Tezos White Paper

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Tezos White Paper
Tezos Whitepaper
​Whitepaper PDF​
Table of Contents
​Self-Amending Cryptoledger​
​Mathematical Representation​
​The Network Shell​
​Economy​
​Proof-of-Stake Algorithm​
​Smart Contracts​
​Conclusion​
Introduction
In the first part of this paper, we will discuss the concept of abstract blockchains and the implementation of a self-amending crypto-ledger. In the second part, we will describe our proposed seed protocol.
Self-amending Cryptoledger
A blockchain protocol can be decomposed into three distinct protocols:
The network protocol discovers blocks and broadcasts transactions.
The transaction protocol specifies what makes a transaction valid.
The consensus protocol forms consensus around a unique chain.
Tezos implements a generic network shell. This shell is agnostic to the transaction protocol and to the consensus protocol. We refer to the transaction protocol and the consensus protocol together as a "blockchain protocol". We will first give a mathematical representation of a blockchain protocol and then describe some of the implementation choices in Tezos.
Mathematical representation
A blockchain protocol is fundamentally a monadic implementation of concurrent mutations of a global state. This is achieved by defining "blocks" as operators acting on this global state. The free monoid of blocks acting on the genesis state forms a tree structure. A global, canonical, state is defined as the minimal leaf for a specified ordering.
This suggests the following abstract representation:
Let $(\mathbf{S},\leq)$ be a totally ordered, countable, set of possible states.
Let $\oslash \notin \mathbf{S}$ represent a special, invalid, state.
Let $\mathbf{B} \subset \mathbf{S}^{\mathbf{S} \cup {\oslash}}$ be the set of blocks. The set of valid blocks is $\mathbf{B} \cap \mathbf{S}^{\mathbf{S}}$.
The total order on $\mathbf{S}$ is extended so that $\forall s \in \mathbf{S}, \oslash < s$. This order determines which leaf in the block tree is considered to be the canonical one. Blocks in $\mathbf{B}$ are seen as operators acting on the state.
All in all, any blockchain protocol (be it Bitcoin, Litecoin, Peercoin, Ethereum, Cryptonote, etc) can be fully determined by the tuple:
​(S,≤,⊘,B⊂SS∪{⊘})\left(\mathbf{S},\leq,\oslash, \mathbf{B} \subset \mathbf{S}^{\mathbf{S} \cup \{\oslash\}}\right)(S,≤,⊘,B⊂SS∪{⊘})
​
The networking protocol is fundamentally identical for these blockchains. "Mining" algorithms are but an emergent property of the network, given the incentives for block creation.
In Tezos, we make a blockchain protocol introspective by letting blocks act on the protocol itself. We can then express the set of protocols recursively as P={(S,≤,⊘,B⊂S(S×P)∪{⊘})}\mathcal{P} = \left\{\left(\mathbf{S},\leq,\oslash,\mathbf{B} \subset \mathbf{S}^{(\mathbf{S} \times \mathcal{P})\cup \{\oslash\}} \right)\right\}P={(S,≤,⊘,B⊂S(S×P)∪{⊘})}
​
The network shell
This formal mathematical description doesn't tell us how to build the block tree. This is the role of the network shell, which acts as an interface between a gossip network and the protocol.
The network shell works by maintaining the best chain known to the client. It is aware of three type of objects. The first two are transactions and blocks, which are only propagated through the network if deemed valid. The third are protocols, OCaml modules used to amend the existing protocol. They will be described in more details later on. For now we will focus on transaction and blocks.
The most arduous part of the network shell is to protect nodes against denial-of-service attacks.
Clock
Every block carries a timestamp visible to the network shell. Blocks that appear to come from the future are buffered if their timestamps are within a few minutes of the system time and rejected otherwise. The protocol design must tolerate reasonable clock drifts in the clients and must assume that timestamps can be falsified.
Chain selection algorithm
The shell maintains a single chain rather than a full tree of blocks. This chain is only overwritten if the client becomes aware of a strictly better chain.
Maintaining a tree would be more parsimonious in terms of network communications but would be susceptible to denial-of-service attacks where an attacker produces a large number of low-scoring but valid forks.
Yet, it remains possible for a node to lie about the score of a given chain, a lie that the client may only uncover after having processed a potentially large number of blocks. However, such a node can be subsequently ignored.
Fortunately, a protocol can have the property that low scoring chains exhibit a low rate of block creation. Thus, the client would only consider a few blocks of a "weak" fork before concluding that the announced score was a lie.
Network level defense
In addition, the shell is "defensive". It attempts to connect to many peers across various IP ranges. It detects disconnected peers and bans malicious nodes.
To protect against certain denial of service attacks, the protocol provides the shell with context dependent bounds on the size of blocks and transactions.
Functional representation
Validating the chain
We can efficiently capture almost all the genericity of our abstract blockchain structure with the following OCaml types. To begin with, a block header is defined as:
type raw_block_header = {
  pred: Block_hash.t;
  header: Bytes.t;
  operations: Operation_hash.t list;
  timestamp: float;
}
We are purposefully not typing the header field more strongly so it can represent arbitrary content. However, we do type the fields necessary for the operation of the shell. These include the hash of the preceding block, a list of operation hashes and a timestamp. In practice, the operations included in a block are transmitted along with the blocks at the network level. Operations themselves are represented as arbitrary blobs.
type raw_operation = Bytes.t
The state is represented with the help of a Context module which encapsulates a disk-based immutable key-value store. The structure of a key-value store is versatile and allows us to efficiently represent a wide variety of states.
module Context = sig
   type t
   type key = string list
​
   val get: t -> key -> Bytes.t option Lwt.t
   val set: t -> key -> Bytes.t -> t Lwt.t
   val del: t -> key -> t Lwt.t
   (*...*)
end
To avoid blocking on disk operations, the functions use the asynchronous monad Lwt[@LWT]. Note that the operations on the context are purely functional: get uses the option monad rather than throwing an exception while set and del both return a new Context. The Context module uses a combination of memory caching and disk storage to efficiently provide the appearance of an immutable store.
We can now define the module type of an arbitrary blockchain protocol:
type score = Bytes.t list
module type PROTOCOL = sig
   type operation
   val parse_block_header : raw_block_header -> block_header option
   val parse_operation :  Bytes.t -> operation option
​
   val apply :
     Context.t ->
     block_header option ->
     (Operation_hash.t * operation) list ->
     Context.t option Lwt.t
​
   val score : Context.t -> score Lwt.t
   (*...*)
end
We no longer compare states directly as in the mathematical model, instead we project the Context onto a list of bytes using the score function. List of bytes are ordered first by length, then by lexicographic order. This is a fairly generic structure, similar to the one used in software versioning, which is quite versatile in representing various orderings.
Why not define a comparison function within the protocol modules? First off it would be hard to enforce the requirement that such a function represent a total order. The score projection always verifies this (ties can be broken based on the hash of the last block). Second, in principle we need the ability to compare states across distinct protocols. Specific protocol amendment rules are likely to make this extremely unlikely to ever happen, but the network shell does not know that.
The operations parse_block_header and parse_operation are exposed to the shell and allow it to pass fully typed operations and blocks to the protocol but also to check whether these operations and blocks are well-formed, before deciding to relay operations or to add blocks to the local block tree database.
The apply function is the heart of the protocol:
When it is passed a block header and the associated list of operations, it computes the changes made to the context and returns a modified copy. Internally, only the difference is stored, as in a versioning system, using the block's hash as a version handle.
When it is only passed a list of operations, it greedily attempts to apply as many operations as possible. This function is not necessary for the protocol itself but is of great use to miners attempting to form valid blocks.
Amending the protocol
Tezos's most powerful feature is its ability to implement protocol capable of self-amendment. This is achieved by exposing two procedures functions to the protocol:
set_test_protocol which replaces the protocol used in the testnet with a new protocol (typically one that has been adopted through a stakeholder voter).
promote_test_protocol which replaces the current protocol with the protocol currently being tested
These functions transform a Context by changing the associated protocol. The new protocol takes effect when the following block is applied to the chain.
module Context = sig
   type t
   (*...*)
   val set_test_protocol: t -> Protocol_hash.t Lwt.t
   val promote_test_protocol: t -> Protocol_hash.t -> t Lwt.t
end
The protocol_hash is the sha256 hash of a tarball of .ml and .mli files. These files are compiled on the fly. They have access to a small standard library but are sandboxed and may not make any system call.
These functions are called through the apply function of the protocol which returns the new Context.
Many conditions can trigger a change of protocol. In its simplest version, a stakeholder vote triggers a change of protocol. More complicated rules can be progressively voted in. For instance, if the stakeholder desire they may pass an amendment that will require further amendments to provide a computer checkable proof that the new amendment respects certain properties. This is effectively and algorithmic check of "constitutionality".
RPC
In order to make the GUI building job's easier, the protocol exposes a JSON-RPC API. The API itself is described by a json schema indicating the types of the various procedures. Typically, functions such as get_balance can be implemented in the RPC.
type service = {
  name : string list ;
  input : json_schema option ;
  output : json_schema option ;
  implementation : Context.t -> json -> json option Lwt.t
}
The name is a list of string to allow namespaces in the procedures. Input and output are optionally described by a json schema.
Note that the call is made on a given context which is typically a recent ancestor of the highest scoring leaf. For instance, querying the context six blocks above the highest scoring leaf displays the state of the ledger with six confirmations.
The UI itself can be tailored to a specific version of the protocol, or generically derived from the JSON specification.
Seed protocol
Much like blockchains start from a genesis hash, Tezos starts with a seed protocol. This protocol can be amended to reflect virtually any blockchain based algorithm.
Economy
Coins
We suggest that a single coin be referred to as a "tez" and that the smallest unit simply as a cent. We also suggest to use the symbol ꜩ (\ua729, "Latin small letter tz") to represent a tez. During the fundraiser, a total of 763,306,929.69 ꜩ was generated and distributed to the contributors to the fundraiser.
Mining and signing rewards
Principle
We conjecture that the security of any decentralised currency requires to incentivize the participants with a pecuniary reward. As explained in the position paper, relying on transaction costs alone suffers from a tragedy of the commons. In Tezos, we rely on the combination of a bond and a reward.
Bonds are 5-cycle security deposits purchased by miners and endorsers. In the event of a double signing, these bonds are forfeited.
After a cycle, the miners and endorsers receive a reward along with their bond to compensate for their opportunity cost. The security is primarily being provided by the value of the bond and the reward need only be a small percentage of that value.
The purpose of the bonds is to diminish the amount of reward needed, and perhaps to use the loss aversion effect to the network's advantage.
Specifics
In the seed protocol, mining a block offers a reward of 16 ꜩ and requires a bond of 512 ꜩ per block created. Signing a block offers a reward of up to 2 ꜩ depending on the priority of the block which was created. There are up to 32 signatures per block and signing requires a bond of 64 ꜩ.
The reward schedule implies at most a 5.5%. nominal inflation rate. Nominal inflation is neutral, it neither enrishes nor impoverishes anyone.
Note that the period of a year is determined from the block's timestamps, not from the number of blocks. This is to remove uncertainty as to the length of the commitment made by miners.
Lost coins
If an address is inactive, it will not be selected to create blocks (which would slow down the consensus algorithm), and it will not be allowed to vote until it is reactivated (to avoid uncertainty about participation rate).
Amendment rules
Amendments are adopted over election cycles lasting $N = 2^{17} = 131072$ blocks each. Given the a one minute block interval, this is about three calendar months. The election cycle is itself divided in four quarters of $2^{15} = 32768$ blocks. This cycle is relatively short to encourage early improvements, but it is expected that further amendments will increase the length of the cycle. Protocol upgrade votes will be much more frequent in the first year in order to allow for rapid iteration. As a security measure, the Tezos foundation will have a veto power expiring after twelve months, until we rule out any kinks in the voting procedure. Adoption requires a certain quorum to be met. This quorum starts at $Q = 80\%$ but dynamically adapts to reflect the average participation. This is necessary if only to deal with lost coins.
First quarter
Protocol amendments are suggested by submitting the hash of a tarball of .ml and .mli files representing a new protocol. Stakeholders may approve of any number of these protocols. This is known as "approval voting", a particularly robust voting procedure.
Second quarter
The amendment receiving the most approval in the first quarter is now subject to a vote. Stakeholders may cast a vote for, against or can choose to explicitely abstain. Abstentions count towards the quorum.
Third quarter
If the quorum is met (including explicit abstentions), and the amendment received $80\%$ of yays, the amendment is approved and replaces the test protocol. Otherwise, it is rejected. Assuming the quorum reached was $q$, the minimum quorum $Q$ is updated as such: Q←0.8Q+0.2q.Q \leftarrow 0.8 Q + 0.2 q.Q←0.8Q+0.2q.
​
The goal of this update is to avoid lost coins causing the voting procedure to become stuck over time. The minimum quorum is an exponential moving average of the quorum reached over each previous election.
Fourth quarter
Assuming the amendment was approved, it will have been running in the testnet since the beginning of the third quarter. The stakeholders vote a second time to confirm they wish to promote the test protocol to the main protocol. This also requires the quorum to be met and an $80\%$ supermajority.
We deliberately chose a conservative approach to amendments. However, stakeholders are free to adopt amendments loosening or tightening this policy should they deem it beneficial
Proof-of-stake mechanism
Overview
Our proof-of-stake mechanism is a mix of several ideas, including Slasher[@Slasher], chain-of-activity[@CoA], and proof-of-burn. The following is a brief overview of the algorithm, the components of which are explained in more details below.
Each block is mined by a random stakeholder (the miner) and includes multiple signatures of the previous block provided by random stakeholders (the signers). Mining and signing both offer a small reward but also require making a 5-cycle safety deposit to be forfeited in the event of a double mining or double signing.
The protocol unfolds in cycles of 4,096 blocks. At the beginning of each cycle, a random seed is derived from numbers that block miners chose and committed to in the penultimate cycle, and revealed in the last. Using this random seed, a follow the coin strategy is used to allocate migning rights and signing rights to a specific addresses for the next cycle.
Four cycles of the proof-of-stake mechanism: 
​
Clock
The protocol imposes minimum delays between blocks. In principle, each block can be mined by any stakeholder. However, for a given block, each stakeholder is subject to a random minimum delay. The stakeholder receiving the highest priority may mine the block one minute after the previous block. The stakeholder receiving the second highest priority may mine the block two minutes after the previous block, the third, three minutes, and so on.
This guarantees that a fork where only a small fraction of stakeholder contribute will exhibit a low rate of block creation. If this weren't the case, a CPU denial of service attacks would be possible by tricking nodes into verifying a very long chain claimed to have a very high score.
Generating the random seed
Every block mined carries a hash commitment to a random number chosen by the miner. These numbers must be revealed in the next cycle under penalty of forfeiting the safety bond. This harsh penalty is meant to prevent selective whitholding of the numbers which could be sued to attack the entropy of the seed.
Malicious miners in the next cycle could attempt to censor such reveals, however since multiple numbers may be revealed in a single block, they are very unlikely to succeed.
All the revealed numbers in a cycle are combined in a hash list and the seed is derived from the root using the scrypt key derivation function. The key derivation should be tuned so that deriving the seed takes on the order of a fraction of a percent of the average validation time for a block on a typical desktop PC.
Follow-the-coin procedure
In order to randomly select a stakeholder, we use a follow the coin procedure.
Principle
The idea is known in bitcoin as follow-the-satoshi. The procedures works "as-if" every satoshi ever minted had a unique serial number. Satoshis are implicitly ordered by creation time, a random satoshi is drawn and tracked through the blockchain. Of course, individual cents are not tracked directly. Instead, rules are applied to describe what happens when inputs are combined and spent over multiple output.
In the end, the algorithm keeps track of a set of intervals associated with each key. Each intervals represents a "range" of satoshis. Unfortunately, over time, the database becomes more and more fragmented, increasing bloat on the client side.
Coin Rolls
We optimize the previous algorithm by constructing large "coin rolls" made up of tez. There are thus about one million rolls in existence. A database maps every roll to its current owner.
Each address holds a certain set of specific rolls as well as some loose change. When we desire to spend a fraction of a full roll, the roll is broken and its serial number is sent in a LIFO queue of rolls, a sort of "limbo". Every transaction is processed in a way that minimizes the number of broken rolls. Whenever an address holds enough coins to form a roll, a serial number is pulled from the queue and the roll is formed again.
The LIFO priority ensures that an attacker working on a secret fork cannot change the coins he holds by shuffling change between accounts.
A slight drawback of this approach is that stake is rounded down to the nearest integer number of rolls. However, this provides a massive improvement in efficiency over the follow-the-satoshi approach.
While the rolls are numbered, this approach does not preclude the use of fungibility preserving protocols like Zerocash. Such protocols can use the same "limbo" queue technique.
Motivation
This procedure is functionally different from merely drawing a random address weighted by balance.
Indeed, in a secretive fork, a miner could attempt to control the generation of the random seed and to assign itself signing and minting rights by creating the appropriate addresses ahead of time. This is much harder to achieve if rolls are randomly selected, as the secretive fork cannot fake ownership of certain rolls and must thus try to preimage the hash function applied to the seed to assign itself signing and minting rights.
Indeed, in a cycle of length $N=2048$, someone holding a fraction $f$ of the rolls will receive on average $f N$ mining rights, and the effective fraction received, $f_0$ will have a standard deviation of 1N1−ff.\sqrt{\frac{1}{N}}\sqrt{\frac{1-f}{f}}.N1​​f1−f​​.
​
If an attacker can perform a brute-force search through $W$ different seeds, then his expected advantage is at most (2log⁡(W)N1−ff)fN\left(\sqrt{\frac{2\log(W)}{N}}\sqrt{\frac{1-f}{f}}\right)fN(N2log(W)​​f1−f​​)fN
​
blocks. For instance, an attacker controlling $f = 10\%$ of the rolls should expect to mine about $205$ blocks per cycle. In a secret fork where he attempts to control the seed, assuming he computed over a trillion hashes, he could assign itself about $302$ blocks, or about $14.7\%$ of the blocks. Note that:
The hash from which the seed is derived is an expensive key derivation function, rendering brute-force search impractical.
To make linear gains in blocks mined, the attacked needs to expend a quadratically exponential effort.
Mining blocks
The random seed is used to repeatedly select a roll. The first roll selected allows its stakeholder to mine a block after one minute, the second one after two minutes --- and so on.
When a stakeholder observes the seed and realizes he can mint a high priority block in the next cycle, he can make a security deposit.
To avoid a potentially problematic situation were no stakeholder made a safety deposit to mine a particular block, after a 16 minutes delay, the block may be mined without a deposit.
Bonds are implicitely returned to their buyers immediately in any chain where they do not mine the block.
Signing blocks
As it is, we almost have a working proof of stake system. We could define a chain's weight to be the number of blocks. However, this would open the door to a form of selfish mining.
We thus introduce a signing scheme. While a block is being minted, the random seed is used to randomly assign 32 signing rights to 32 rolls.
The stakeholders who received signing rights observe the blocks being minted and then submit signatures of that blocks. Those signatures are then included in the next block, by miners attempting to secure their parent's inclusion in the blockchain.
The signing reward received by signers is dependent on the priority of the miner who mined the block. The higher the priority, the higher the signing reward. Signer thus have a strong incentive to sign what they genuinely believe to be the best block produced at one point. They also have a strong incentive to agree on which block they will sign as signing rewards are only paid if the block ends up included in the blockchain.
If the highest priority block isn't mined (perhaps because the miner isn't on line), there could be an incentive for signers to wait for a while, just in case the miner is late. However, other signers may then decide to sign the best priority block, and a new block could include those signatures, leaving out the holdouts. Thus, miners are unlikely to follow this strategy.
Conversely, we could imagine an equilibrium where signers panic and start signing the first block they see, for fear that other signers will do so and that a new block will be built immediately. This is however a very contrived situation which benefits no one. There is no incentive for signers to think this equilibrium is likely, let alone to modify the code of their program to act this way. A malicious stakeholder attempting to disrupt the operations would only hurt itself by attempting to follow this strategy, as others would be unlikely to follow suit.
Weight of the chain
The weight is the number of signatures.
Denunciations
In order to avoid the double minting of a block or the double signing of a block, a miner may include in his block a denunciation.
This denunciation takes the form of two signatures. Each minting signature or block signature signs the height of the block, making the proof of malfeasance quite concise.
While we could allow anyone to denounce malfeasance, there is really no point to allow anyone else beyond the block miner. Indeed, a miner can simply copy any proof of malfeasance and pass it off as its own discovery.
Once a party has been found guilty of double minting or double signing, the safety bond is forfeited.
Smart contracts
Contract type
In lieu of unspent outputs, Tezos uses stateful accounts. When those accounts specify executable code, they are known more generally as contracts. Since an account is a type of contract (one with no executable code), we refer to both as \"contracts\" in full generality.
Each contract has a "manager\", which in the case of an account is simply the owner. If the contract is flagged as spendable, the manager may spend the funds associated with the contract. In addition, each contract may specify the hash of a public key used to sign or mine blocks in the proof-of-stake protocol. The private key may or may not be controlled by the manager.
Formally, a contract is represented as:
type contract = {
  counter: int; (* counter to prevent repeat attacks *)
  manager: id; (* hash of the contract's manager public key *)
  balance: Int64.t; (* balance held *)
  signer: id option; (* id of the signer *)
  code: opcode list; (* contract code as a list of opcodes *)
  storage: data list; (* storage of the contract *)
  spendable: bool; (* may the money be spent by the manager? *)
  delegatable: bool; (* may the manager change the signing key? *)
}
The handle of a contract is the hash of its initial content. Attempting to create a contract whose hash would collide with an existing contract is an invalid operation and cannot be included in a valid block.
Note that data is represented as the union type.
type data =
  | STRING of string
  | INT of int
where INT is a signed 64-bit integer and string is an array of up to bytes. The storage capacity is limited to bytes, counting the integers as eight bytes and the strings as their length.
Origination
The origination operation may be used to create a new contract, it specifies the code of the contract and the initial content of the contract's storage. If the handle is already the handle of an existing contract, the origination is rejected (there is no reason for this to ever happen, unless by mistake or malice).
Transactions
A transaction is a message sent from one contract to another contract, this messages is represented as:
type transaction = {
  amount: amount; (* amount being sent *)
  parameters: data list; (* parameters passed to the script *)
  (* counter (invoice id) to avoid repeat attacks *)
  counter: int;
  destination: contract hash;
}
Such a transaction can be sent from a contract if signed using the manager's key or can be sent programmatically by code executing in the contract. When the transaction is received, the amount is added to the destination contract's balance and the destination contract's code is executed. This code can make use of the parameters passed to it, it can read and write the contract's storage, change the signature key and post transactions to other contracts.
The role of the counter is to prevent replay attacks. A transaction is only valid if the contract's counter is equal to the transaction's counter. Once a transaction is applied, the counter increases by one, preventing the transaction from being reused.
The transaction also includes the block hash of a recent block that the client considers valid. If an attacker ever succeeds in forcing a long reorganization with a fork, he will be unable to include such transactions, making the fork obviously fake. This is a last line of defense, TAPOS is a great system to prevent long reorganizations but not a very good system to prevent short term double spending.
The pair (account_handle, counter) is roughly the equivalent of an unspent output in Bitcoin.
Storage fees
Since storage imposes a cost on the network, a minimum fee of ꜩ 1 is assessed for each byte increase in the storage. For instance, if after the execution of a transaction, an integer has been added to the storage and ten characters have been appended to an existing string in the storage, then ꜩ 18 will be withdrawn from the contract's balance and destroyed.
Code
The language is stack based, with high-level data types and primitives and strict static type checking. Its design is insipired by Forth, Scheme, ML and Cat. A full specification of the instruction set is available in[@language]. This specification gives the complete instruction set, type system and semantics of the language. It is meant as a precise reference manual, not an easy introduction.
Fees
So far, this system is similar to the way Ethereum handles transaction. However, we differ in the way we handle fees. Ethereum allows arbitrarily long programs to execute by requiring a fee that increases linearly with the program's executing time. Unfortunately, while this does provide an incentive for one miner to verify the transaction, it does not provide such an incentive to other miners, who must also verify this transaction. In practice, most of the interesting programs that can be used for smart contracts are very short. Thus, we simplify the construction by imposing a hard cap on the number of steps we allow the programs to run for.
If the hard cap proves too tight for some programs, they can break the execution in multiple steps and use multiple transactions to execute fully. Since Tezos is amendable, this cap can be changed in the future, or advanced primitives can be introduced as new opcodes.
If the account permits, the signature key may be changed by issuing a signed message requesting the change.
Conclusion
We feel we've built an appealing seed protocol. However, Tezos's true potential lies in putting the stakeholders in charge of deciding on a protocol that they feel best serves them.
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On this page
Tezos Whitepaper
Introduction
Self-amending Cryptoledger
Mathematical representation
The network shell
Functional representation
Seed protocol
Economy
Proof-of-stake mechanism
Smart contracts
Conclusion