Quantum Encryption: Private Keys

Abstract

The practical security of cryptocurrency wallets reduces to the unpredictability of secret keys and signing nonces. In deployed systems these values are derived from classical pseudorandom number generators (PRNGs) seeded by operating-system pools. Determinism elevates the seed to a single point of cryptographic failure: if effective min-entropy is transiently low, biased, or duplicated across devices, attacks become feasible despite standard assumptions about elliptic-curve hardness. We present a systems architecture in which on-device quantum random numbers are subjected to online health testing, conservative entropy estimation, and extractor-based post-processing prior to seeding any cryptographic primitive. We formalize the adversary, prove that (i) seed min-entropy shortfalls collapse key search, and (ii) biased or reused ECDSA nonces leak the private key via the Hidden Number Problem; and we show that when the extractor output is ε-close to uniform with a certified entropy bound, these attack classes are eliminated up to ε, independently of implementation details of the quantum source. The resulting seeds are standards-compatible (e.g., BIP-39/BIP-32), enabling our device to supply trustworthy entropy to existing wallet ecosystems.

Introduction

A cold wallet is secure if and only if the random variables that instantiate its master seed and its signing nonces are unpredictable to any computationally bounded adversary. Classical PRNGs provide efficient expansion but are deterministic functions of their seeds; consequently, the problem of breaking the wallet reduces to learning or constraining the seed. At scale, operating conditions create correlations and low-entropy states (cloned firmware images, synchronized virtual machines, early-boot scarcity), and these effects have repeatedly surfaced as catastrophic failures in the form of nonce reuse or weak key generation. The thesis of this paper is that anchoring the origin of entropy in a quantum process, coupled with extractor-backed post-processing and continuous liveness tests, closes the dominant avenues of attack that exploit determinism at the origin.

Model

A wallet generates a master seed and derives keys and (for ECDSA) nonces from a DRBG GGG. The adversary knows firmware and primitives, controls the host OS/VM at seed time, can inject classical signals (RF/EM/power/timing), and observes unlimited signatures; secrets inside the secure element are not readable during normal operation.

For the quantum device, at discrete times the comparator input is

where Qt∼N are balanced (differential) quantum fluctuations after whitening; Z is arbitrary classical side information under adversarial control; μ(Z) is a Z-measurable residual offset bounded by the analog rejection ∣μt∣≤Δ; and ∣Wt∣≤η|Wt| models bounded nonidealities. Health tests enforce fail-closed behavior on anomalies.

Why classical determinism fails

Theorem 1 (Seed search).

Let H∞(S)=h. All wallet secrets are deterministic in (S,T), with public transcript T. An attacker recovers the seed by enumerating 2^{h} candidates and checking a derived public key or a known signature: expected time O(2^{h}).
Consequence. Any deployment that permits correlated boot states, cloning, or seed reuse risks h≪ℓ; curve hardness is irrelevant once the seed space is small.

Theorem 2 (ECDSA nonce defects).

With hash m and secret x, ECDSA signs using nonce k: r=(kG)x mod ns ≡ k−1(m+rx) mod ns.
(a) Reuse: If two signatures reuse k, then x is recovered algebraically from public transcripts.
(b) Bias/low entropy: If nonces​ lie in small intervals or are otherwise low-entropy, x is recovered from polynomially many signatures via lattice methods for the Hidden Number Problem.
Consequence. Any PRNG state reuse or seed correlation that affects nonces is catastrophic.

Lemma 1 (Exact bias bound under arbitrary Z).

Conditioned on any Z and ∣Wt∣≤η,

where Φ is the standard normal CDF.
Reason. Worst case is ∣μt∣=Δ and Wt=±η; with Qt∼N(0,σ2), this is exact.

The per-sample min-entropy lower bound can thus be defined as:

Lemma 2 (Conditional min-entropy accumulation).

Without assuming independence across time and for arbitrary Z,

This occurs, as we arrive at the sum through chain rule.

Theorem 3 (Extractor closeness to uniform).

Let k:=H∞(Bn∣Z). For a strong seeded extractor Ext:{0,1}n×{0,1}d ⁣→ ⁣{0,1}m,

with Um uniform and Δ statistical distance (strong leftover-hash lemma).

Interpretation. Extraction is proven conditioned on any classical signals Z; signals count only by tightening Δ and thus δ. This demonstrates the robustness to signal injection by QRNG-based cryptographic protocols.

Quantum Correlations

A raw QRNG stream is not automatically usable. Real devices introduce nonidealities such as detector bias, analog drift, and classical electromagnetic contamination. A wallet that claims quantum strength must therefore implement three layers of assurance: online health testing, conservative entropy estimation, and cryptographic extraction.

Health tests operate continuously and at startup. Their purpose is not to “prove” randomness but to rapidly detect pathological behavior such as a stuck symbol, a sudden change in bias, or long-range correlations inconsistent with the calibrated model. The statistical machinery in NIST SP 800-90B provides a useful template, including repetition-count and adaptive-proportion tests that bound the probability of undetected faults. When a test fails, the device fails closed: key generation and signing halt rather than continuing under degraded conditions.

Entropy estimation converts raw symbol statistics into a defensible lower bound​ per output bit under worst-case modeling assumptions. Conservative estimation treats unexplained correlations as adversarial rather than benign. The wallet firmware keeps a running lower bound for the min-entropy in recent buffers and never extracts more near-uniform bits than the bound permits.

Extraction then maps the partially unpredictable stream to bits that are close to uniform in the information-theoretic sense. A seeded extractor based on standard cryptographic primitives, such as an AES-based construction or an extendable-output function, removes bias and renders any residual correlations useless to an adversary who does not know the extractor seed. The seed itself is established as device-unique material during provisioning and is periodically refreshed from the QRNG so that even an attacker with transient access to firmware state cannot use old observations to predict future outputs. The extractor output in turn seeds a deterministic random bit generator (DRBG) to satisfy application throughput, with regular reseeding so that compromise of DRBG state does not retroactively endanger past keys.

In security terms, the decisive property is the quantum-to-classical correlation of the entropy source: the fraction of variability at the digitizer that originates in irreducible quantum fluctuations versus variability that is traceable to classical, potentially reproducible influences (power ripple, clocks, firmware timing, RF/EM pickup, VM/snapshot state). Our wallet architecture is engineered so that quantum fluctuations are the primary driver of threshold crossings after the entire analog chain—balanced detection with high common-mode rejection, shielding, calibrated offset nulling, bandwidth-limited filtering, and whitening—has suppressed classical contributions to a strictly bounded residue. We then certify that dominance at run time via health tests (stationarity, bias and noise-floor monitors, autocorrelation windows) and enforce a fail-closed policy: if the measured classical residue ever threatens to influence bit decisions beyond the allowed envelope, no mnemonic is produced. This regime has a concrete security consequence: the resulting seed bits are not functionals of operating-system state, deployment regularities, or broadcast environmental conditions; they are the extractor image of measurements whose unpredictability persists even when an adversary knows the entire classical environment. By contrast, in a classical RNG pipeline the “quantum share” is, by construction, zero: all variability is classical and therefore entangled with provisioning practices and software state. Routine operations—golden-image cloning, synchronized first boots, container schedulers, or library misuses—imprint structure into the seed material and create predictable correlations across time and across fleets that an attacker can exploit through offline enumeration and transcript checks. Maintaining a high quantum-to-classical dominance, continuously verified and backed by extraction and fail-closed gating, removes that attack surface: there is no stable classical footprint for an adversary to model or replay, and the only viable path would be to defeat the extractor’s indistinguishability guarantee or the downstream cryptography under uniform seeding—both strictly stronger requirements than the well-documented seed-space narrowing that compromises classical wallet creation in practice.

In prevailing wallet implementations, entropy is purely classical: seeds and nonces derive from operating-system state, device drivers, timer readings, and on-chip TRNGs that are themselves deterministic circuits modulo environmental jitter. Consequently, their outputs inherit reproducible structure from routine operational practices—golden-image cloning, VM snapshot restoration, synchronized first-boot windows, container scheduling, and library/API misuses—which compress the effective seed space and create cross-device correlations exploitable by offline enumeration and transcript checks. Our design changes the causal origin of the seed: it introduces a high-integrity quantum source whose fluctuations dominate the decision boundary after balanced detection, shielding, calibrated offset nulling, bandwidth filtering, and whitening; the remaining classical residue is continuously bounded by online health tests, and seed emission is gated (fail-closed) if those bounds are ever threatened. A strong extractor then maps the measured randomness to a mnemonic that is indistinguishable from uniform up to a specified margin, so the resulting seed is not a functional of host or deployment state and cannot be reconstructed from public clues about provisioning or timing. In short, classical wallets are vulnerable precisely because their entropy is entangled with reproducible system conditions, whereas the QRNG-first pipeline ensures the seed is anchored in device-local quantum measurements and is released only when instrumentation verifies that classical influence is negligible—removing the practical attack surface created by purely classical seeding.

Conclusion

One might argue that carefully engineered classical entropy sources suffice. In small deployments with meticulous operational security, they can. The challenge is longevity and scale. Cryptographic assets now persist for decades, while software stacks and operating environments churn continuously. Each churn event reopens the possibility of low-grade entropy regressions that go unnoticed for months. Quantum-origin entropy breaks this feedback loop because it does not depend on operating-system state, boot order, or virtualization idiosyncrasies. Two devices built a year apart and provisioned in different factories do not accidentally share a seed simply because a daemon restarted at the wrong time. The source of unpredictability is anchored in physical law rather than in the contingent dynamics of a software pool.

For cryptocurrencies, this anchoring changes economic incentives. Attackers prefer scalable defects: a misconfigured library or a biased OS RNG that exposes many wallets at once. A sealed QRNG module with online testing denies that leverage. Compromise becomes individualized and expensive, which is precisely the direction one wants risk to flow when defending bearer instruments that cannot be revoked. The doctrine that “crypto is easy; randomness is hard” has been vindicated repeatedly by real-world failures. The move from classical PRNG seeding to quantum-origin entropy with extraction is not a cosmetic upgrade; it is a categorical change in what must go wrong for an attacker to succeed. By beginning with physically unpredictable events, bounding their min-entropy conservatively, extracting near-uniform bits, and only then entrusting a DRBG with throughput, a cold wallet regains the assumptions that textbook reductions require. The details of our QRNG module remain intentionally low-key and unpublished, yet the argument for its necessity is public and falsifiable: if one can guarantee high min-entropy at the origin and preserve it through extraction and reseeding, the avenues that historically break wallets—seed prediction, nonce reuse, silent bias—close decisively. For digital assets intended to outlive operating systems, cloud images, and even cryptographic fashions, such guarantees are not a luxury. They are the foundation.

Quantum Shield

Our wallets generate the 24-word mnemonic directly from extractor-processed quantum entropy, not from a software PRNG whose state can be nudged or synchronized by external signals. Classical generators have repeatedly exhibited correlations induced by shared boot timing, VM snapshots, RF or EM interference, power-cycling patterns, and identical factory images; these exogenous signals constrain the seed space and enable scalable enumeration. In contrast, the entropy that underlies our mnemonic originates in a calibrated quantum process whose outcomes are information-theoretically unpredictable given any classical side information. Raw measurements are subjected to continuous health tests and conservative min-entropy estimation; only when lower bounds are satisfied are bits passed through a strong cryptographic extractor, which provably removes residual bias and renders any environment-induced structure useless. The extractor seed is device-unique and refreshed from the QRNG, reseeding the DRBG that feeds mnemonic construction; host-OS state, clocks, and I/O timing do not influence the seed path, and the device fails closed on any liveness anomaly. Consequently, the 24-word phrase encodes near-uniform bits with a quantified entropy bound, eliminating the class of “signal-driven correlation” attacks that compromise classical wallet generators while remaining fully compatible with BIP-39/BIP-84 ecosystems.