis, and the prospective mathematical directions explored near the end of the book. The monograph was therefore written as a scientific exploration rather than as an attempt to replace established cryptographic theory. It follows a simple operation as far as its mathematics will lead, asking questions, checking them analytically and computationally, and observing how deterministic structure and randomness interact. In that sense, the book is both a synthesis and an exploration written for the joy of understanding.