Anonymous One-Time Pads Explained

• Anonymous one-time pads are encryption schemes that ensure perfect secrecy by using untraceable, randomly generated key material to protect message content and source identity.
• They integrate diverse methods from classical dynamical systems to quantum protocols, thereby enabling secure logic operations, scalable key distribution, and robust physical-layer implementations.
• Research in this field addresses practical issues such as key randomness, physical and unconventional randomness sources, and interdisciplinary strategies for combating adversaries.

Anonymous one-time pads (OTPs) represent a class of information-theoretically secure encryption schemes in which the secrecy of the message, the identity (or source) of the communicating parties, or both, are protected by ensuring that the pad remains unknown or untraceable to eavesdroppers, infrastructure providers, and even potentially to the entity distributing the pad material. Research across cryptography, communications, and dynamical systems has developed a variety of frameworks for implementing, understanding, and applying anonymous OTPs in both classical and quantum regimes. These advances unify themes in encryption, lossless compression, logic computation, quantum cryptography, and physical-layer security.

1. The Dynamical Systems Perspective and Classical OTPs

A dynamical systems view interprets classical OTP encryption as the process of backward iteration on a binary map with random switching determined by the key—a symbolic sequence M is mapped by switching between the standard and "dual" binary maps at each position according to the pad (0803.0046). This construction shows that encryption is equivalent to determining the initial condition on a switched dynamical system, with the symbol-table for one pad bit and one message bit precisely replicating the XOR operation:

K (pad bit)	M (msg bit)	First bit of x₀ (initial condition)
0	0	0
0	1	1
1	0	1
1	1	0

The corresponding system can be written: $x_0 = \text{GLS-coding}(M, K)$ where “GLS-coding” applies backward iteration using either the standard binary map

$$T(x) = \begin{cases} 2x, & 0 \leq x < 1/2 \ 2x - 1, & 1/2 \leq x < 1 \ \end{cases}$$

or its dual (exchanging subintervals) at each step, selected by the pad bit.

There exist 16 distinct such switching systems, all achieving perfect secrecy in the binary case. Each represents an equally secure but different implementation of the OTP. The dynamical framework generalizes beyond XOR, enabling the construction of arbitrary logic gates (e.g., XOR, AND, OR, NOT, NAND, NOR, XNOR) as iterated map compositions—linking source coding, encryption, and Boolean computation within a unified perspective.

2. Robustness, Key Randomness, and Practical Generation

Perfect secrecy (as defined by Shannon) requires that the key be uniformly random, as long as the message, and used only once. However, practical constraints often motivate the paper of OTP robustness:

• If the key sequence deviates slightly from perfect uniformity (e.g., generated by a biased Bernoulli source), secrecy is only slightly diminished. Specifically, if the key process Y has entropy h(Y) = 1 – r_y, then

$$\liminf_{t\to\infty} \frac{1}{t} \log |\mathcal{V}(Z)| \geq h(X) + h(Y) - 1$$

where 𝒱(Z) is the set of likely plaintexts given ciphertext Z, and h(X) the entropy of the plaintext process (Ryabko, 2013). For small redundancy r_y, the attacker's uncertainty remains exponential in message length.

• Physical and unconventional randomness sources are increasingly used to generate pads:
  • Physical Unclonable Functions (PUFs) based on volumetric optical scattering create gigabits of random key material by probing unique material structures with spatial light modulators. Shared randomness is established by both parties probing their devices with identical input patterns; keys are mixed and made public via XOR to enable anonymous pad distribution without electronic storage (Horstmeyer et al., 2013).
  • The cosmic microwave background (CMB) power spectrum serves as a physically irreproducible random bit generator, producing keys via digitization, binning, and mapping schemes. Keys extracted in this way are unpredictable—even if the general parameters are known, repeated measurement is infeasible for eavesdroppers, and keys are FIPS 140-2 "potentially conforming" (Lee et al., 2015).
• Passphrase-based extraction from mathematical constants (e.g., digits of π) can produce OTPs combinatorially as secure as honest random pads—assuming strong, secret passphrases, and using algorithms such as BBP for unpredictable digit location selection (Gualtieri, 2021). However, the underlying randomness is only empirical, and the entropy bottleneck is managed via multiple-round mixing (XOR) and careful address space selection.

3. Distribution, Scalability, and Physical-Layer Keys

Anonymity in OTP deployment is often challenged by key distribution. Several approaches have been advanced for scalable and/or anonymous key delivery:

• The "hub model" supports practical, scalable distribution: one central authority ("hub") stores the main pad reserve; on-demand distribution of sub-pad material to users enables n-party scalability with minimal pre-shared key requirements. End-to-end session pads are then established for specific communications (Abel, 2012).
• Physical-layer and wireless schemes realize anonymous OTPs without explicit key exchange:
  • In the "KeyBITS" platform, optical shot noise is sampled using a high-rate physical random bit generator. The randomly generated bits are transmitted over classical channels, with the addition of recorded quantum noise (generated and added to the signal), and only parties sharing basis information can decode the key. Privacy amplification removes any information that might have leaked to eavesdroppers, and fresh pads are repeated secured in each transmission batch, eliminating courier-based delivery (Barbosa, 2019).
  • The "untappable key distribution system" uses phase-modulated optical signals with secret base indices and high intrinsic phase noise, followed by privacy amplification. The collision entropy (Rényi entropy) for Eve is ensured to be maximal if the noise width overlaps multiple bases (Barbosa et al., 2014).
• In wireless multi-user settings, "anonymous" one-time pads are achieved by aligning message and artificial noise signals at the eavesdropper (real interference alignment + cooperative/ blind jamming). The cooperative jamming signal plays the role of a one-time pad, masking the message so that the eavesdropper's mutual information is effectively zero, without explicit key sharing (Xie et al., 2015). Key implementation principles include controlling the entropy of jamming signals and aligning them to maximize secrecy without penalizing throughput at legitimate receivers.

4. Quantum Anonymous One-Time Pads and Reusability

Quantum mechanics enables several new possibilities for anonymous or reusable OTPs:

• Quantum one-time pad (QOTP) encryption applies random unitary operations (e.g., random Pauli operators) to a quantum state. When Alice and Bob share quantum correlations (potentially even when partially known to an eavesdropper), the effective rate at which classical or quantum messages can be communicated securely—without revealing information to Eve—takes the single-letter form

$$Q(ABE) = \sup_{A \to a\alpha} \left[ I(a : B | \alpha) - I(a : E | \alpha) \right]$$

where I denotes the conditional mutual information and the optimization is over all quantum instruments (partitions) from A to (a, α) (Brandão et al., 2010). This framework supports direct quantum message encryption or key distillation even in the presence of adversarial entanglement.

• Quantum physical principles—specifically, the no-cloning theorem and eavesdropping-detection (disturbance upon measurement)—permit the safe reuse of classical OTPs in quantum transmission media (Bennett et al., 2014, Deng et al., 2019). Secure transmission is maintained as long as no eavesdropping is detected; once compromised, the key is refreshed by using clean transmission capacity to send new pad bits. This property holds whether message encoding uses BB84 polarizations, strong quantum error correction, or other settings. Consequently, practical schemes can multiplex key replacement and message transmission, increasing pad lifetimes.
• Single-use quantum money or token schemes can be "recycled" as anonymous pads: a quantum token (a minted pure state) is measured to yield a classical string $R$ that becomes the OTP key. Tokens are constructed to be indistinguishable (ensuring user anonymity) and unforgeable (enforced by the no-cloning theorem and quantum query lower bounds), and the actual transmission and validation can occur over classical channels, eliminating the need for long-term quantum memory (Gavinsky et al., 7 Oct 2025). Anonymity is enforced via swap tests between tokens, making detectable any attempt by the issuer to flag or track pad material.

5. Novel OTP Constructions: Permutations, Steganography, and Integrity

Variants to standard positionwise-XOR OTPs enable advanced properties:

• Permutational OTP ("non-degenerate OTP"): Messages and keys are encoded as permutations via their Lehmer code representation (after conversion to factoradic), and encryption is a key-dependent cyclic permutation operation. The diffusive property ensures that any bit flip in the ciphertext translates to a non-local and unpredictable permutation change in plaintext, complicating tampering and error analysis (Shafarenko, 10 Apr 2024). Redundancy is injected using Pseudo Foata Injection, ensuring that only a negligible fraction of ciphertexts could plausibly pass an integrity check, providing unconditional integrity assurance.
• Quantum permutation pads (QPP): Over the discrete Hilbert sphere (DHS), the pad becomes a randomly selected permutation from the full symmetric group S_{2ⁿ}. This offers a large keyspace (entropy ∼ n·log₂(n!)), supports reusability (due to noncommutativity preventing key cancellation), and generalizes the "one-to-one" mapping property of the original OTP to a "1-(2ⁿ–1)!-1" joint mapping, vastly enhancing secrecy and resisting brute-force attack, even under repeated usage (Kuang et al., 2023).
• Steganographic OTP ("T.E.C."): Here, randomness not only in the pad but also in placement—extra bits are inserted into the plaintext's binary representation according to a secret, non-repeating key, derived from mathematical constants (e.g., transcendental numbers), identities, and timestamps. The result is a ciphertext of variable length and structure, complicating cryptanalysis and file recovery, and significantly elevating brute-force complexity in the context of password protection (Zirkind, 2013).

6. Short Key Approximations and Practical Variants

Given the storage and distribution limitations inherent in conventional OTPs, substantial research has focused on key-length reduction strategies:

• In the "Short Key Dream" paradigm, a much shorter, shared secret string Z is used to extract positions in a large public randomness pool R (e.g., biological databases, π digits, or other astrophysical sources). Both parties can then independently generate a long pad from multiple public substrings indexed by Z, potentially combining them via XOR (Cerruti, 2021). Provided that the adversary is storage-constrained (cannot store all of R), near-OTP security is achievable even for long messages [see also (Starossek, 2016)].

Approach	Secret Shared	Pad Generation Method	Security Condition
Classical OTP	O(n) bits	Pre-distributed truly random bits	Shannon's theorem; key used once
Maurer Short-Key	O(log n) bits	Indexed substrings of public randomness	Storage-bounded adversary; key expansion
QPP	O(M log₂(n!))	Random permutation schedule	Shannon perfect secrecy for permutations

Streaming and high-throughput applications are facilitated by multiple-time pad (MTP) ciphers, which use publicly shared libraries of basic keys. Each session picks a combination (XOR) of subset keys, specified by a short, securely transmitted keyword, drastically reducing key distribution requirements while enabling blockwise, stateless encryption (Starossek, 2016).

7. Interdisciplinary Connections and Impact

Anonymous one-time pads unify core principles and techniques from dynamical systems, information theory, communication complexity, and quantum information. Techniques that were originally invented in the context of arithmetic coding, logic computation, and physical-layer security now integrate seamlessly with advanced cryptographic protocols in both theory and practice. As both quantum and classical hardware continue to evolve, schemes emphasizing physical, statistical, or combinatorial properties of the pad (as opposed to computational assumptions) are likely to become central to next-generation private communication systems, secure voting, and anonymous authentication.

Research continues to address the tradeoffs among secrecy, integrity, efficiency, key management practicality, and anonymity—exploring boundaries between perfect and nearly-perfect secrecy, and the implications of physical law for cryptographic protocol design.