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In mathematics , the probabilistic method is a nonconstructive method, primarily used in combinatorics and pioneered by Paul Erdős , for proving the existence of a prescribed kind of mathematical object. It works by showing that if one randomly chooses objects from a specified class, the probability that the result is of the prescribed kind is strictly greater than zero. Although the proof uses probability, the final conclusion is determined for certain , without any possible error.

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53-415: Monero ( / m ə ˈ n ɛr oʊ / ; Abbreviation : XMR ) is a cryptocurrency which uses a blockchain with privacy-enhancing technologies to obfuscate transactions to achieve anonymity and fungibility . Observers cannot decipher addresses trading Monero, transaction amounts, address balances, or transaction histories. The protocol is open source and based on CryptoNote v2, a concept described in

106-516: A JavaScript implementation of a Monero miner that was embedded in websites and apps, in some cases by hackers. Coinhive generated the script as an alternative to advertisements; a website or app could embed it, and use website visitor's CPU to mine the cryptocurrency while the visitor is consuming the content of the webpage, with the site or app owner getting a percentage of the mined coins. Some websites and apps did this without informing visitors, or in some cases using all possible system resources. As

159-526: A "hard space" ( non-breaking space ) and the amount: and for texts in Bulgarian , Croatian , Czech , Danish , Dutch , Estonian , Finnish , French , German , Greek , Hungarian , Italian , Lithuanian , Polish , Portuguese , Romanian , Slovak , Slovene , Spanish , and Swedish the order is reversed; the amount is followed by a non-breaking space and the ISO ;4217 code: As illustrated,

212-430: A 2013 white paper authored by Nicolas van Saberhagen. Developers used this concept to design Monero, and deployed its mainnet in 2014. The Monero protocol includes various methods to obfuscate transaction details, though users can optionally share view keys for third-party auditing. Transactions are validated through a miner network running RandomX, a proof-of-work algorithm. The algorithm issues new coins to miners and

265-588: A bitcoin ransom paid in the Colonial Pipeline cyber attack . The May 2021 hack forced the pipeline to pay a $ 4.4M ransom in bitcoin, though a large portion was recovered by the United States federal government the following month. The group behind the attack, DarkSide , normally requests payment in either bitcoin or Monero, but charge a 10–20% premium for payments made in bitcoin due to its increased traceability risk. Ransomware group REvil removed

318-456: A collection of objects fails to have a certain property, then the probability that a random object chosen from the collection has that property is zero. Similarly, showing that the probability is (strictly) less than 1 can be used to prove the existence of an object that does not satisfy the prescribed properties. Another way to use the probabilistic method is by calculating the expected value of some random variable . If it can be shown that

371-456: A coloring satisfying the condition that the number of monochromatic r -subgraphs is strictly less than 1 . The number of monochromatic r -subgraphs in this random coloring is a non-negative integer , hence it must be 0 ( 0 is the only non-negative integer less than 1 ). It follows that if (which holds, for example, for n = 5 and r = 4 ), there must exist a coloring in which there are no monochromatic r -subgraphs. By definition of

424-434: A coloring. The problem of finding such a coloring has been open for more than 50 years. A 1959 paper of Erdős (see reference cited below) addressed the following problem in graph theory : given positive integers g and k , does there exist a graph G containing only cycles of length at least g , such that the chromatic number of G is at least k ? It can be shown that such a graph exists for any g and k , and

477-518: A design decision which was based on Monero project's opposition to mining centralisation which ASIC mining creates, but has also resulted in Monero's popularity among malware -based non-consensual miners. Monero's privacy features have made it popular for illicit purposes. Monero is a common medium of exchange on darknet markets . In August 2016, dark market AlphaBay permitted its vendors to start accepting Monero as an alternative to bitcoin. The site

530-618: A list of all currently pegged currencies. Despite having no presence or status in the standard, three letter acronyms that resemble ISO 4217 coding are sometimes used locally or commercially to represent de facto currencies or currency instruments. The following non-ISO codes were used in the past. Minor units of currency (also known as currency subdivisions or currency subunits) are often used for pricing and trading stocks and other assets, such as energy, but are not assigned codes by ISO 4217. Two conventions for representing minor units are in widespread use: A third convention

583-451: A network observer. These privacy features are enforced on the network by default. Monero uses Dandelion++, a protocol which obscures the IP address of devices producing transactions. This is done through a method of transaction broadcast propagation; new transactions are initially passed to one node on Monero's peer-to-peer network, and a repeated probabilistic method is used to determine when

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636-478: A number of reasons for this: In addition to codes for most active national currencies ISO 4217 provides codes for "supranational" currencies, procedural purposes, and several things which are "similar to" currencies: The use of an initial letter "X" for these purposes is facilitated by the ISO ;3166 rule that no official country code beginning with X will ever be assigned. The inclusion of EU (denoting

689-572: A proof of a lower bound for the Ramsey number R ( r , r ) . Suppose we have a complete graph on n vertices . We wish to show (for small enough values of n ) that it is possible to color the edges of the graph in two colors (say red and blue) so that there is no complete subgraph on r vertices which is monochromatic (every edge colored the same color). To do so, we color the graph randomly. Color each edge independently with probability 1/2 of being red and 1/2 of being blue. We calculate

742-626: A result, the script was blocked by companies offering ad blocking subscription lists, antivirus services, and antimalware services. Coinhive had been previously found hidden in Showtime - owned streaming platforms and Starbucks Wi-Fi hotspots in Argentina. Researchers in 2018 found similar malware that mined Monero and sent it to Kim Il-sung University in North Korea . Monero is sometimes used by ransomware groups. According to CNBC , in

795-445: A sender's outputs with other decoy outputs. Encryption of transaction amounts began in 2017 with the implementation of ring confidential transactions (RingCTs). Developers also implemented a zero-knowledge proof method, "Bulletproofs", which guarantee a transaction occurred without revealing its value. Monero recipients are protected through "stealth addresses", addresses generated by users to receive funds, but untraceable to an owner by

848-402: Is The sum of expectations is the expectation of the sum ( regardless of whether the variables are independent ), so the expectation of the sum (the expected number of all monochromatic r {\displaystyle r} -subgraphs) is Consider what happens if this value is less than 1 . Since the expected number of monochromatic r -subgraphs is strictly less than 1 , there exists

901-543: Is a standard published by the International Organization for Standardization (ISO) that defines alpha codes and numeric codes for the representation of currencies and provides information about the relationships between individual currencies and their minor units. This data is published in three tables: The first edition of ISO 4217 was published in 1978. The tables, history and ongoing discussion are maintained by SIX Group on behalf of ISO and

954-513: Is also the ISO 3166-1 code for "US" (United States). The following is a list of active codes of official ISO 4217 currency names as of 1 January 2024 . In the standard the values are called "alphabetic code", "numeric code", "minor unit", and "entity". According to UN/CEFACT recommendation 9, paragraphs 8–9 ECE/TRADE/203, 1996: A number of currencies had official ISO 4217 currency codes and currency names until their replacement by another currency. The table below shows

1007-411: Is gaining increased use in illicit activities such as money laundering , darknet markets , ransomware , cryptojacking , and other organized crime . The United States Internal Revenue Service (IRS) has posted bounties for contractors that can develop Monero-tracing technologies. Monero's roots trace back to CryptoNote v2, a cryptocurrency protocol first introduced in a white paper published by

1060-421: Is present in G with probability p . Hence by Markov's inequality we have Proof. Let Y be the size of the largest independent set in G . Clearly, we have when For sufficiently large n , the probability that a graph from the distribution has both properties is positive, as the events for these properties cannot be disjoint (if they were, their probabilities would sum up to more than 1). Here comes

1113-633: Is similar to the second one but uses an upper-case letter, e.g. ZAC for the South African Cent. Cryptocurrencies have not been assigned an ISO 4217 code. However, some cryptocurrencies and cryptocurrency exchanges use a three-letter acronym that resemble an ISO 4217 code. Probabilistic method This method has now been applied to other areas of mathematics such as number theory , linear algebra , and real analysis , as well as in computer science (e.g. randomized rounding ), and information theory . If every object in

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1166-418: Is the same color, and 0 otherwise. Note that the number of monochromatic r {\displaystyle r} -subgraphs is the sum of X ( S r ) {\displaystyle X(S_{r})} over all possible subsets S r {\displaystyle S_{r}} . For any individual set S r i {\displaystyle S_{r}^{i}} ,

1219-481: The C ( n , r ) {\displaystyle C(n,r)} possible subsets we could have chosen, i.e. i {\displaystyle i} ranges from 1 to C ( n , r ) {\displaystyle C(n,r)} . So we have that the sum of E [ X ( S r i ) ] {\displaystyle E[X(S_{r}^{i})]} over all S r i {\displaystyle S_{r}^{i}}

1272-526: The European Union ) in the ISO 3166-1 reserved codes list allows the euro to be coded as EUR rather than assigned a code beginning with X, even though it is a supranational currency. ISO 4217 also assigns a three-digit numeric code to each currency. This numeric code is usually the same as the numeric code assigned to the corresponding country by ISO 3166-1 . For example, USD ( United States dollar ) has numeric code  840 which

1325-410: The Ramsey number , this implies that R ( r , r ) must be bigger than n . In particular, R ( r , r ) must grow at least exponentially with r . A weakness of this argument is that it is entirely nonconstructive . Even though it proves (for example) that almost every coloring of the complete graph on (1.1) vertices contains no monochromatic r -subgraph, it gives no explicit example of such

1378-590: The Swiss Association for Standardization . The ISO 4217 code list is used in banking and business globally. In many countries, the ISO 4217 alpha codes for the more common currencies are so well known publicly that exchange rates published in newspapers or posted in banks use only these to delineate the currencies, instead of translated currency names or ambiguous currency symbols . ISO 4217 alpha codes are used on airline tickets and international train tickets to remove any ambiguity about

1431-436: The expected value of X ( S r i ) {\displaystyle X(S_{r}^{i})} is simply the probability that all of the C ( r , 2 ) {\displaystyle C(r,2)} edges in S r i {\displaystyle S_{r}^{i}} are the same color: (the factor of 2 comes because there are two possible colors). This holds true for any of

1484-695: The Bitcoin Lightning Network , or other "layer 2" protocol. The contract was awarded to blockchain analysis groups Chainalysis and Integra FEC. Monero uses a proof-of-work algorithm, RandomX, to validate transactions. The method was introduced in November 2019 to replace the former algorithm CryptoNightR. Both algorithms were designed to be resistant to ASIC mining, which is commonly used to mine other cryptocurrencies such as bitcoin . Monero can be mined somewhat efficiently on consumer-grade hardware such as x86 , x86-64 , ARM and GPUs ,

1537-491: The ISO currency codes of former currencies and their common names (which do not always match the ISO 4217 names). That table has been introduced end 1988 by ISO. The 2008 (7th) edition of ISO 4217 says the following about minor units of currency: Requirements sometimes arise for values to be expressed in terms of minor units of currency. When this occurs, it is necessary to know the decimal relationship that exists between

1590-669: The United States dollar and the Bahraini dinar , for which the column headed "Minor unit" shows "2" and "3", respectively. As of 2021 , two currencies have non-decimal ratios, the Mauritanian ouguiya and the Malagasy ariary ; in both cases the ratio is 5:1. For these, the "Minor unit" column shows the number "2". Some currencies, such as the Burundian franc , do not in practice have any minor currency unit at all. These show

1643-488: The creation of Monero. Monero translates to coin in Esperanto . Both van Saberhagen and thankful_for_today remain anonymous. Monero has the third-largest community of developers, behind bitcoin and Ethereum . The protocol's lead maintainer was previously South African developer Riccardo Spagni. Much of the core development team chooses to remain anonymous. Improvements to Monero's protocol and features are, in part,

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1696-482: The currency concerned and its minor unit. This information has therefore been included in this International Standard and is shown in the column headed "Minor unit" in Tables A.1 and A.2; "0" means that there is no minor unit for that currency, whereas "1", "2" and "3" signify a ratio of 10:1, 100:1 and 1000 :1 respectively. The names of the minor units are not given. Examples for the ratios of 100 :1 and 1000 :1 include

1749-509: The currency's full English name, this is not always the case, as currencies such as the Algerian dinar , Aruban florin , Cayman dollar , renminbi , sterling , and the Swiss franc have been assigned codes which do not closely resemble abbreviations of the official currency names. In some cases, the third letter of the alpha code is not the initial letter of a currency unit name. There may be

1802-417: The expected number of monochromatic subgraphs on r vertices as follows: For any set S r {\displaystyle S_{r}} of r {\displaystyle r} vertices from our graph, define the variable X ( S r ) {\displaystyle X(S_{r})} to be 1 if every edge amongst the r {\displaystyle r} vertices

1855-503: The first half of 2018, Monero was used in 44% of cryptocurrency ransomware attacks. The perpetrators of the 2017 WannaCry ransomware attack , which was attributed by the US government to North Korean threat actors, attempted to exchange the ransom they collected in Bitcoin to Monero. Ars Technica and Fast Company reported that the exchange was successful, but BBC News reported that

1908-409: The formation of new countries, treaties between countries on shared currencies or monetary unions, or redenomination from an existing currency due to excessive inflation. As a result, the list of codes must be updated from time to time. The ISO 4217 maintenance agency is responsible for maintaining the list of codes. In the case of national currencies, the first two letters of the alpha code are

1961-532: The number "0", as with currencies whose minor units are unused due to negligible value. The ISO standard does not regulate either the spacing, prefixing or suffixing in usage of currency codes. The style guide of the European Union 's Publication Office declares that, for texts issued by or through the Commission in English , Irish , Latvian , and Maltese , the ISO 4217 code is to be followed by

2014-833: The option of paying ransom in bitcoin in 2021, demanding only Monero. Ransomware negotiators, groups that help victims pay ransoms, have contacted Monero developers to understand the technology. Despite this, CNBC reported that bitcoin was still the currency of choice demanded in most ransomware attacks, as insurers refuse to pay Monero ransom payments because of traceability concerns. The attribution of Monero to illicit markets has influenced some exchanges to forgo listing it. This has made it more difficult for users to exchange Monero for fiat currencies or other cryptocurrencies. Exchanges in South Korea and Australia have delisted Monero and other privacy coins due to regulatory pressure. In 2018, Europol and its director Rob Wainwright wrote that

2067-512: The order is determined not by the currency but by the native language of the document context. The US dollar has two codes assigned: USD and USN ("US dollar next day"). The USS (same day) code is not in use any longer, and was removed from the list of active ISO 4217 codes in March 2014. A number of active currencies do not have an ISO 4217 code, because they may be: These currencies include: See Category:Fixed exchange rate for

2120-541: The presumed pseudonymous Nicolas van Saberhagen in October 2013. In the paper, the author described privacy and anonymity as "the most important aspects of electronic cash" and characterized bitcoin 's traceability as a "critical flaw". A Bitcointalk forum user known as " thankful_for_today " implemented these ideas into a coin they called BitMonero. However, other forum users disagreed with thankful_for_today' s direction for BitMonero and decided to fork it in 2014, leading to

2173-642: The price. In 1973, the ISO Technical Committee 68 decided to develop codes for the representation of currencies and funds for use in any application of trade, commerce or banking. At the 17th session (February 1978), the related UN / ECE Group of Experts agreed that the three-letter alphabetic codes for International Standard ISO 4217, "Codes for the representation of currencies and funds", would be suitable for use in international trade. Over time, new currencies are created and old currencies are discontinued. Such changes usually originate from

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2226-410: The proof is reasonably simple. Let n be very large and consider a random graph G on n vertices, where every edge in G exists with probability p = n . We show that with positive probability, G satisfies the following two properties: Proof. Let X be the number cycles of length less than g . The number of cycles of length i in the complete graph on n vertices is and each of them

2279-438: The random variable can take on a value less than the expected value, this proves that the random variable can also take on some value greater than the expected value. Alternatively, the probabilistic method can also be used to guarantee the existence of a desired element in a sample space with a value that is greater than or equal to the calculated expected value, since the non-existence of such element would imply every element in

2332-634: The right output in a ring signature could potentially be easier than previously thought. In 2018, researchers presented possible vulnerabilities in a paper titled "An Empirical Analysis of Traceability in the Monero Blockchain". In September 2020, the United States Internal Revenue Service 's criminal investigation division (IRS-CI), posted a $ 625,000 bounty for contractors who could develop tools to help trace Monero, other privacy-enhanced cryptocurrencies,

2385-639: The sample space is less than the expected value, a contradiction. Common tools used in the probabilistic method include Markov's inequality , the Chernoff bound , and the Lovász local lemma . Although others before him proved theorems via the probabilistic method (for example, Szele's 1943 result that there exist tournaments containing a large number of Hamiltonian cycles ), many of the most well known proofs using this method are due to Erdős. The first example below describes one such result from 1947 that gives

2438-526: The service the criminals attempted to use, ShapeShift , denied any such transfer. The Shadow Brokers , who leaked the exploits which were subsequently used in WannaCry but are unlikely to have been involved in the attack, began accepting Monero as payment later in 2017. In 2021, CNBC , the Financial Times , and Newsweek reported that demand for Monero was increasing following the recovery of

2491-652: The task of the Monero Research Lab (MRL), some of whom are anonymous. Monero's key features are those around privacy and anonymity. Even though it is a public and decentralized ledger, all transaction details are obfuscated. This contrasts to bitcoin, where all transaction details, user addresses, and wallet balances are public and transparent. These features have given Monero a loyal following among crypto anarchists , cypherpunks , and privacy advocates. The transaction outputs, or notes, of users sending Monero are obfuscated through ring signatures , which groups

2544-518: The transaction should be sent to just one node or broadcast to many nodes in a process called flooding. In April 2017, researchers highlighted three major threats to Monero users' privacy. The first relies on leveraging the ring signature size of zero, and ability to see the output amounts. The second, "Leveraging Output Merging", involves tracking transactions where two outputs belong to the same user, such as when they send funds to themselves ("churning"). Finally, "Temporal Analysis", shows that predicting

2597-627: The trick: since G has these two properties, we can remove at most n /2 vertices from G to obtain a new graph G′ on n ′ ≥ n / 2 {\displaystyle n'\geq n/2} vertices that contains only cycles of length at least g . We can see that this new graph has no independent set of size ⌈ n ′ k ⌉ {\displaystyle \left\lceil {\frac {n'}{k}}\right\rceil } . G′ can only be partitioned into at least k independent sets, and, hence, has chromatic number at least k . This result gives

2650-445: The two letters of the ISO 3166-1 alpha-2 country code and the third is usually the initial of the currency's main unit. So Japan 's currency code is JPY : "JP" for Japan and "Y" for yen . This eliminates the problem caused by the names dollar , franc , peso , and pound being used in dozens of countries, each having significantly differing values. While in most cases the ISO code resembles an abbreviation of

2703-667: The year would see criminals shift from using bitcoin to using Monero, as well as Ethereum, Dash , and Zcash . Bloomberg and CNN reported that this demand for Monero was because authorities were becoming better at monitoring the Bitcoin blockchain. On 20 February 2024, the cryptocurrency exchange Binance delisted Monero, citing regulatory compliance. On 11 April 2024, Kraken announced they would be delisting Monero for users located in Ireland and Belgium on June 10, 2024. As of 10 May 2024 Monero deposits and trades have been suspended. ISO 4217#Cryptocurrencies ISO 4217

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2756-442: Was designed to be resistant against application-specific integrated circuit (ASIC) mining. Monero's privacy features have attracted cypherpunks and users desiring privacy measures not provided in other cryptocurrencies. A Dutch–Italian study published in 2022 decisively concluded "For now, Monero is untraceable. However, it is probably only a matter of time and effort before it changes." Due to its perceived untraceability Monero

2809-500: Was taken offline by law enforcement in 2017, but it was relaunched in 2021 with Monero as the sole permitted currency. Reuters reported in 2019 that three of the five largest darknet markets accepted Monero, though bitcoin was still the most widely used form of payment in those markets. Hackers have embedded malware into websites and applications that hijack victim CPUs to mine Monero (sometimes called cryptojacking ). In late 2017, malware and antivirus service providers blocked Coinhive,

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