The sole distinction between public and private blockchain is related to who is allowed to participate in the network, execute the consensus protocol and maintain the shared ledger. A public blockchain network is completely open and anyone can join and participate in the network. The network typically has an incentivizing mechanism to encourage more participants to join the network. Bitcoin is one of the largest public blockchain networks in production today.
Send your Bitcoins to a specially formed Bitcoin address. The address is specially designed so that the coins will now be out of your control… and out of the control of anybody else either. They’re completely immobilized and can only be unlocked if somebody can prove they’re no longer being used elsewhere (I’ll explain what I mean by this in a minute).   In other words, you’ve used the core bitcoin transaction rules I described above to lay down a specific condition that the future owner – whoever it ends up being – needs to fulfil in order to take control
The public blockchain is open to anyone who wants to deploy smart contracts and have their executions performed by public mining nodes. Bitcoin is one of the largest public blockchain networks today. As such, there is limited privacy in the public blockchain. Mining nodes in the public blockchain requires a substantial amount of computational power to maintain the distributed ledger at a large scale. In the Ethereum public blockchain, smart contract codes can be viewed openly.
“Blockchain could significantly reduce time delays and human mistakes, and monitor cost, labor, waste and emissions at every point in the supply chain. In the food sector, a manufacturer could automatically identify contaminated products in a matter of seconds and wouldn’t need to pull an entire product line from store shelves in the case of contamination.”
^ Jump up to: a b c d Bhaskar, Nirupama Devi; Chuen, David Lee Kuo (2015). "3 – Bitcoin Mining Technology". In Cheun, David Lee Kuo. Handbook of Digital Currency: Bitcoin, Innovation, Financial Instruments, and Big Data. Academic Press. pp. 47–51. ISBN 978-0-12-802117-0. Archived from the original on 25 October 2016. Retrieved 2 December 2016 – via ScienceDirect.

The main point of a side-chain is to allow cryptocurrency networks to scale and interact with one-another. For example alt-coins and Bitcoin run on separate chains, however side chains allow for these separate currencies to be transferred through these two-way 'portal's or interfaces via a fixed conversion amount. Added benefits of side-chains are different asset classes like,stocks, bonds etc being integrated through a converted price onto the main chain, along with additional functionality like smart contracts,unique D-Apps, micro-payments and security updates that can be later incorporated into the primary network from these side-chains.
Sidechains solve a lot of problems, but at what cost? The introduction of sidechains makes things even more complex and much harder to understand for those who are not actively involved in the blockchain space. This also divides assets, no more “one chain, one asset” adage, which further complicates things. And on a network level there are multiple independent unsynchronised blockchains interacting with each other.
Put simply, sidechaining is any mechanism that allows tokens from one blockchain to be securely used within a completely separate blockchain but still moved back to the original chain if necessary. By convention the original chain is normally referred to as the "main chain", while any additional blockchains which allow users to transact within them in the tokens of the main chain are referred to as "sidechains". For example, a private Ethereum-based network that had a linkage allowing ether to be securely moved from the public Ethereum main chain onto it and back would be considered to be a sidechain of the public network.
Sometimes separate blocks can be produced concurrently, creating a temporary fork. In addition to a secure hash-based history, any blockchain has a specified algorithm for scoring different versions of the history so that one with a higher value can be selected over others. Blocks not selected for inclusion in the chain are called orphan blocks.[22] Peers supporting the database have different versions of the history from time to time. They keep only the highest-scoring version of the database known to them. Whenever a peer receives a higher-scoring version (usually the old version with a single new block added) they extend or overwrite their own database and retransmit the improvement to their peers. There is never an absolute guarantee that any particular entry will remain in the best version of the history forever. Because blockchains are typically built to add the score of new blocks onto old blocks and because there are incentives to work only on extending with new blocks rather than overwriting old blocks, the probability of an entry becoming superseded goes down exponentially[23] as more blocks are built on top of it, eventually becoming very low.[1][24]:ch. 08[25] For example, in a blockchain using the proof-of-work system, the chain with the most cumulative proof-of-work is always considered the valid one by the network. There are a number of methods that can be used to demonstrate a sufficient level of computation. Within a blockchain the computation is carried out redundantly rather than in the traditional segregated and parallel manner.[26]
As you can see, several of these real-world demands for the evolution of the initial Bitcoin implementation are still highly relevant. Trade-offs between scalability and decentralization are demonstrated with Ethereum’s focus on decentralization first and resulting complexities in developing scalable solutions. The increased emphasis on smart contract functionality, pegging real-world assets to blockchains, and experimentation of altcoins that are currently ongoing also represent the forward-thinking ideas outlined in the paper.
@tradles – thanks for taking the time to explain this. OK – so I get the debate around blockchain bloat and the (grudging) inclusion of OP_RETURN, etc., but what I’m missing is that I can only really see one scenario where embedding any identity data into the blockchain makes sense…. and that’s when I want to *associate* an identity with a transaction I’m performing.

A blockchain is a continuously growing list of records called blocks, these blocks are linked and secured using cryptographic algorithms. Each block typically contains a hash (a link to a previous block), a timestamp as well as transaction data. Full nodes validate all the transactions, but are unable to settle the disagreements in regards to the order in which they were received. To prevent double-spending, the entire network needs to reach global consensus on the transaction order. It achieves this by using centralised parties or a decentralised proof of work or proof of stake algorithm (and its derivatives).
Of course, the drawbacks of public and private blockchains are still very much present in the case consortium chains. This all depends on the way each consortium is constructed: a more public consortium chain will bear the burdens of public chains, while a more private one might suffer from the relative lack of openness and disintermediation. The right configuration depends on the needs and vision for each specific chain. Strategy and tailoring are always necessary to get the best solution.
A consortium blockchain is often said to be semi-decentralized. It, too, is permissioned but instead of a single organization controlling it, a number of companies might each operate a node on such a network. The administrators of a consortium chain restrict users' reading rights as they see fit and only allow a limited set of trusted nodes to execute a consensus protocol.

“The reason why you put up private blockchains is potentially because you want to have control over the participants in the blockchain. So as we have banks and financial institutions, who have to worry heavily about regulations, they can’t use the public blockchains right now because they are open and permission-free, and anyone can participate, and that’s contradictory to the regulations to which they must abide.
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These in-channel payments would be instant, unlike current Bitcoin payments, which require an hour to be fully verified on the blockchain. What’s more, payments would be routable across multi-hop paths, like packets across the Internet — so instead of having to create a channel to every new counterparty, you could maintain a few channels to a small number of well-connected secure intermediaries and send/receive money through them.

This is justified by observing that, in our pre-sidechain world, miners always want things to be correct. In theory, the incentives of miners and investors are very strongly aligned: both are compensated most when the exchange rate is highest. And, in practice, we do not see large reorganizations (where miners can “steal”, by first depositing BTC to major exchanges, then selling that BTC for fiat (which they withdraw), and finally rewriting the last 3 or 4 days of chain history, to un-confirm the original deposits). These reorgs would devastate the exchange rate, as they would cast doubt on the entire Bitcoin experiment. The thesis of Drivechain is that sidechain-theft would also devastate the exchange rate, as it would cast doubt on the entire sidechain experiment (which would itself cast doubt on the Bitcoin experiment, given the anti-competitive power of sidechains).

Congratulations! You’ve just educated yourself on the most common advanced topics in blockchain that you’ll hear about. By understanding these concepts, you have a firmer grasp on the fundamental tradeoffs and latest research on the blockchain than most industry “experts”! Better yet, next time you hear your colleagues around the water cooler talking about state channels, the Lightning Network and Byzantine fault tolerance, not only will you know what they’re talking about but you might be able to teach them a thing or two!
Let me explain. The Lightning Network allows for the creation of “micropayment channels” across which multiple Bitcoin transactions can be securely performed without interacting with the blockchain, except for the initial transaction that initiates the channel. There is no counterparty risk: if any party ceases to cooperate, and/or does not respond within an agreed-on time limit, the channel can be closed and all its outstanding transactions kicked up to the blockchain to be settled there.
This comparison might make you think that private blockchains are more reasonable to use as they are faster, cheaper, and protect the privacy of their members. However, in certain cases, transparency is more crucial than the speed of transaction approval. So, every company interested in moving their processes to a blockchain evaluates the needs and goals and only then selects a particular type of distributed ledger.

Peer-to-peer blockchain networks lack centralized points of vulnerability that computer crackers can exploit; likewise, it has no central point of failure. Blockchain security methods include the use of public-key cryptography.[4]:5 A public key (a long, random-looking string of numbers) is an address on the blockchain. Value tokens sent across the network are recorded as belonging to that address. A private key is like a password that gives its owner access to their digital assets or the means to otherwise interact with the various capabilities that blockchains now support. Data stored on the blockchain is generally considered incorruptible.[1]
My take is that the Bitcoin architecture is a solution to the problem of how to maintain consensus about a ledger when the participants are unknown and many of them are adversarial (I know this is loose language… computer scientists working in the consensus space are more precise but I think this captures the essence…. i.e. we’re explicitly in a world where there is no “leader” and no identities for those providing the consensus services).

I have a hard time swallowing that Bitcoin “isn’t a ledger”. That’s like saying “Bitcoin isn’t the blockchain”, and if you take the blockchain away from Bitcoin, you aren’t really left with much (including, sidechains). Perhaps Bitcoin isn’t a ledger *from the perspective* of individual transactions, but by the same logic, nothing that isn’t transaction data is.
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