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...element is intractable if and only if the ECDLP is intractable. A simple cryptographic goal, such as sharing a key, can be achieved using only a single generator P ∈ G; in the Diffie-Hellman protocol =-=[5]-=- the desired key is just (zu)P , where z and u are the two users’ secret exponents. However, a coin is much more complicated. It must contain elements that 4 NEAL KOBLITZ AND ALFRED J. MENEZES implici...

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...y systems based on cryptographic oneway functions were due to David Chaum [4] over 30 years ago. They were based on the RSA function. We shall give a more efficient construction, due to Stefan Brands =-=[2]-=-, that can be implemented using Elliptic Curve Cryptography (ECC). The main purpose of these systems is to duplicate the anonymity and untraceability properties of cash while allowing remote electroni...

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...ansactions, and that U ’s signature verifies using her ECDSA public key — before sending it on through the network. Forming the blockchain. The most important contribution of Nakamoto’s seminal paper =-=[15]-=- was to propose a method of establishing a chronological sequence of transactions that all users can agree on — without, of course, relying on a trusted authority. The idea was to organize transaction...

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...the identity of U can be computed either as I = u1P1 or as I = s −1A− P2. Security results. Assuming that E(Fq) has an intractable discrete log problem and H has good hash function properties, Brands =-=[2, 3]-=- showed that the above construction provides certain security guarantees. We shall not go into detail or be very precise about these results, but rather just give a brief informal list: CRYPTOCASH, CR...

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...d national security agencies can no longer follow the money trail. Money-laundering, tax evasion, terrorist activities, and purchase of child pornography all become harder to stop. In 1996 an article =-=[12]-=- by three NSA researchers explained the mathematics of e-cash systems and concluded with some comments about the problem they pose for law enforcement and a recommendation to implement such a system o...

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... CRYPTOCONTRACTS 17 4. Cryptocontracts The idea of a peer-to-peer system of contracts that are enforced cryptographically rather than through any government authority goes back at least 20 years (see =-=[17]-=-). But it was largely the success of Bitcoin that provided the impetus for putting the idea into practice. We shall discuss the most important project to do this, called Ethereum [8, 9]. The main desi...

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...tic Curve Discrete Log Problem (ECDLP) is the problem, given two elements P,Q ∈ G, of finding s ∈ Z/pZ such that Q = sP . We assume that the ECDLP is intractable for appropriate choices of E(Fq) (see =-=[10]-=-). We shall use capital letters to denote elements of G and lower-case letters to denote elements of the exponent space Z/pZ. In particular, the letter h will denote values of a hash function H that m...

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...rnment unwittingly provided a strong argument in favor of the cypherpunk viewpoint when it blatantly violated the political neutrality of the banking system. As described in a column in Forbes online =-=[13]-=-, Following a massive release of secret U.S. diplomatic cables in November 2010, donations to WikiLeaks were blocked by Bank of America, VISA, MasterCard, PayPal and Western Union on December 7th, 201...

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...o open bank accounts because of banks’ reluctance to risk inadvertent violation of U.S. sanctions against countries that were on the U.S. State Department’s list of “State Sponsors of Terrorism;” see =-=[1]-=-. This meant, for example, that Cubans living in the U.S. whose Cuban passports had expired were unable to get new passports so that they could visit family in Cuba. Among U.S. states, about half have...

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...cause the central government in Washington D.C. still lists marijuana as an illegal drug, and banks fear that opening an account for a marijuana business could run afoul of money-laundering laws (see =-=[11]-=-). In both cases Bitcoin could have been a solution, although it would have been only a partial solution because use and acceptance of Bitcoin are still on a relatively small scale. 3. Peer-to-Peer Cr...

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...to be unreliable. For example, in 2013 the largest exchange company was Mt. Gox. 16 NEAL KOBLITZ AND ALFRED J. MENEZES Based in Tokyo and headed by a Frenchman named Mark Karpelès, who ran it poorly =-=[16]-=-, the company lost 744,408 customer bitcoins (worth about $400 million at the time) shortly before declaring bankruptcy in February 2014. • Anonymity and untraceability of currency transactions — basi...

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...and for any given transaction to be included in a confirmed block. In the seven years that Bitcoin has existed, there has been only one time when a major fork occurred that lasted several blocks (see =-=[18]-=-). This was in March 2013, and it was caused by a software update that introduced an incompatibility between two versions. Users with version 0.7 were accepting a different block than those using vers...