  While cryptocurrencies and blockchain applications continue to gain
popularity, their energy cost is evidently becoming unsustainable. In most
instances, the main cost comes from the required amount of energy for the
Proof-of-Work, and this cost is inherent to the design. In addition, useless
costs from discarded work (e.g., the so-called Forks) and lack of scalability
(in number of users and in rapid transactions) limit their practical
effectiveness.
  In this paper, we present an innovative scheme which eliminates the nonce and
thus the burden of the Proof-of-Work which is the main cause of the energy
waste in cryptocurrencies such as Bitcoin. We prove that our scheme guarantees
a tunable and bounded average number of simultaneous mining whatever the size
of the population in competition, thus by making the use of nonce-based
techniques unnecessary, achieves scalability without the cost of consuming a
large volume of energy. The technique used in the proof of our scheme is based
on the analogy of the analysis of a green leader election. The additional
difference with Proof-of-Work schemes (beyond the suppression of the nonce
field that is triggering most of the waste), is the introduction of (what we
denote as) "empty blocks" which aim are to call regular blocks following a
staircase set of values. Our scheme reduces the risk of Forks and provides
tunable scalability for the number of users and the speed of block generation.
We also prove using game theoretical analysis that our scheme is resilient to
unfair competitive investments (e.g., "51 percent" attack) and block nursing.
