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Brief History of Cryptography
The development of civilization provided the need for encryption of data and its secret transmission. Thus, cryptography was used in early states such as ancient Egypt or Mesopotamia. At that time, the primary method of encrypting information was the substitution of symbols (Dooley, 2018). In more recent times in ancient Greece, cryptography was used to protect sensitive military information. For example, the Spartans used special Scytales cylinders around which parchment was wrapped. Such a message could only be decrypted with exactly the same cylinder; it was encrypted on the color. Spies in India also used encrypted messages to transmit essential data. However, the most advanced method of cryptography was developed in ancient Rome. The famous Caesar cipher assumed shifting letters by a fixed number of places in the alphabet to create a secret message. The recipient of such a message could decipher it only by knowing the number of places required for the shift.
In the Middle Ages, cryptography did not evolve significantly, but cryptanalysis began to develop, aimed at decoding secret messages. Al-Kindi, an Arab mathematician, created a frequency analysis that made it easy to decode ciphers created by rearranging letters (Dooley, 2018). Thus, cryptographic techniques have had to evolve to remain useful and relevant. In response to Al-Kindis analysis in the 15th century, a polyalphabetic cipher was developed, which used two different alphabets to encode, increasing the security of secret messages. During the Renaissance, Francis Bacon created a binary encoding method. More recently, a cipher wheel was developed, which allowed the creation of complex ciphers and was used by the US Army until World War II. Later, the Enigma machine was used, based on the same principle as the wheel. The messages encoded on it were impossible to identify without using another Enigma. Modern computer technology makes it possible to create much more complex ciphers than in the analog era.
Cryptography and Mathematics
Cryptography, the development of computer technology, and the principles of mathematics became the basis for the creation of cryptocurrency. Modern cryptography is used to ensure the security of data, as well as to verify their transmission (Fleming, 2019). Cryptocurrency is a virtual means of payment that is created on the basis of these principles. Such a currency is almost impossible to double-use or counterfeit due to complex encryption methods. The virtual currency uses the principles of cryptography as well as encryption keys and the concept of signatures. In this case, mathematical methods are used to ensure the safe transfer of information, access to which only verified users have.
Prime Numbers and RSA Algorithm
Cryptocurrencies, in particular Bitcoin, ensure the safety of transactions even between unfamiliar users due to the exceptional mathematics of operations. Modern cryptography uses a variety of principles, one of which is RSA (named after the first letters of the names of the creators), which is based on prime numbers. Prime numbers are numbers greater than 1, which are divisible only by themselves and 1. The RSA algorithm is based on creating a cipher from two large prime numbers and using them to encode a message. (Pachghare, 2019). The recipient can decipher the received information only by knowing the original prime numbers. Thus, transmission security is ensured by the problem of factoring in these numbers.
Elliptic Curve and ECC Algorithm
Another modern mathematical cryptographic algorithm uses an elliptic curve to encrypt the message. Elliptic Curve Cryptography (ECC) is used in particular to create virtual signatures, which are the backbone of cryptocurrency (Bolfing, 2020). This approach is an alternative to RSA but has become more popular due to its higher security and performance. ECC uses information about how elliptic curves are structured algebraically over finite fields ( Elliptic curve cryptography, n. d.). In particular, in Bitcoin and blockchain, the cryptographic algorithm is based on the discrete logarithm for elliptic curves on finite fields (Jimenez-Ayala & Medina, n. d.). Despite the fact that the two algorithms use different principles, they are based on private and public keys, which are used to encrypt and decrypt data.
Private and Public Keys
The basis of transactions using cryptocurrency is private and public keys. Transactions are possible between two peers within the cryptocurrency system, each of which has its own address. Each of these addresses address is associated with a public key and a private key (Guri, 2018, p. 1). In the RSA algorithm, keys are large primes; in the ECC algorithm, 64-bit and 32-bit numbers. Public keys are available to all users, so transactions, which are signed by a private key, can be verified by anyone using the corresponding public key (Guri, 2018, p. 1). In this case, the difference between the two algorithms is noteworthy, since ECC provides the same level of data protection with much shorter keys ( Elliptic curve cryptography, n. d.). A pair of private and public keys are used to create a virtual signature that ensures data security. When sent, the message is encrypted using a private digital key, which the recipient can then verify using a public one.
A message can be encrypted with the recipients public key, but it can only be decrypted utilizing the private key. Cryptocurrency security is built on the mathematical one-way function principle, which implies that keys can be easily generated but are difficult to invert. Thus, messages encoded with a public key cannot be decoded without a corresponding private key. Thus, the digital signature and keys are used to identify the sender and confirm its authenticity.
References
Bolfing, A. (2020). Cryptographic primitives in blockchain technology A mathematical introduction. Oxford University Press.
Dooley, J. F. (2018). History of cryptography and cryptanalysis codes, ciphers, and their algorithms. Springer International Publishing.
Elliptic curve cryptography. (n. d.). AVI Networks. Web.
Fleming, S. (2019). A brief history of cryptography and why it matters. World Economic Forum.
Guri, M. (2018). BeatCoin: Leaking private keys from air-gapped cryptocurrency wallets [PDF document].
Jimenez-Ayala, E. S., & Medina, J. (n. d.). The mathematics behind cryptocurrencies and blockchain [PDF document].
Pachghare, V. K. (2019). Cryptography and information security (3rd ed.). PHI Learning Pvt. Ltd.
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