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An Introduction to Public Key Cryptosystems with RSA

In the past few years alone, the world has seen many massive data breaches. Most recently, Marriott Hotels exposed the data of 500 million people (7% of the world’s population!).Security is important in every single facet of the internet. Imagine if Amazon couldn’t keep your credit card secure, or if your Facebook messages were public, or if anyone could read the emails from your healthcare provider. A world without secure communications is a scary one. But most of us have no idea why our communication is secure.This article begins to answer this question. We’ll look at public key cryptosystems by means of an introduction and an example. Next, we’ll discuss a specific public key cryptosystem — the RSA (Rivest–Shamir–Adleman) algorithm. This algorithm is named after Ron Rivest (1942), Adi Shamir (1952), and Leonard Adleman (1945). RSA has been around for 50 years and it is still used today!Public Key CryptosystemsA cryptosystem is a set of processes used to securely transfer information between parties. A great example of a cryptosystem is writing. You and I agree that a set of squiggles is the encryption of an object. These squiggles are more commonly referred to as letters.Some common squiggle-object pairsNow, you and I can send messages about that object. These messages are secure against everyone who cannot read our language. This certainly seems trivial now. But if you are Julius Caesar and only 5% of Romans can read or write, this just might do the trick.What about this public key business? What is that? In a cryptosystem, a key is a piece of information (usually a number) that is inputted into a function that produces a specific output. A public key is a key whose value is open knowledge. Some cryptosystems use a private key, which is a key whose value remains secret.RSA is a public key cryptosystem because it uses a public encryption key and a private decryption key. As you can guess, an encryption key takes messages in plaintext and converts them to secure ciphertext. And a decryption key undoes this process.An ExampleMany of us have used a public key cryptosystem in the form of a ballot box. We’ll use a modified ballot box below so we can use our terms from above. Let’s assume we have an outer box that is locked and inside this outer box is our ballot box.We first start with the outer box. The outer box is currently locked. But the key is on the outside of the box, so anyone can open it. The key on the outside of the outer box is our public key. Everyone has access to it and everyone can use it.The outer box (Image source here)So, we open the outer box and we find a ballot box inside. The ballot box is below. Let’s assume the slit at the top is perfectly one way (i.e. you can’t pull any ballots out). There is a lock on this box but there is no key. The key to the inner lock is the private key. It must be kept private or anyone can read the ballots.The inner boxAnd this is precisely a public key cryptosystem. Assuming the private key stays private, anyone can drop off a ballot safely and securely. It can only be read by the person with the key to the inner box.In RSA Encryption, anyone can send a message safely and securely using the public key but these messages can only be read by the person with the private key.The Benefit of Public Key CryptosystemsThe reason public key cryptosystems are so important is that they do not require any shared prior information. You and your local ballot official do not agree to share a secret private key, you do not exchange any information, and you don’t even have to meet. This is invaluable for modern systems as establishing a private key is impractical.Imagine having to go to a brick-and-mortar Amazon store, get a secret password in person, go back to your computer, and then start shopping. And imagine having to do that for Facebook, Google, Twitter, Venmo, iMessage, Outlook, and all the rest. It would be impossible. Public key cryptosystems solve this problem.4 Key Concepts in the RSA AlgorithmBefore we can dive into the RSA algorithm itself, we need to cover four key concepts: modular arithmetic, the greatest common divisor, Euler’s Totient Function, and Euler’s Theorem.(1) Modular arithmetic is often referred to as clock arithmetic. A standard clock only has 12 hours. The 13th hour doesn’t exist. The clock just circles back to the first hour. Clocks work with the modulus 12. It’s equivalent to say clocks work modulo 12. Let’s say I have some number x = 376. And say we want to know what x is congruent to modulo 59. We look at the remainder of x when it is divided by our modulus. Our result is shown below. Therefore x = 376 ≡ 22 mod 59.An example of modular arithmeticAnother way to think of this is as 376 = 59•6+22. In fact, any number can be written as x = n•q+r for integers q and r with 0 ≤ r < n. Here, x is congruent to r modulo n. This is an important component of RSA encryption.(2) The greatest common divisor of two numbers is the largest integer that divides both numbers. It is also known as the greatest common factor. For example, gcd(15, 21) = 3 since 3 is a factor of both 15 and 21 and it is the largest such factor. Similarly, gcd(8, 12) = 4. Although 2 is also a factor of both 8 and 12, 4 is larger. If the two numbers share no factors, like 14 and 25, then the gcd(14,25) = 1. For more on gcd, see here.(3) When two numbers don’t share any factors (like 14 and 25), we say these numbers are relatively prime and their gcd equals 1. For example, 19 and 8 are relatively prime. And 271 and 22 are as well. Next, we define a function ϕ(n). This function returns the number of integers less than n that are relatively prime to n. Some examples are below. This function is called Euler’s Totient Function. It is named after Leonhard Euler (1707). Note that if n is prime, then ϕ(n) = (n-1) as all numbers less than n are relatively prime to n.Some examples of the ϕ(n) function(4) The final key concept comes from Euler as well. Euler’s Theorem (also known as the Fermat-Euler Theorem) underlies the success of the RSA algorithm. The theorem is stated below. The proof contains a little too much number theory for this article. But for those interested in the proof, see here.Euler’s TheoremThe RSA AlgorithmAlright! Now we have all the tools we need to discuss the RSA algorithm. Let’s say Alice wants to send a message to Bob. We’ll call this message m.We can translate m to a number using ASCII values. For example, the message “Hello” in hexadecimal ASCII is “48 65 6C 6C 6F “. We can then convert from hexadecimal (0x48656C6C6F) to base 10 (310939249775). Now our message m is 310939249775. Once we have a valid message, Bob has to do a little work before Alice can send Bob a message. The RSA algorithm is outlined below.RSA AlgorithmHow do we know that last step will return the original message? We rely on Euler’s Theorem as shown below. For our purposes, we can assume gcd(m, n) = 1 as the only factors of n are p and q. The odds m = p or m = q are very small.Euler’s Theorem and the verification of the RSA algorithmAn Implementation of RSA Using PythonThe following code uses RSA to allow for the encryption and decryption of a given message m (so long as m, when converted to an integer, is less than n). This code is certainly not robust and is meant only as an example. It should not be used as a means of secure communication. For those looking to edit the codebase, the GitHub repository is here.We’ll first need two methods — one to convert words to integers and one to convert integers to words i.e. “ Hello” ⇆ 310939249775. These two functions are below.Conversion Function from Strings to IntegersConversion Function from Integers to StringsThe next step is to handle the setup. We’ll need to define n and ϕ(n). This function takes a parameter num_digits, which represents the number of digits in our primes p and q. The code below uses 50 digit primes. This implies n is approximately 100 digits long. Real RSA encryption uses an n with ~600 digits.Generation of n and ϕ(n)Bob is almost there! All he needs to do is define e and d, which is done in the below methods. The generation of d requires modular inverses. Rosetta Code has a great implementation that we use. We use an e with 25 digits, which should be sufficient for our purposes.Generation of e and dAnd we’re all set. We can now encrypt and decrypt a message m using the two methods below.RSA Encryption and Decryption AlgorithmsWe implement a main() method below to drive our code and we’re all set.Our main() method for our RSA implementationAnd we see it works!The results of our RSA algorithm.Concluding NoteThis article was only meant as an introduction to public key cryptosystems and RSA. But there is certainly much more to cover! There are restrictions and standard protocols for choosing the values of p, q, e, d, and m. There are efficient ways of performing the above calculations. For those interested in learning and exploring more, I recommend Introduction To Cryptography (2nd Edition).While building RSA is certainly interesting, breaking RSA is even more interesting. The strength of RSA lies in our inability to factor numbers that are a product of two large, distinct primes. The two methods that attempt to do this are the Quadratic Sieve and the General Number Field Sieve. These are fascinating methods and we’ll hopefully have an article on them soon.An Introduction to Public Key Cryptosystems with RSA was originally published in Hacker Noon on Medium, where people are continuing the conversation by highlighting and responding to this story.

Decentralized Public Key Infrastructure (DPKI): What is it and why does it matter?

Photo by Burst on UnsplashYou may have heard the term decentralized public key infrastructure (DPKI). But you probably don’t know what it is or how it works, let alone why it matters so much.In this post, we will examine current approaches to the traditional centralized PKI, explore the basics of decentralized public key infrastructure (DPKI), and then show how blockchain-based DPKI can make an impact as the industry transitions to the next generation of PKI.What is Centralized PKI?In today’s world, the most commonly employed approach to public key infrastructures (PKIs) is the Web PKI. It’s a Certificate Authority (CA) based system that adopts a centralized trust infrastructure. Communications over the internet are secured through the safe delivery of public keys and the corresponding private keys.Third-parties such as CAs are responsible for facilitating the authentication and distribution of public keys. What a CA does is that it functions as a trusted third party that distributes and manages digital certificates for a network of users. Most web services are secured through the creation of the keys signed by CAs.Problems with Centralized PKIsThe centralized PKIs such as the CA-based system has its problems and limitations generally because it relies on a central trusted party. In a centralized PKI system, you don’t get to choose your own online identity; instead, your identity is defined by trusted third parties the CAs, sometimes private companies and sometimes governments.This is a big problem because it leaves the door open for attackers to conduct MITM (Man-in-the-Middle) attacks. Currently, there are about 3,675 trusted CAs around the world that have been appearing as attractive targets for cyber-criminals. Each of these thousands have the ability to create alternative identities for you.There are different forms of MITM attacks — ARP spoofing, IP spoofing, DNS spoofing, HTTPS spoofing, and Man in the Browser (MITB), and more. Numerous incidents have already shown that you can increase the risk of MITM attacks when you place too much trust in CAs.In practice, attackers can trick the CA into thinking they are someone else, or they can go so far as to compromise the CA to get it to issue a rogue certificate. For instance, the DigiNotar incident that happened in 2011 when fraudulent certificates from the Dutch certificate authority company were issued as a result of an attack.Another incident happened in 2017 where hackers took control of Brazilian banks DNS server and tricked a CA into issuing a valid certificate to them.The Internet Engineering Task Force (IETF) responsible for Web PKI itself has created a memo describing current issues of PKI; independently, a group of researchers around Rebooting the Web of Trust (including Vitalik Buterin) assessed its weaknesses in their publication, all agreeing that the current implementation of Web PKI has problems that shouldn’t be ignored.The out-of-date PKI design poses high security risks because a single point of failure can be used to open any encrypted online communication. Centralized PKI systems are struggling to keep up with the evolving digital landscape; the modern world is desperate for a better designed, decentralized approach to PKIs.Decentralized PKI (DPKI)Decentralized Public Key Infrastructure, or DPKI, is an alternative approach to designing better PKI systems. Pretty Good Privacy (PGP), an encryption program developed by Phil Zimmermann, is a decentralized trust system that was created when blockchain didn’t exist.It has issues with establishing trust relations between all parties. But today there is no need for the third-parties. Blockchain is a novel approach to build a more competent, secure PKI system.But how blockchain is going to improve PKI? In decentralized PKI, blockchain acts as a decentralized key-value storage. It is capable of securing the data read to prevent MITM attacks, and to minimize the power of third parties. By bringing the power of blockchain technology to the systems, DPKI resolves the issues with traditional PKI systems.The decentralized nature of the management framework can tackle the problems with the CA systems through certificate revocation, eliminating single points of failure, and reacting fast to misuses of CAs. Blockchain is able to make the process transparent, immutable, and prevent attackers from breaking in, thus effectively avoiding the MITM attacks.In 2015, Allen et al. explored in a publication titled “Decentralized Public Key Infrastructure,” that unlike the traditional approach, DPKI ensures no single third-party can compromise the integrity and security of the system as a whole. In blockchain-powered DPKI, the new third parties become miners or validators.The trust is established and maintained based on consensus protocols. Third-parties, the miners or validators, will have to follow the rules of the protocol, that would financially reward and punish these third-parties to effectively preventing misbehavior in the blockchain and limiting their roles.“Trust is decentralized through the use of technologies that make it possible for geographically and politically disparate entities to reach consensus on the state of a shared database,” the authors wrote in the 2015 paper, “blockchains allow for the assignment of arbitrary data such as public keys to these identifiers and permit those values to be globally readable in a secure manner that is not vulnerable to the MITM attacks that are possible in PKIX.”Furthermore, researchers argued that the logic of key management can be implemented on smart contract of blockchain, and “Privacy based decentralized Public Key Infrastructure (PKI) implementation using Smart contract in Blockchain,” a 2017 publication by Sivakumar P and Dr. Kunwar Singh had successfully implemented it.Nevertheless, blockchain is not perfect yet because it requires a device to synchronize a full copy of consensus data. Today’s Geth (Go-Ethereum) client provides multiple types of sync mode: full sync, fast sync, light sync. Diode, a Taiwan-based and U.S.-based blockchain initiative, recently developed a light client protocol called BlockQuick that aims to establish decentralized trust at a low bandwidth.The following table is a comparison of different types of sync mode, trust model, bandwidth, and duration for Geth, FlyClient, BlockQuick, traditional Web PKI client.As the table shows, a standard sync of Geth client takes up to 400GB of your disk — that’s a huge user experience downgrade, compared to a traditional Web PKI client’s ~5kb size that is needed for a standard TLS certificate handshake. In addition, 400GB raises a bar too high for IoT devices that are generally resource-constrained with limited computing power.The transformation of PKI is inevitable and it looks to be picking up speed. This is a good time to start increasing the efforts to create awareness of PKI, and to help more people to navigate the fast moving digital landscape. Public Key Infrastructure (DPKI): What is it and why does it matter? was originally published in Hacker Noon on Medium, where people are continuing the conversation by highlighting and responding to this story.

WordPress Blogging CMS Platform Now Uses Monero-Like Cryptographic Public Key System

WordPress has introduced several new security features for content management with update 5.2. The recent update would ensure that WordPress now uses Ed25519 public-key signature system, quite similar to the privacy crypto token Monero. WordPress is currently being used on 34 percent of all the websites available on the internet, and thus the recent update […]
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Public Key Cryptography Simply Explained

Photo by Liam Macleod on UnsplashPublic key cryptography seems magical to everyone, even those who understand it. In this post, I’m going to explain public key cryptography. Public Key Cryptography is based on asymmetric cryptography, so first let us talk about symmetric cryptography. CryptographyYour front door is usually locked by a key. This key unlocks & locks your front door. With symmetric cryptography, you have one key which you use to unlock and lock things.Only people with the key or a copy of the key can unlock the door. Now, imagine you’re on holiday in Bali. You want to invite your friend around to look after your cat 😺 while you’re on the beautiful beaches 🏖️.Before the holiday, you give your friend the key to your door. Your friend is then robbed, so someone else has your front door key now. Or your friend leaves it laying around and someone clones it. The problem with symmetric key cryptography is that this one key is easy to clone, it’s easy to attack your house in many different ways.Let’s take this from an analogy to a real-life example of symmetric cryptography.Caeser’s CipherJulius Caeser used a cipher to send messages that no one else could read other than the intended recipient. Mainly because no one could read back in 100 BC, and those that could wouldn’t understand a random string of letters. That’s the whole point of cryptography. To create ways to communicate without third parties listening in. This cipher is Caeser’s Cipher. Given an alphabet and a key (the key is an integer between 1 and 25), shift all of the alphabet letters by key.With a shift of 3, as seen in the image above, A becomes D, B becomes E and so on until it wraps around with X = A. The original message is called the plaintext and the encrypted message is called the ciphertext.The easiest way to perform Caesar’s Cipher is to turn all of the letters into numbers, a = 1, b = 2, c = 3 and so on.To encrypt, E, you calculate this for every letter (where s is the shift):This is called a function. You put an input into it, and an output comes out. A lot of functions are known as two-way functions. You can encrypt by using the function above, and it makes sense that to decrypt you just do the opposite. Given a function that doubles a number, if you have a doubled number and you want to reverse the function do the opposite of multiplying by 2, divide the number by 2.mod is the modulus operator. It’s the remainder of dividing. We do modulus because there isn’t a 27th letter in the alphabet, you just wrap around from “z” back to “a”. We’ll talk more about modular on in this article. Look at this small example below:Because 4 divided by 3 has a remainder of 1.To decrypt Caesar’s cipher, D, you calculate this for every letter:As you can tell, it’s not very secure. With 25 total shifts you just have to shift the text 25 times until you find the decrypted code, this is called a brute force attack. You take the encrypted text and shift it all 25 times until you find the decrypted text. But let’s imagine for a second that this was a hard cipher — that brute force isn’t feasible.How do you tell your friend you’re using a shift of 9, for example? You have to communicate it to them somehow. Any and all forms of communication can be listened in on — whether that’s writing a letter or going to a hidden forest in Switzerland 30 miles from the nearest town and telling your friend.The latter isn’t very feasible, but it is a lot more secure than telling your friend in Times Square, New York 🗽 what the shift is.Now, imagine you brought your lunch to work in a special lunchbox — the same you’ve had since nursery school. Someone steals your food and your lunchbox. You don’t mind losing the food, but you do want the lunchbox back. You want a way for them to securely return your lunchbox without you knowing who took it — because that takes the pressure off of them.You place a box in the staff room with a lock & key. You give copies of keys to everyone in the office and hope for the best — that someone will return the lunchbox by placing it in the box.Unfortunately, the keys everyone has also unlocks the box as well as locks it. This means that someone could unlock the box and re-steal your lunchbox.We need to find a way to get rid of this idea of sharing keys, get rid of the idea of ‘any key can lock and unlock’, and this is where asymmetric cryptography comes in.Asymmetric cryptographyYou install an extraordinary lock on this box, one that has two separate keys. The first key 🔑 can only turn clockwise, from A (locked) to B (unlocked) to C (locked).The second key 🗝️ can only turn anti-clockwise, from C to B to A. You pick the first key and keep it to yourself. This is called a private key. The second key is called the public key. This key is given out to everyone in the office. You want everyone to have this key.When someone returns your prized lunchbox, they can leave it in this box. All the public keys can do is lock the box. Your private key is the only one that can open it.This is public key cryptography. Everyone knows that if they put something in the box and lock it, only you can open it with your private key.With symmetric cryptography, everyone could open your box if they had the key. Now, no one apart from you can open the box.Public key cryptography was first formulated by Whitfield-Diffie or James Ellis (Ellis discovered first, but he didn’t publish it. Whitfield-Diffie published first). Both Ellis and Whitfield-Diffie enjoyed that public key cryptography could work in theory, but never managed to figure out how it would work in practice.The production of a working Public Key Encryption system is attributed to Rivest–Shamir–Adleman (RSA) or Clifford Cocks. Like above, Cocks discovered first, but he didn’t publish it. Rivest–Shamir–Adleman published first.Let’s look at how this works mathematically.The maths behind public key cryptographyWhile the box analogy was something physical, we’re going to go back to encrypting messages much like we did with Caeser Cipher.When you apply the public key (K+) to the encrypted message, and then the private key (K-)to the encrypted message you get the plaintext message. We are also looking for these attributes:It is computationally easy to:Generate a set of keysEncrypt / Decrypt using these keysBut it is also computationally infeasible to:Turning messages into numbersWe want to turn a message into numbers. Previously we assigned a number to each letter, A = 1 and so on. The American Standard Code for Information Interchange (ASCII) is a table of all English letters and most symbols along with their associated ASCII code & Binary output.When you press a key on the keyboard, the keyboard converts this to Ascii as numbers are easier to work with than letters for a computer. If you want to learn more about ASCII, check out this video.Let’s encrypt the word “cats”. In binary, according to Ascii, this is:If you add them all together and convert to base 10, you get 4430123. For our example, we’re going to look at how Rivest–Shamir–Adleman (RSA), a public key cipher, calculates public & private keys. We’re also going to use much smaller numbers, so the maths isn’t as hard to read.Below is a calculator I created for turning ASCII into Binary. View it better on my website ( ).trinket: run code anywhereRSAChoose 2 large prime numbers, p & q.Prime numbers are numbers that only have 2 factors, 1 and itself. We’re going to pick 5 & 7, not large prime numbers but small for brevity.2. Compute n = pq, z = (p-1)(q-1)3. Choose e (with e < z) such that e has no common factors with z.e = 55 has no common factors with 24, and it is smaller than 24.4. Choose d such that ed — 1 is exactly divisible by z.The easiest way to do this would be to loop over all possible values of d in code. This code is written in Functional Python, but the language and paradigm doesn’t matter. we’re using such small numbers, we have overlap. Both e and d are 5. Let’s set d to 29, just so we don’t have this overlap.d = 295. The public key is (n, e). The private key is (n, d)Below is code to generate RSA keys. Note that we have overlap on d with p = 5 and q = 7, as discussed above.trinket: run code anywhereTo send an encrypted message, Bob computes C = m^e mod n for message m and key e. To decrypt the message, Alice computes m = c^d mod n.Encrypting “cats” gives us 42⁷⁵ mod 35 = 7.Decrypting “cats” gives us 4430123.Then to send a message m, Bob computes c=m^e (mod N) and sends it to Alice and Alice decrypts the message using her private key d with m=c^d (mod N)Why does it work?Prime factorisation. It’s easy to multiply two prime numbers together, but it’s incredibly hard to find out what prime numbers were used to make that number. You can easily multiply these two together:But if I gave you 992,474,117 and told you to find the prime numbers that were used to make this number, it’s not computationally feasible. Even more so when you realise the prime numbers used are very, very large.This is known as a trap-door function or a one-way function. While it is easy to go through one way, it is computationally infeasible to go the other way. Boiling an egg is a one-way function because it is easy to boil an egg, but it is not possible to un-boil an egg. Let’s go deeper into the mathematics and explore modular arithmetic.Back to modular arithmeticImagine a finite range of numbers, for example, 1 to 12. These numbers are arranged in a circle, much like a clock (modular arithmetic is sometimes called clock arithmetic because of this)Count 13 around this clock. You get to 12 and then you need to count 1 more — so you go back to 1. Modular arithmetic is still defined as the remainder of division, however it can also be defined (and is more commonly defined) as a clock.Functions using modular arithmetic tend to perform erratically, which in turn sometimes makes them one-way functions. Let’s see this with an example by taking a regular function and seeing how it works when it becomes a modular arithmetic function.3^xWhen we insert 2 into this function, we get ³² = 6. Insert 3 and we get ³³ = 9This function is easy to reverse. If we’re given 9, we can tell that the function had an input of 3, because of ³³ = 9.However, with modular arithmetic added, it doesn’t behave sensibly.Image we had this formula:3^x mod 7 = 1How would you find out what x is? You can’t put the mod on the other side, because there isn’t really an inverse of modular arithmetic. What about guessing? Let’s input 5:³⁵ mod 7 = 5Okay, that was too big. You might want to go lower, maybe 4 or 3 but actually this is the wrong direction. When x is 6, it is equal to 1.In normal arithmetic, we can test numbers and get a feel for whether we are getting warmer or colder, but this isn’t the case with modular arithmetic.Often the easiest way to reverse modular arithmetic is to compile a table for all values of x until the right answer is found. Although this may work for smaller numbers, it is computationally infeasible to do for much larger numbers. This is often why modular arithmetic is known as a one-way function.If I gave you a number such as 5787 and told you to find the function for it, it would be infeasible. In fact, if I gave you the ability to input any number into the function it would still be hard. It took me a mere few seconds to make this function, but it’ll take you hours or maybe even days to work out what x is.RSA is a one-way function. While it is relatively easy to carry out this function, it is computationally infeasible to do the reverse of the function and find out what the keys are. Although, it is possible to reverse an RSA encryption if you know some numbers such as N.Let’s talk about NRecall earlier where I detailed how the RSA algorithm worked. There was one number, $n$. This n is special because under some circumstances n can make this one-way function reversibleN is a product of 2 prime numbers. As we saw earlier, if we take $5$ and $7$ and multiply them together, we get:n = 35In order for Bob to send Alice a message, he encrypts the message using Alice’s public key. Now that the message is encrypted, there has to be some way for Alice to decrypt it. There has to be some way for Alice to reverse this, but only for Alice to reverse it. You can’t have Eve or Niamh or Hannah reversing it — because that beats the point of encrypting it.RSA is designed so the person who knows P and Q (the two prime numbers that are multiplied together to give N) can decrypt the message.Although Alice has told the world her public key is n = 35, no one apart from Alice knows that P = 7, Q = 5. Note that the prime numbers are intentionally small for brevity.You may be thinking “it’s easy to guess that 35’s prime factors are 5 and 7” and you would be right. But, with large enough numbers it is virtually impossible to find p and q.In fact, with large enough numbers multiplying p and q are essentially one way functions. It is possible that in the future, perhaps the near future (with the invention of quantum computers) that factoring numbers becomes easy. Mathematicians have tried and failed for thousands of years to find an efficient way to factor numbers, so for now it is considered secure.Let’s look more into the mathsOkay, let’s look at how modulus works in all of this. You understand why multiplication works, and how modulus works. But what about the other equations? What are they for?Let’s demonstrate the deciphering algorithm using an identity due to Euler and Fermate:For any integer (M), M is relatively prime to n:This is the Euler totient function giving the number of positive integers less than n which are relatively prime to n. Relatively prime is where 2 numbers only share the factor 1 with each other. In modern day we use Carmichael’s function over Euler’s function, as Euler’s function can sometimes produce numbers too large to use. However, we’re using Euler’s totient function as it is what the original RSA paper used.This sounds confusing, but let’s break it down. By elementary properties of the totient function:Since d is relatively prime to ϕ i (n), it has a multiplicative inverse e in the ring of integers modulo $ϕ (n). What this means is that the formula we used for RSA can be reversed (the trap door can be reversed) given some knowledge about the numbers used.Without this special mathematical property it wouldn’t be possible to reverse the encryption and find out the ciphertext if you know some of the numbers used.The modular multiplicative inverse of the encryption algorithm c = m^e mod n is m = c^d mod n. All of this maths has built up to this. Modular arithmetic and one-way functions are heavily involved here. In order to encrypt, you calculate c. In order to decrypt, you calculate m. Both of these require knowledge of n, which is the special number we talked about earlier.If you want to learn more about the maths of RSA, I highly reccomend the readable, origianl RSA paper.AuthenticationHow do you prove that a message sent by Bob was actually sent by Bob, and not sent by Eve? You need a way to authenticate them. In the real world, we authenticate using signatures. Although these can be forged, you can authenticate using a biometric scanner, but your fingerprints can be lifted and copied.You can use a passcode, but again much like how Caeser cipher and its single key is useless, authentication methods that use single keys aren’t as perfect.You can use a passcode, but again much like how Caeser’s cipher and its single key is useless, authentication methods that use single keys aren’t as perfect.Let’s say Bob wants to prove to Alice that Bob wrote the message he sent her. Bob sends his original message with an encrypted version of the message with his private key (K-). Alice uses Bob’s public key (K+)which, using the formula above, turns the encrypted message back into the normal message. Then Alice checks the message Bob sent with the message she got from the encrypted message. If they match, she can be sure that someone with Bob’s private key (probably Bob) sent it.This method sucks for encrypting because if Bob encrypts his message with his private key, anyone can read it with his private key. Also, it’s computationally expensive to prove that Bob sent something. This is why we create a digest of the message and encrypt that instead to verify Bob. This digest of a message is done using a hash function.To learn more about hash functions, I wrote a sister article which explains them here.Back to cryptographyBy encrypting the hash of the message we speed up the process of encrypting it, which makes authentication a lot faster. Now, let’s play a prank on Bob.We create an e-mail order to a pizza shop asking for 4 pepperoni pizzas. We sign this email with our private key. We send the pizza store our public key, but we tell them that Bob’s phone is dead and that our public key is actually Bob’s public key.The pizza store verifies the signature and sends 4 pepperoni pizzas 🍕 to Bob. The worst part is, Bob doesn’t even like pepperoni. This is where a certification authority comes into play.Certificate authorities (CA) bind a public key to a specific entity. This entity provides proof of identity to the CA, the CA then creates a certificate binding the entity to its public key. The idea is to take the trust out of trusting an individual for public keys. You still have to trust an organisation, but many people find trusting an organisation is better than trusting an individual.The certificate containing the entities public key is digitally signed by the CA. This signing is the CA saying “this is the entities public key”.When Alice want’s Bob’s public key, she gets Bob’s certificate. She then applies the CA’s public key to Bob’s certificate to get Bob’s public key.Cloudflare has an amazing article on certificate authorities here.Secure Email with Pretty Good PrivacyPhil Zimmerman invented Pretty Good Privacy (PGP), the de facto standard for email encryption. Zimmerman used RSA in PGP. RSA is patented and he did not have permission from RSA inc (the company that holds the patent) to publish another cipher using RSA.Zimmerman was also a target of a 3-year U.S federal investigation because at the time cryptography programs were considered munitions under U.S law. When asked whether all of the trouble was worth it to publish PGP, he said he had “no regrets”. Let’s look at how this used to be illegal algorithm works.When Alice wants to send a confidential email to Bob, she:Generates random symmetric private key, K-.Encrypts her email with K-(for efficiency)Also encrypts K-with Bob’s public key.Alice digitally signs the encrypted message.Alice sends Bob bothand her digital signature.In total, Alice uses three keys. Her private key, Bob’s public key, and the newly created symmetric key.This idea of encrypting a symmetric key with a public key is called a Hybrid Cryptosystem. Some email messages can be incredibly large, encrypting these with a public key system would take a very long time.Use a symmetric key system such as AES, which is incredibly hard to break (but not as hard as RSA). Encrypt the AES key (and only the key, not the whole email) with the public key. This way, the receiver can apply their private key and find out the AES symmetric key to decrypt the email.Not many people use PGP, because of how difficult it is to set up. At most, you need to download a program you trust to correctly implement PGP. In 2018 it was shown that email clients such as Apple Mail, Thunderbird, and Outlook — who have settings to enable PGP can be forced to show the non-encrypted versions.Not to mention how suspicious it looks for one person to send encrypted emails on a network of non-encrypted emails. The only email client (and address provider) which enables PGP by default is ProtonMail, but even then it’s only for Proton-to-Proton emails and you have to trust the company to implement it correctly.body[data-twttr-rendered="true"] {background-color: transparent;}.twitter-tweet {margin: auto !important;}@camfassett Most of them do a good job, but we understand your point. We built ProtonMail to make PGP encryption accessible to non-technical people. We will make sure this goal is reached 100%. ;) Thanks again! — @ProtonMailfunction notifyResize(height) {height = height ? height : document.documentElement.offsetHeight; var resized = false; if (window.donkey && donkey.resize) {donkey.resize(height); resized = true;}if (parent && parent._resizeIframe) {var obj = {iframe: window.frameElement, height: height}; parent._resizeIframe(obj); resized = true;}if (window.location && window.location.hash === "#amp=1" && window.parent && window.parent.postMessage) {window.parent.postMessage({sentinel: "amp", type: "embed-size", height: height}, "*");}if (window.webkit && window.webkit.messageHandlers && window.webkit.messageHandlers.resize) {window.webkit.messageHandlers.resize.postMessage(height); resized = true;}return resized;}'rendered', function (event) {notifyResize();});'resize', function (event) {notifyResize();});if (parent && parent._resizeIframe) {var maxWidth = parseInt(window.frameElement.getAttribute("width")); if ( 500 < maxWidth) {window.frameElement.setAttribute("width", "500");}}ConclusionCryptography has been used for thousands of years, almost as long as mankind has held secrets. In our constant effort to keep our secrets secret to everyone apart from a select few we’ve found this magical algorithm that works pretty well. No doubt, in 300 or 400 years it will have been broken much like how Caeser thought his cipher would never be broken.Hey 👋 Want to subscribe to my blog and stay up to date with posts similar to this one? Subscribe to my email list below. I won’t spam you. I will only send you posts similar to this one 😊✨ you’re feeling extra generous, I have a PayPal and even a Patreon. I’m a university student who writes these articles in my spare time. This blog is my full time job, so any and all donations are appreciatedPublic Key Cryptography Simply Explained was originally published in Hacker Noon on Medium, where people are continuing the conversation by highlighting and responding to this story.

Generating RSA Private and Public Keys

We use SSH, HTTPS, etc., on a daily basis. These programs depend on RSA asymmetric key encryption and decryption for providing security.Asymmetric key encryption involves two keys, public key and private key. Public key is used for encrypting the message and Private key is used for decrypting the message.In this post, we will look into how a public key and private key pair are generated using simple mathematics.We will use small numbers for simplicity.Public Key ( e, n )Public key is made up of two numbers called e and n.Generation of nGenerate two prime numbers.Prime number 1, p = 7Prime number 2, q = 17n = p x qn = 7 x 17 = 119Thus n = 119Generation of eCompute totient of n, ϕ(n) = ( p -1) x (q -1)Choose a random prime number that has a greatest common divisor (gcd) of 1 with ϕ(n)ϕ(n) = ( 7 — 1 ) x ( 17–1 ) = 6 x 16 = 96Prime numbers between 1 and 96 are,2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89Lets us choose a random prime number that has a GCD of 1 with 96We cannot use 2, since 2 is the GCD for 96 and 2.We cannot use 3, since 3 is the GCD for 96 and 3.13 is a good number, since 1 is the GCD for 96 and 13.Now, we have got, e = 13Public Key ( e, n ) = ( 13, 119 )Private Key ( d, n )We have already generated n, which is 119. Now, we need to generate d.Generation of dd is the multiplicative inverse of (e) with ϕ(n)ie, find d, which is the multiplicative inverse of e (13) with 96e = 13, ϕ(n) = 96d * e ≡ 1 mod ϕ(n)d * 13 ≡ 1 mod 96i.e., ( d * 13 )% 96 should yield a remainder of 1This requires computing numbers one by one, until we find the right number. This is hard to do by hand, so let’s use a small python program to generate d,# Python program to find modular # inverse of a under modulo m # A naive method to find modulor # multiplicative inverse of 'e' under modulo 'm'def modInverse(e, m) : e = e % m; for x in range(1, m) : if ((e * x) % m == 1) : return x return 1 # Driver Programe = 13m = 96print(modInverse(e, m))37Computed value of d is 37Verify dd * e ≡ 1 mod ϕ(n)d = 37d * e = 37 * 13 = 48196 * 5 = 480481 % 96 = 1thus d * e ≡ 1 mod ϕ(n)Private Key ( d, n ) = ( 37, 119 )So FarWe have generated a public key and private key, using simple mathematics.Public Key ( e, n ) = ( 13, 119)Private Key ( d, n ) = ( 37, 119 )Generating RSA Private and Public Keys was originally published in Hacker Noon on Medium, where people are continuing the conversation by highlighting and responding to this story.
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BlockchainBrad | Blockvera | Perlin | Crypto Interview | Leaderless PoS | IEO Binance| WASM | Dorjee

BlockchainBrad sits down with Perlin Co-founder, Dorjee Sun, for a Blockvera Deep Dive in an EXCLUSIVE crypto interview. They discuss all aspects of Perlin, including its leaderless ledger powering WASM smart contracts, their Initial Exchange Offering (IEO) on the Binance Launchpad and much more. This is a Free Interview and there is a very strong focus on business & enterprise. In just over an hour, Brad & Dorjee discuss topics such as Perlin’s technical solution, why it’s scalability, speed and finality are real advantages, their strategic positioning in Singapore, upcoming partnerships and collaboration with governments, the value and utility of their token and the details of their IEO are all discussed at length. This video was completely free - neither Brad nor Blockvera were compensated in any way, not in $, token or any other form of payment. ●▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬● 👇 MORE INFO ON PERLIN Website - Telegram Community - Medium - Twitter - ●▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬● ⏳Timestamps 0:52 A brief rundown of Perlin 4:14 Scalability, finality & potential to become a global settlement system 7:53 The value & credentials of their project team 10:30 How many crypto startups will still be around in a few years? 12:56 The future of utility tokens 16:20 The advantages of operating in Singapore 18:34 Partnerships 25:05 The network effect of belief with top 100 coins 30:03 Snowball Protocol 36:47 Does Perlin make it cheaper for businesses to operate? 39:00 The value & utility of the token 42:05 Would the increase in value of the token be a problem in the future? 43:40 Team tokens & private sale terms 47:25 Difference in IEO price vs Private sale 49:30 Why Binance? 51:10 How can you avoid “pump and dumps” in these situations? 55:33 Government collaborations worldwide 59:00 Outlook for the next year 1:05:44 Final thoughts ●▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬● ►Join The BCB Tele ► Follow on twitter: ► Join BlockVera on twitter: @BlockVera ► Check out BlockVera on youtube: ●▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬● PAYMENT/DONATION DISCLOSURE: This interview was entirely, 100% free in every way, not tokens, no under the table deals. I believe that sponsored content is not a problem if disclosed openly to the community. This is not Financial Advice. Please #DYOR ●▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬● INVESTMENT DISCLOSURE: I have not invested in Perlin. I did consider investing in Perlin, but I decided I may invest later after listing. This is not Financial Advice, but I really believe in this project. #DYOR always. ●▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬▬● BlockVera General Disclaimer: This BlockVera Episode was entirely free. BlockVera is about providing more #Truth, #Trust and #Transparency in the Crypto Space. We will be bringing to you several free and unsponsored pieces of content on the BV platform, however to sustain our presence, we will also be doing some sponsored pieces too. BlockVera will always disclose sponsored content. This interview is not sponsored in any way. The BlockVera interviewer was not paid in any way to conduct this interview - there was also no sponsorship given to BlockVera. This was entirely free and for all of us. The information provided is not to be considered as a recommendation to buy or invest in certain assets or currencies and is provided solely as an educational and information resource to help traders make their own decisions. Past performance is no guarantee of future success. It is important to note that no system or methodology has ever been developed that can guarantee profits or ensure freedom from losses. No representation or implication is being made that using the attached material will guarantee profits or ensure freedom from losses. BlockVera shall not be liable to the participant for any damages, claims, expenses or losses of any kind (whether direct or indirect) suffered by the participant arising from or in connection with the information obtained from this website or directly from the website owner.
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Titles and Designations Among Industry Participants

Those that follow developments within the digital securities sector may have come across a variety of titles/designations given to industry participants. While a select few companies have set their sights on attaining a full scope of designations, most specialize in one area or another. This necessitates a high level of cooperation among companies, as issuing digital securities requires utilizing various services. With Securitize recently attaining the title of ‘transfer agent’, now is as good a time as any to take a brief look at what positions, such as this, entail. Here are a few designations typically associated with digital securities, and a superficial look at the roles which they play. Placement Agent Companies tasked with completing the roles of a placement agent typically function as a conduit for raising capital. A placement agent is usually hired by a company looking to raise capital through an STO/DSO or some other means of fundraising. Throughout this process, the placement agent will attempt to connect appropriate, and interested, parties (issuers & investors). In doing so, investors gain access to pre-vetted opportunities in their ‘wheelhouse’, while issuers benefit from access to a larger pool of investors. Beyond simply providing token issuers access to their contact book, placement agents are able to provide certain levels of clout to relatively unknown companies through mere affiliation. In addition, they are often tasked with helping develop marketing strategies for token issuers, to more efficiently connect appropriate parties. The following companies are examples of participants within the digital securities sector which hold the title of a placement agent. US Capital Global Entoro BlockPass Issuance Platform The entire process of selling and distributing digital securities is contingent on finding a competent issuance platform. Digital securities require specific traits to be built into their coding, as they are required to be compliant with securities laws imposed by regulatory bodies, such as the SEC. This is done when they are created, using issuance protocols based on blockchain technologies, such as Polymath’s well known ST-20. The following companies are examples of participants within the digital securities sector which act as issuance platforms. Harbor Fintelum Swarm Broker-Dealer A broker-dealer refers to a licenced company which buys and sells securities. A broker-dealer has the ability to act on behalf of, either, themselves or a client. This is a fluctuating designation which is broken down as follows: When securities are traded on behalf of a client, the company is assuming the role of a broker. When securities are traded on behalf of the company, itself, the company is assuming the role of a dealer. The following companies are examples of participants within the digital securities sector which hold the title of a broker-dealer. StartEngine Gemini Dinosaur Financial Group Custodian In a world which is becoming increasingly connected, new challenges regarding security measures are arising every day. This places increased importance on companies assuming the roles of custodians. Custodians within the digital securities sector are tasked with safely storing digital assets. While their means for achieving this may vary, their presence within the sector is extremely important. Warranted or not, blockchain based assets are often viewed together. This means that when an unregulated exchange with poor security measures is hacked, it paints a bleak picture of similar assets. To continue the upwards trajectory of blockchain based assets (digital securities), regulated custodians are of key importance. Through stringent security measures, they are able to provide a safe home for valuable assets, as well as piece of mind for their holders. The following companies are examples of participants within the digital securities sector which provide custodial services. PrimeTrust Copper TokenSoft Marketplace Provider For participating parties to benefit from the oft-touted liquidity associated with digital securities, these assets need a place to call home. Marketplace providers offer this, as they facilitate secondary market trading of digital securities. By facilitating the buying/selling of digital securities, investors can now easily enter and exit their positions. The following companies are examples of participants within the digital securities sector which act as Marketplace Providers. Archax OpenFinance TokenMarket Transfer Agent For companies which undergo the tokenization process and distribute tokens, a transfer agent is vital. Companies which assume this role are typically tasked with accurately tracking the activity and ownership of distributed assets. This means providing token issuers with an accurate picture of who is in possession of their digital assets, and in some instances doling out dividends to holders. The SEC breaks down the roles of a transfer agent into the following 3 main categories. Issue and cancel certificates to reflect changes in ownership. Act as an intermediary for the company. Handle lost, destroyed, or stolen certificates The following companies are examples of participants, within the digital securities sector, which hold the title of a transfer agent. Securitize VStock Transfer Horizon Globex Jockeying for Position While there are many roles and designations within the sector, these are a few of the most prominent and important found in digital securities. With the digital securities sector still in a nascent stage of growth, there are various companies jockeying for position as the ‘go-to’ entity for their specialities. In time, we will eventually see the cream rise to the top, as select companies stand out from the pack with the services they offer. The post Titles and Designations Among Industry Participants appeared first on

Surprise! Binance Researchers Prefer Binance Chain

The scalability trilemma continues to dog blockchain economies, according to the latest Binance Research. Ethereum, while clinging to the crown as the dominant token economy, is facing stiff competition from a range of scrappy competitors. And you’ll never guess which blockchain Binance researchers consider to be a top contender. The Binance report recognizes Ethereum’s success as  “the most used blockchain worldwide for developers to issue new tokens.” As the dominant network, Ethereum has introduced a large variety of fully developed token standards, including newer innovations such as security tokens and non-fungible tokens. But despite Ethereum’s large range of offerings, Binance Research says,  the “vast majority of these tokens are worthless,” and tokens on other blockchains also hold little value. The big exception is Binance Chain, which has “the second largest amount of positively-valued tokens,” after Ethereum. The study explains that “newer blockchains have begun to compete in different segments” as Ethereum suffers from issues with scalability and gas fees. In addition to Binance Chain, which allows users to pay fees “in any valuable asset,” popular competing blockchains mentioned in the report include “EOS, Ontology, and TRON or second layers running on blockchains like Simple Ledger Protocol for Bitcoin Cash.”   source: Binance Research   Presenting a detailed comparison of token-focused blockchain solutions, the study examines some of the distinctions between the various networks. DApp availability justifies the growth in use-case for token ecosystems and therefore is a key factor for consideration, according to the report, and speed and fees are important considerations as well. The researchers also consider “easiness to build,” along with security and the extent to which a blockchain is decentralized. In terms of DApp activity, EOS and TRON are the favorite blockchains for casino-style gaming. Ontology is a favorite among gamers, while exchange dApps have a strong presence on NEO.  Ethereum is more diversified in its offerings, with growth in a wider range of applications in finance and exchange. The blockchains seeing the most activity are “Ethereum, Binance Chain, EOS, Tron, and NEO.”    source: Binance Research   Binance Research points out that many blockchains offer a “compelling value proposition” for the issuance of tokens, which may eventually overtake Ethereum’s dominant position. Binance Chain in particular is singled out for “the creation of tokens natively” giving it an advantage over others that rely on smart-contract deployments, like Ethereum and a number of competitors. With the relatively low number of use-cases and users across the industry, the report concludes that even though Ethereum currently dominates, it is “too early to rule out” potential competitors. “In the long run,” the study says,  “a wide variety of programmable blockchains will likely coexist if interoperability solutions across chains develop and prove to be secure and usable.”    The post Surprise! Binance Researchers Prefer Binance Chain appeared first on Crypto Briefing.

TriveAcademy Awarded the Bloconomic Excellence Award at the Bloconomic Expo 2019

TriveAcademy, a player in building the blockchain technology infrastructure which also conducts training classes and consultation processes was awarded for the “Bloconomic Excellence Award – Best Blockchain Technology Developer Award” at the  Bloconomic Expo 2019. The Bloconomic Expo 2019 is organized by the Malaysian Blockchain Association and Alphacap Sdn. Bhd. As a sponsor partner for Bloconomic event, TriveAcademy has presented their latest technology and applications of Trivechain 2.0 to the public at the expo. Trivechain just launched TRVC App which all the speakers’ and volunteers’ certificates is been blockchain in their TRVC app. After a successful fork on April 22, 2019, the new version of Trivechain 2.0 has been successfully forked, deployed and is running steadily. Trivechain 2.0 has include major changes such as adjustments to their Proof-of-Work algorithm from X11 to X16R and Proof-of-Stake collateral from 1,000 TRVC to 10,000 TRVC. The mining hash rate and the number of masternode needs to catch up slowly and be supported by a new community. The hash power and number of masternodes is increasing gradually every day indicating a significant increase from the date it was forked. In an interview with Tan, he said that “Trivechain 2.0, as a blockchain platform, will create a highly compatible community to attract developers and entrepreneurs around the world to become part of the Trivechain community. This community along with a number of open source products will offer and create a more conducive ecosystem for developers. This allows the chain to provide the most favorable conditions for its users to develop its application.” Trivechain 2.0 is offering another alternative open source platform for developers to develop their new blockchain base business ventures. The platform is ready for deployment and for those who are interested to catch new mining trend. Come and join the Trivechain community! You are invited to apply for the development fund through the DAO governance system to build a friendly and efficient development ecosystem in the blockchain environment. Visit the official website at for more details. About Trivechain (TRVC): Trivechain (TRVC) is a games and entertainment public blockchain protocol managed by Decentralized autonomous organization (DAO) which focuses on games and entertainment to enter the new era digital age with implementation of blockchain-based technology and DApps (decentralized applications). Facebook: The post TriveAcademy Awarded the Bloconomic Excellence Award at the Bloconomic Expo 2019 appeared first on Bitcoin Garden.
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Wanchain, Civic, Aion and Tael Top All Cryptos; Coins in Aggregate Up 3.13% Overall, 34 Coins Cross Key Moving Average

The Big Winners From Yesterday Over the past day, the top performing coin out of the 133 coins we are tracking was Wanchain, which offered a day-over-day return 90.53%. Rounding out the top four currencies for the day were Civic, Aion, and Tael, which provided holders with returns of 27.02%, 26.53%, and 22.7% for the day. These moves were quite significant, in the sense that they were well outside of the volatility each of the respective coins had seen for the past two weeks. Crypto brokers to trade the currencies mentioned here: Gate, Yobit, Stex, Binance, DDEX, ETHfinex The Crypto Big Picture Overall, the average change in coin price for the coins we’re tracking was up 3.1253%. On a more granular level, 65% of the coins we’re tracking were up while 35% of the coins were down. Below we can see the average daily change for the coins we are tracking our index over time. 34 coins are especially close to their 20 day moving average, and thus may be worth watching for technical traders who view the 20 day moving average as a key support/resistance level. Crypto brokers to trade the currencies mentioned here: Gate, Yobit, Stex, Binance, DDEX, ETHfinex Currencies With Significant Price Moves Here’s a list of the specific coins that crossed their key moving average level: Status, district0x, Loopring, 0x, SingularDTV, SONM, IOTA, Verge, AirSwap, Request, Viberate, Power Ledger, Ripio Credit Network, Agrello, BlockMason Credit Protocol, Aeron, Genesis Vision,, Tierion, Tael, Time New Bank, Waves, OST, NavCoin, Lunyr, AppCoins, VIBE, Nucleus Vision, POA Network, Groestlcoin, Skycoin, Civic, Streamr DATAcoin, Dock. Also of note is that 66 of the 133 we track have contracting volatility. Volatility contraction often precedes a breakout, so this may be something to watch. Below is a chart that zooms in a bit more, showing 4 coins trading below their 20 day moving average and exhibiting contracting volatility. Are these coins ready for a rally? Crypto brokers to trade the currencies mentioned here: Gate, Yobit, Stex, Binance, DDEX, ETHfinex Article by SixJupiter The post Wanchain, Civic, Aion and Tael Top All Cryptos; Coins in Aggregate Up 3.13% Overall, 34 Coins Cross Key Moving Average appeared first on DecentralPost.
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