# Bitcoin elliptic curve parameters statistics

The actual supported algorithms and features depends on their availability in the actual libcrypto used. See the crypto App about dependencies. This option handles padding in the last block. If not set, no padding is done and any bytes in the last unfilled block is silently discarded. Where E is the public exponent, N is public modulus and D is the private exponent. The longer key format contains redundant information that will make the calculation faster.

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Content:

- Mastering Bitcoin, 2nd Edition by Andreas M. Antonopoulos
- Maths Behind Bitcoin
- A (relatively easy to understand) primer on elliptic curve cryptography
- Elliptic Curve Digital Signature Algorithm
- Subscribe to RSS
- Digital signature scheme for information non-repudiation in blockchain: a state of the art review
- Elliptic Curve Cryptography
- The Elliptic Curve Digital Signature Algorithm (ECDSA)
- An Analysis of Non-standard Transactions
- The Math Behind the Bitcoin Protocol

**WATCH RELATED VIDEO: Bitcoins Elliptic Curve Algorithm Explained safe-crypto.me math along the SECP256k1 curve**

## Mastering Bitcoin, 2nd Edition by Andreas M. Antonopoulos

Since that time, this paper has taken on a life of its own In the earlys, when the commercial Internet was still young!

Many thoiught that increased security provided comfort to paranoid people while most computer professionals realized that security provided some very basic protections that we all needed? Cryptography for the masses barely existed at that time and was certainly not a topic of common discourse. Security and privacy impacts many applications, ranging from secure commerce and payments to private communications and protecting health care information.

One essential aspect for secure communications is that of cryptography. But it is important to note that while cryptography is necessary for secure communications, it is not by itself sufficient. The reader is advised, then, that the topics covered here only describe the first of many steps necessary for better security in any number of situations.

This paper has two major purposes. The first is to define some of the terms and concepts behind basic cryptographic methods, and to offer a way to compare the myriad cryptographic schemes in use today. The second is to provide some real examples of cryptography in use today.

See Section A. Such mention is for example purposes only and, unless explicitly stated otherwise, should not be taken as a recommendation or endorsement by the author.

Some experts argue that cryptography appeared spontaneously sometime after writing was invented, with applications ranging from diplomatic missives to war-time battle plans. It is no surprise, then, that new forms of cryptography came soon after the widespread development of computer communications. In data and telecommunications, cryptography is necessary when communicating over any untrusted medium, which includes just about any network, particularly the Internet.

In cryptography, we start with the unencrypted data, referred to as plaintext. Plaintext is encrypted into ciphertext , which will in turn usually be decrypted back into usable plaintext. The encryption and decryption is based upon the type of cryptography scheme being employed and some form of key.

For those who like formulas, this process is sometimes written as:. Given this, there are other functions that might be supported by crypto and other terms that one might hear:. In many of the descriptions below, two communicating parties will be referred to as Alice and Bob; this is the common nomenclature in the crypto field and literature to make it easier to identify the communicating parties.

If there is a third and fourth party to the communication, they will be referred to as Carol and Dave, respectively. A malicious party is referred to as Mallory, an eavesdropper as Eve, and a trusted third party as Trent. Finally, cryptography is most closely associated with the development and creation of the mathematical algorithms used to encrypt and decrypt messages, whereas cryptanalysis is the science of analyzing and breaking encryption schemes.

Cryptology is the umbrella term referring to the broad study of secret writing, and encompasses both cryptography and cryptanalysis. There are several ways of classifying cryptographic algorithms.

For purposes of this paper, they will be categorized based on the number of keys that are employed for encryption and decryption, and further defined by their application and use.

The three types of algorithms that will be discussed are Figure 1 : Secret Key Cryptography SKC : Uses a single key for both encryption and decryption; also called symmetric encryption. Primarily used for privacy and confidentiality. Public Key Cryptography PKC : Uses one key for encryption and another for decryption; also called asymmetric encryption. Primarily used for authentication, non-repudiation, and key exchange. Hash Functions: Uses a mathematical transformation to irreversibly "encrypt" information, providing a digital fingerprint.

Primarily used for message integrity. Secret key cryptography methods employ a single key for both encryption and decryption. As shown in Figure 1A, the sender uses the key to encrypt the plaintext and sends the ciphertext to the receiver. The receiver applies the same key to decrypt the message and recover the plaintext. Because a single key is used for both functions, secret key cryptography is also called symmetric encryption. With this form of cryptography, it is obvious that the key must be known to both the sender and the receiver; that, in fact, is the secret.

The biggest difficulty with this approach, of course, is the distribution of the key more on that later in the discussion of public key cryptography. Secret key cryptography schemes are generally categorized as being either stream ciphers or block ciphers.

Stream ciphers operate on a single bit byte or computer word at a time and implement some form of feedback mechanism so that the key is constantly changing.

Stream ciphers come in several flavors but two are worth mentioning here Figure 2. Self-synchronizing stream ciphers calculate each bit in the keystream as a function of the previous n bits in the keystream. It is termed "self-synchronizing" because the decryption process can stay synchronized with the encryption process merely by knowing how far into the n -bit keystream it is. One problem is error propagation; a garbled bit in transmission will result in n garbled bits at the receiving side.

Synchronous stream ciphers generate the keystream in a fashion independent of the message stream but by using the same keystream generation function at sender and receiver. While stream ciphers do not propagate transmission errors, they are, by their nature, periodic so that the keystream will eventually repeat.

A block cipher is so-called because the scheme encrypts one fixed-size block of data at a time. In a block cipher, a given plaintext block will always encrypt to the same ciphertext when using the same key i. The most common construct for block encryption algorithms is the Feistel cipher , named for cryptographer Horst Feistel IBM. As shown in Figure 3, a Feistel cipher combines elements of substitution, permutation transposition , and key expansion; these features create a large amount of " confusion and diffusion " per Claude Shannon in the cipher.

One advantage of the Feistel design is that the encryption and decryption stages are similar, sometimes identical, requiring only a reversal of the key operation, thus dramatically reducing the size of the code or circuitry necessary to implement the cipher in software or hardware, respectively. One of Feistel's early papers describing this operation is " Cryptography and Computer Privacy " Scientific American , May , 5 , DES is a Feistel block-cipher employing a bit key that operates on bit blocks.

DES has a complex set of rules and transformations that were designed specifically to yield fast hardware implementations and slow software implementations, although this latter point is not significant today since the speed of computer processors is several orders of magnitude faster today than even twenty years ago.

DES was based somewhat on an earlier cipher from Feistel called Lucifer which, some sources report, had a bit key.

This was rejected, partially in order to fit the algorithm onto a single chip and partially because of the National Security Agency NSA. By combining 64 additional key bits to the plaintext prior to encryption, effectively increases the keylength to bits. The algorithm can use a variable block length and key length; the latest specification allowed any combination of keys lengths of , , or bits and blocks of length , , or bits. It's code has not been made public although many companies have licensed RC2 for use in their products.

Described in RFC RC4: A stream cipher using variable-sized keys; it is widely used in commercial cryptography products. More detail about RC4 and a little about Spritz can be found below in Section 5. RC5 : A block-cipher supporting a variety of block sizes 32, 64, or bits , key sizes, and number of encryption passes over the data. Key lengths can vary from 32 to bits in length. Twofish : A bit block cipher using , , or bit keys. Designed to be highly secure and highly flexible, well-suited for large microprocessors, 8-bit smart card microprocessors, and dedicated hardware.

Anubis : Anubis is a block cipher, co-designed by Vincent Rijmen who was one of the designers of Rijndael. Anubis is a block cipher, performing substitution-permutation operations on bit blocks and employing keys of length to bits in bit increments.

Anubis works very much like Rijndael. ARIA : A bit block cipher employing , , and bit keys to encrypt bit blocks in 12, 14, and 16 rounds, depending on the key size. Developed by large group of researchers from academic institutions, research institutes, and federal agencies in South Korea in , and subsequently named a national standard.

Camellia has some characteristics in common with AES: a bit block size, support for , , and bit key lengths, and suitability for both software and hardware implementations on common bit processors as well as 8-bit processors e.

Also described in RFC CLEFIA is one of the new-generation lightweight blockcipher algorithms designed after AES, offering high performance in software and hardware as well as a lightweight implementation in hardware. FPE schemes are used for such purposes as encrypting social security numbers, credit card numbers, limited size protocol traffic, etc.

FFX can theoretically encrypt strings of arbitrary length, although it is intended for message sizes smaller than that of AES 2 points. The FFX version 1. Use of this scheme is reportedly one of the reasons that the National Security Agency NSA can easily decode voice and data calls over mobile phone networks.

KASUMI is the intended confidentiality and integrity algorithm for both message content and signaling data for emerging mobile communications systems. Using simple arithmetic operations, the algorithms offers fast encryption and decryption by use of efficient implementations. KCipher-2 has been used for industrial applications, especially for mobile health monitoring and diagnostic services in Japan.

LED is designed for RFID tags, sensor networks, and other applications with devices constrained by memory or compute power. MARS employs bit blocks and a variable key length from to bits. The MARS document stresses the ability of the algorithm's design for high speed, high security, and the ability to efficiently and effectively implement the scheme on a wide range of computing devices.

Designed for hardware and software implementations, and is resistant to differential and linear cryptanalysis. Salsa20 uses a pseudorandom function based on bit whole word addition, bitwise addition XOR , and rotation operations, aka add-rotate-xor ARX operations. Salsa20 uses a bit key although a bit key variant also exists.

In , Bernstein published ChaCha , a new family of ciphers related to Salsa SEED : A block cipher using bit blocks and bit keys. Serpent : Serpent is another of the AES finalist algorithms. Serpent supports , , or bit keys and a block size of bits, and is a round substitution—permutation network operating on a block of four bit words.

The Serpent developers opted for a high security margin in the design of the algorithm; they determined that 16 rounds would be sufficient against known attacks but require 32 rounds in an attempt to future-proof the algorithm.

As a hash function, SHA repeatedly calls on a compression scheme to alter the state of the data blocks. While SHA like other hash functions is irreversible, the compression function can be used for encryption by maintaining appropriate state information.

Simon and Speck : Simon and Speck are a pair of lightweight block ciphers proposed by the NSA in , designed for highly constrained software or hardware environments.

While both cipher families perform well in both hardware and software, Simon has been optimized for high performance on hardware devices and Speck for performance in software.

Both are Feistel ciphers and support ten combinations of block and key size:.

## Maths Behind Bitcoin

Metrics details. Blockchain, as one of the most promising technology, has attracted tremendous attention. The interesting characteristics of blockchain are decentralized ledger and strong security, while non-repudiation is the important property of information security in blockchain. A digital signature scheme is an effective approach to achieve non-repudiation. In this paper, the characteristics of blockchain and the digital signature to guarantee information non-repudiation are firstly discussed. Secondly, the typical digital signature schemes in blockchain are classified and analyzed, and then the state-of-the-art digital signatures are investigated and compared in terms of application fields, methods, security, and performance. Lastly, the conclusions are given, and some future works are suggested to stir research efforts in this field.

## A (relatively easy to understand) primer on elliptic curve cryptography

This system is used in Bitcoin to allow people to receive and send bitcoins. The ability to create digital signatures has been around since the s thanks to the invention of RSA. In , DSA was released as the standard for digital signature systems. ECDSA is just an implementation of DSA using elliptic curve cryptography, as the mathematics of elliptic curves allow for more efficient signature creation and verification. All they needed to know is that it does work, and that they could use it as the mechanism for sending and receiving money in the system they were building. The first version of Bitcoin used the OpenSSL library to provide the functionality for creating and verifying digital signatures. In summary, public keys and signatures are just points on an elliptic curve. Satoshi chose the secpk1 curve, which has the following parameters:. This is due to the fact that the curve used in Bitcoin is over a finite field of whole numbers i.

## Elliptic Curve Digital Signature Algorithm

ECC focuses on pairs of public and private keys for decryption and encryption of web traffic. RSA achieves one-way encryption of things like emails, data, and software using prime factorization. It generates security between key pairs for public key encryption by using the mathematics of elliptic curves. RSA does something similar with prime numbers instead of elliptic curves, but ECC has gradually been growing in popularity recently due to its smaller key size and ability to maintain security. This trend will probably continue as the demand on devices to remain secure increases due to the size of keys growing, drawing on scarce mobile resources.

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This library is primarily a reference implementation for the CurveZMQ specification but may also be used for end-to-end security. CurveZMQ creates encrypted sessions "connections" between two peers using short term keys that it securely exchanges using long term keys. When the session is over, both sides discard their short term keys, rendering the encrypted data unreadable, even if the long term keys are captured. It is not designed for long term encryption of data. A library for effective private and public keychain management. This library provides a powerful key derivation interface that is based on the creation of private and public keychains.

## Digital signature scheme for information non-repudiation in blockchain: a state of the art review

One reason bitcoin can be confusing for beginners is that the technology behind it redefines the concept of ownership. To own something in the traditional sense, be it a house or a sum of money, means either having personal custody of the thing or granting custody to a trusted entity such as a bank. With bitcoin the case is different. Bitcoins themselves are not stored either centrally or locally and so no one entity is their custodian. They exist as records on a distributed ledger called the block chain, copies of which are shared by a volunteer network of connected computers. What grants this ability? What does that mean and how does that secure bitcoin? With bitcoin, the data that is signed is the transaction that transfers ownership.

## Elliptic Curve Cryptography

Elliptic-curve cryptography ECC is an approach to public-key cryptography based on the algebraic structure of elliptic curves over finite fields. ECC allows smaller keys compared to non-EC cryptography based on plain Galois fields to provide equivalent security. Elliptic curves are applicable for key agreement , digital signatures , pseudo-random generators and other tasks. Indirectly, they can be used for encryption by combining the key agreement with a symmetric encryption scheme.

## The Elliptic Curve Digital Signature Algorithm (ECDSA)

However, the creation of ad-hoc scripts to lock and unlock transactions allows for also generating non-standard transactions, which can be nevertheless broadcast and mined as well. In this work, we explore the Bitcoin block-chain with the purpose to analyze and classify standard and non-standard transactions, understanding how much the standard behavior is respected. Money transactions do not require a third-party intermediary, with no traditional financial-institution involved in transactions. Therefore, the Bitcoin network is completely decentralized, with all the transaction components performed by the users of the system.

## An Analysis of Non-standard Transactions

ECDSA relies on the math of the cyclic groups of elliptic curves over finite fields and on the difficulty of the ECDLP problem elliptic-curve discrete logarithm problem. Elliptic curves, used in cryptography, define:. The private key is generated as a random integer in the range [ The proof s is by idea verifiable using the corresponding pubKey. ECDSA signatures are 2 times longer than the signer's private key for the curve used during the signing process. For example, for bit elliptic curves like secpr1 the ECDSA signature is bits 64 bytes and for bit curves like secpr1 the signature is bits.

## The Math Behind the Bitcoin Protocol

Author Nick Sullivan worked for six years at Apple on many of its most important cryptography efforts before recently joining CloudFlare, where he is a systems engineer. He has a degree in mathematics from the University of Waterloo and a Masters in computer science with a concentration in cryptography from the University of Calgary. This post was originally written for the CloudFlare blog and has been lightly edited to appear on Ars.

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