Coin selection by Random Draw according to the Boltzmann distribution

Coin selection refers to the problem of choosing a set of tokens to fund a transaction in token-based payment systems such as, e.g., cryptocurrencies or central bank digital currencies (CBDCs). In this paper, we propose the Boltzmann Draw that is a probabilistic algorithm inspired by the principles of statistical physics. The algorithm relies on drawing tokens according to the Boltzmann distribution, serving as an extension and improvement of the Random Draw method. Numerical results demonstrate the effectiveness of our method in bounding the number of selected input tokens as well as reducing dust generation and limiting the token pool size in the wallet. Moreover, the probabilistic algorithm can be implemented efficiently, improves performance and respects privacy requirements - properties of significant relevance for current token-based technologies. We compare the Boltzmann draw to both the standard Random Draw and the Greedy algorithm. We argue that the former is superior to the latter in the sense of the above objectives. Our findings are relevant for token-based technologies, and are also of interest for CBDCs, which as a legal tender possibly needs to handle large transaction volumes at a high frequency.

💡 Research Summary

The paper addresses the coin‑selection problem that arises in token‑based payment systems such as Bitcoin‑style UTXO ledgers and prospective central bank digital currencies (CBDCs). Selecting a subset of unspent tokens (coins) to cover a transaction amount V is a classic subset‑sum (NP‑complete) problem, and existing solutions (FIFO, LVF, HVF, Greedy, Knapsack, Branch‑and‑Bound, genetic‑algorithm hybrids, etc.) typically satisfy only a subset of the five research objectives the authors define:

• RO 1 – keep the wallet’s token pool size bounded,
• RO 2 – minimise the number of input tokens per transaction,
• RO 3 – limit the creation of dust (very low‑value tokens),
• RO 4 – preserve a balanced distribution of token values, and
• RO 5 – support high‑throughput, concurrent transaction processing.

A systematic literature review (Table I) shows that no single prior method meets all five goals without incurring high computational cost or requiring fallback mechanisms.

The authors propose a new probabilistic algorithm called Boltzmann Draw (BD). The key idea is to reinterpret a token’s monetary value v as a physical energy level and to draw tokens according to the Boltzmann distribution:

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