Abstract

We lay the foundations for a blockchain scheme, whose consensus is reached via a proof-of-work algorithm based on the solution of consecutive discrete logarithm problems over the point group of elliptic curves. In the considered architecture, the curves are pseudorandomly determined by block creators, chosen to be cryptographically secure and changed at every epoch. Given the current state of the chain and a prescribed set of transactions, the curve selection is fully rigid, therefore trust is needed neither in miners nor in the scheme proposers.

Introduction

A proof-of-work (PoW) is a procedure that allows a prover to demonstrate that it is very likely to having performed a speci c amount of computational work within a prescribed interval of time.

This concept has been formalized in 1999 [19], although previous instances of delaying

functions conceived for similar purposes had appeared earlier [ 2, 9, 12, 14, 17, 20, 30 ].

Since 2008, PoW-methods [ 24 ] have been attracting a considerable interest as Bitcoin [ 27 ] introduced a PoW-based consensus algorithm, which puts miners in competition for solving a cryptographic challenge. Bitcoin's consensus relies on a hashcash system [ 3, 4 ], whose workload may be easily adjusted with a fastly veri able output. Despite their high e ciency and easy implementation, all the hashcash-based protocols share a common limitation: the huge amount of computations employed by nodes becomes useless after the consensus is reached. This aspect has been raising environmental concerns and many solutions have been proposed to reduce these energy-intensive computer calculations.

A promising countermeasure to this issue is the adoption of bread pudding protocols [19].

They face the aforementioned problem by perfoming a computational work that is reusable

either for practical [ 11, 13, 26, 33 ], cryptographical [ 19, 29 ] or mathematical [ 36 ] reasons. Moreover, the latter class of systems encloses several protocols that are meant to be research propellants [ 5 ], namely designed to boost the commitment upon the solution of di cult mathematical problems.

Along the same line, we propose a blockchain architecture with a PoW-consensus algorithm based on the solution of the Discrete Logarithm Problem over the point groups of elliptic curves (ECDLP). This approach does not aim at reducing the energy consumption, but at adding a scienti c scope to the PoW mechanics, meanwhile achieving supplementary security features.

The idea of basing the PoW on ECDLP has already appeared in other works [18, 21], as

this problem is widely studied and applied in cryptographic protocols. However, the considered curves does not usually ful l the standard security criteria [ 6 ], especially for what concerns the fully rigidity : the network has to initially trust an authority that is providing the curve parameters.

In this work we radically solve this issue by designing a PoW-system based on elliptic curves

that are changing over the time. Since the curves are pseudo-randomly constructed and satisfy general security conditions, a malicious user could attack the chain only by breaking the ECDLP for an immense class of elliptic curves, which is currently considered infeasible.

An extended version of this work may be found in [23].

This paper is organized as follows: after a quick summary of the ECDLP in Section 2, we

deline the proposed blockchain architecture in Section 3 and its blocks construction in Section

4. The strong points of this system are discussed in Section 5, while in Section 6 future work directions are suggested.

2

ECDLP

The ECDLP is a renown problem that consists of nding an integer N 2 N such that the N -th

multiple of a base point P of an elliptic curve E over a nite eld equals another given point Q, i.e. Q = N P .

Solving ECDLP for a well-chosen curve E is considered to be a di cult challenge. Currently

the best known general attacks are Baby-Step Giant-Step [ 32 ] and Pollard's Rho - Kangaroo algorithms [ 28 ], which have an asymptotic complexity of O(pjEj), where jEj is the size of E.

The introduction of Semaev's polynomials [31] have suggested the existence of subexponential algorithms to solve ECDLP, however no clear evidence has emerged. Pairings-based attacks [15, 25], Index calculus [1, 22, 35] and Xedni calculus [34] have been recently being studied, but none of them seem to signi cantly reduce the problem complexity so far.

3

A sample blockchain architecture

To show how our PoW works, we introduce a schematic sample ledger architecture, but our

algorithm may easily be adapted for any blockchain scheme. Our architecture is based on two types of blocks: [EB] An Epoch Block contains, aside from the header and a list of transactions, a prime number p, an elliptic curve E de ned over Fp and a base point P of E, all to be determined by the proposing miner.

Moreover, it encloses as PoW an integer N 2 f0; : : : ; jEj 1g to be discovered by the

proposing miner such that N P is the point of E that is deterministically determined from the header of the block.

These EBs occur once every 2016 blocks in the blockchain.

[SB] The Standard Blocks are just a light version of the EB blocks, they are constructed in the same way except for p, E, P , which are inherited from the last EB block of the chain.

SBs constitute the vast majority of the blocks of the chain.

Data #0

EBs basically de ne the setting (curves and base points) on which the discrete logarithm

PoWs will have to be solved in the following epoch. They are slightly heavier to be produced and veri ed but occur rarely (roughly once every two weeks with a BTC-like di culty adjustment).

4

Block construction

In this section we de ne the envisioned speci cations of blocks, whose motivation will be analyzed in Section 5.

First, we need a deterministic function P_Gen to construct a point on a given elliptic curve

E from a prescribed hash digest h, which we treat as an integer for simplicity. The following is a concrete example of such a function.

f u n c t i o n P _ G e n ( h , E ) i = 0 w h i l e #{ p o i n t s of E with x - c o o r d = h + i } = 0: i = i + 1 P = ( h + i , *) p o i n t of E with 0 * < p /2 r e t u r n P

We notice that the points determined by the above function are a ne by construction. The

hash H that we propose to use in the following is SHA3-512 [ 7 ], which provides a satisfying collision resistance even against post-quantum attacks, but one might conceivably replace it with another properly constructed one.

Standard Blocks

A minimal model of a SB consists of a list of valid transactions and a header, which comprises

their Merkel root M, the hash of the previous header hprev and an integer N solving PoW :

P_Gen(H(hprevjjM); E) = N

P; where E and P are the ones de ned in the last EB. h = H(new header) hprev

M

New transactions

T1

T2 Tk Epoch Blocks

An EB is a thick version of a SB, namely it is constructed in a similar fashion but it encloses

three additional data: the prime p, the elliptic curve E over Fp and the base point P of E.

Generating p

The prime number p is the responsible of the expected running time of the PoW. Its size is determined by the di culty parameter d, whose tuning depends on the block production ratio that a designer wants to obtain. Therefore we do not discuss the choice of d but we refer to the BTC implementation [ 8 ] or to more structured models such as personalized di culty adjustments [ 10 ]. Our goal is to produce a prime number of the prescribed size and satisfying known properties of secutity (for a detailed description, see the full paper [ 23 ]).

Given the di culty parameter d and the hash of the previous header h, we propose the generation of such a prime number p as follows. f u n c t i o n p _ G e n ( d , h ) r e p e a t h = H( h ) p = N e x t P r i m e ( h mod 22d ) u n t i l p s a t i s f i e s the r e q u i r e d p r o p e r t i e s r e t u r n p

Generating E

We aim at generating pseudorandom elliptic curves for which no attacks are currently known, i.e. satisfying a number of security properties [6, 23].

Let h be the previous block header, we suggest to generate the curve as follows.

f u n c t i o n E _ G e n ( p , h ) i = 0 r e p e a t i = i + 1

AE = H( h + i ) BE = H(AE )

E d e f i n e d by y2 = x3 + AE x + BE over Fp u n t i l E is an EC s a t i s f y i n g s e c u r i t y p r o p e r t i e s r e t u r n E

Generating P

The base point we prescribe for an EB and its subsequent epoch is

P = P_Gen(H(p jj AE jj BE ); E):

The new epoch parameters are manufactured before the PoW production, which therefore depends on them. [EB]

h = H(new header)

Epoch Data

p = p_Gen (d; hprev) E = E_Gen (p; hprev) P = P_Gen (H(p jj AE jj BE); E)

PoW : N hprev M

New transactions

T1 T2 Tk

Despite the veri cation of SBs is extremely fast, EBs are slower to be checked since veri ers need to test that all the curve parameters involved have been properly constructed, running several types of mathematical algorithms such as primality testing, nite elds operations and points counting.

5

Method discussion

Here we discuss motivation and advantages of the presented choices.

First, this PoW model involves many di erent mathematical algorithms of wide interest, for which this blockchain may represent a concrete research propellant. Furthermore, it might also provide a public collection of cryptographically secure elliptic curves of moderate size.

Apart from its scienti c usefulness, it conveys many desirable security properties. The chal

lenges involved do not rely on a given curve of questionable provenance but on the generic di culty of the ECDLP, which is much more fair to be trusted. Thus, we nd it aims at embracing the decentralization ideals that lead to cryptocurrencies creation: even the mathematical objects involved are publicly manufactured, no trust is required even in the authors or the proposing entities.

As for blocks forgery, we point out that both SBs and EBs comprise a PoW which depends on the entire block, together with the previous one. This means that any counterfeit in any position of the chain results into an incorrect nal block, which may be easily detected from the network.

Moreover, it is hard to conceive shortcuts for the PoW production: for a given di culty

parameter d we expect a d-bits secutity of the general ECDPL by using p 22d, unless attacks outperforming Pollard's rho are discovered. Furthermore, common base eld operations speed ups are avoided by making use of non-exceptional primes, ensuring a fair and general problem to be solved equally for every miner. In fact, neither speci c algorithms nor dedicated hardware may be used for solving such a general problem, of which easy cases are carefully avoided [ 23 ].

Besides security, the curves we propose are fully rigid as de ned in [ 6 ]: their construction is entirely explained in terms of the previous block, which cannot be controlled by a malicious actor since there is no room for miner choices (such as nonces). Even assuming that the transactions of the previous block might be chosen ad hoc, an attacker who wants to impose a particular curve during the next epoch has to brute-force invert the hash H at the cost of one ECDLP solution for each attempt, until a desired hash digest is obtained, within the time needed for the entire network to solve a single ECDLP. We consider this scenario unachievable under realistic assumptions. However, even assuming that a miner succeeds in breaking or nding a shortcut for solving the ECDLP over the current epoch curve, it would bene t from the chance of consistently winning the PoW-challenge only until the next epoch. In any case, this miner would get no advantage in the choice of the next epoch curve. We nd this eventuality a fair reward for such an unlikely and scienti cally unexpected achievement.

As regards the di erence between EBs and SBs, we point out that the bulk of miner's work consists of the ECDLP solution: we expect good parameters to be generated in EBs in a time which is linear in the di culty parameter [16] whereas the asymptotic di culty of ECDLP solution is exponential in it.

Since the curves creation appears not to be computationally demanding when compared to the actual PoW, then lazy miners do not have any substantial advantage in skipping it.

6

Conclusion

We have proposed a new PoW-based blockchain model based on general ECDLP, highlighting the desirable properties that such a scheme provides in terms of scienti c relevance, security and pure decentralization ideals.

The past proposals [18, 21] have the high merit of introducing ECDLP as a problem whose

solution provides consensus, but we felt compelled to remove the suspiscious choice of the curve serving as a common battle eld for miners. The novel multi-curve approach introduced in the current work wipes out this grey area by providing a fully transparent scheme.

It may be interesting to produce an actual implementation of the proposed scheme, obtaining practical time measurments and e ciency considerations. A subsequent engaging project might address the resistance of such a protocol to the known attacks under real-world assumptions, comparing the obtained results with outcomes of existing cryptocurrencies.

Finally, di erent types of PoW might be conceived in a similar fashion, possibly employing problems which are thought to resist even to quantum attacks.

Aknowledgments

The results presented here have been carried on within the EU-ESF activities, call PON Ricerca e Innovazione 2014-2020, project Distributed Ledgers for Secure Open Communities. We thank the Quadrans Foundation for its support.

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