In cryptography two major types of encryption schemes are widely used: symmetric encryption (where a single secret key is used to encrypt and decrypt data) and asymmetric encryption (where a public key cryptosystem is used and encryption and decryption is done using a pair of public and corresponding private key). Let's explain these fundamental crypto-concepts in details.
Symmetric encryption schemes use the same symmetric key (or password) to encrypt data and decrypt the encrypted data back to its original form:
Symmetric encryption usually combines several crypto algorithms into an symmetric encryption scheme, e.g. AES-256-CTR-HMAC-SHA256. The encryption scheme (cipher construction) may include: password to key derivation algorithm (with certain parameters) + symmetric cipher algorithm (with certain parameters) + cipher block mode algorithm + message authentication (MAC) algorithm. This means that the above shown diagram is simplified and does not fully represent the process.
The secret key used to cipher (encrypt) and decipher (decrypt) data is typically of size 128, 192 or 256 bits and is sometimes referred as "encryption key" or "shared key", because both sending and receiving parties should know it.
Most applications use a password-to-key-derivation scheme to extract a secret key from certain password, because users tend to remember passwords easier than binary data. Additionally, message authentication is often incorporated along with the encryption to provide integrity and authenticity (this encryption approach is known as "authenticated encryption").
How does a private key look like? Let's start from a simple example of 256-bit secret key, encoded as hex string:
For example, the above key looks like this in base58:
The same key looks like this in base64:
In decimal system, the above key is the following integer number:
Widely used in modern cryptography symmetric encryption algorithms (ciphers) are: AES (AES-128, AES-192, AES-256), ChaCha20, Twofish, IDEA, Serpent, Camelia and others. Most of them are block ciphers (encrypt data by blocks of fixed size, e.g. 128 bits), while others are stream ciphers (encrypt data byte by byte as a stream). Block ciphers can be turned into stream ciphers by using a technique called "cipher block mode".
We shall give more details and code examples using the AES and ChaCha20 algorithms a bit later.
In order to better understand the idea behind the symmetric encryption, you can play with some online symmetric encryption tool to encrypt and decrypt a sample message by sample secret key (or password). You can play a bit with this site: https://aesencryption.net.
It demonstrates how we can encrypt and decrypt messages, using the AES cipher (with certain settings, more precisely AES-128-CBC) and certain password-to-key-derivation function. In the above example if we encrypt the text "secret msg" by the password "p@ss", we will get the base64-encoded binary data "jVJwOBmH+qMqHdg22KwMyg==" as output. After decryption with the same secret key we get back the original text "secret msg".
Note that the above encrypted text is dependent to many algorithm parameters and settings, so if you encrypt the same at another "AES live example" Web site, the result most likely will be different.
Also note that the above mentioned Web site uses bad practices, old algorithms and weak cipher settings, so don't follow the code examples there.
Before introducing the asymmetric key encryption schemes and algorithms, we should first understand the concept of public key cryptography (asymmetric cryptography).
The public key cryptography uses a different key to encrypt and decrypt data (or to sign and verify messages). Keys always come as public + private key pairs. Asymmetric cryptography deals with encrypting and decrypting messages using a public / private key, signing messages, verifying signatures and securely exchanging keys.
Popular public-key cryptosystems (asymmetric crypto algorithms) like RSA (Rivest–Shamir–Adleman), ECC (elliptic curve cryptography), Diffie-Hellman, ECDH, ECDSA and EdDSA, are widely used in the modern cryptography and we shall demonstrate most of them in practice with code examples.
Asymmetric encryption schemes use a pair of cryptographically related public and private keys to encrypt the data (by the public key) and decrypt the encrypted data back to its original forms (by the private key). Data encrypted by a public key is decrypted by the corresponding private key:
The encrypted data, obtained as result of encryption is called "ciphertext". The ciphertext is a binary sequence, unreadable by humans and by design cannot be decrypted without the decryption key.
Note that the above shown diagram is highly simplified and does not fully represent the asymmetric encryption / decryption process. Typically, public-key cryptosystems can encrypt messages of limited length only and are slower than symmetric ciphers. For encrypting longer messages (e.g. PDF documents) usually a public-key encryption scheme (also known as hybrid encryption scheme) is used, which combines symmetric and asymmetric encryption like this:
For the encryption a random symmetric key
sk is generated, the message is symmetrically encrypted by
sk is asymmetrically encrypted using the recipient's public key.
For decryption, first the
sk key is asymmetrically decrypted using the recipient's private key, then the ciphertext is decrypted symmetrically using
The above process is known as key encapsulation mechanism (KEM): encapsulate an asymmetrically-encrypted random (ephemeral) symmetric key and use symmetric algorithm for the data encryption.
Public key encryption can work also in the opposite scenario: encrypt data by a private key and decrypt it by the public key. Thus someone can prove that he is owner of certain private key, while revealing only its corresponding public key. This approach is used by some digital signature schemes.
In the context of digital signatures, a message signed by a private key (digital signature) is later verified by the corresponding public key.
Digital signatures will be explained in more details later, but in short: a message can be signed by certain private key and the obtained signature can be later verified by the corresponding public key. A signed message cannot be altered after signing. A message signature proves that certain message (e.g. blockchain transaction) is created by the owner of certain public key. Digital signatures provide message authentication, message integrity and non-repudiation.
Digital signatures are widely used in the finance industry for authorizing payments. In operating systems OS components and device drivers are usually digitally signed to avoid injecting insecure code, trojans or viruses in the OS. In blockchain systems, transactions are typically signed by the owner of certain blockchain address (which corresponds to certain public key and has corresponding private key). So a signed blockchain transaction holds a proof of authorship: it is guaranteed mathematically that the signature is created by the holder of certain blockchain address and the transaction was not modified after the signing. This works perfectly for the scenario of digital payments and digital signing of documents and contracts.
The public key cryptography uses a pair of keys: public key + private key. These keys are mathematically connected and are used together as pair.
In some public key cryptosystems (like the Elliptic-Curve Cryptography - ECC), the public key can be calculated from the private key. In other cryptosystems (like RSA), the public key and the private key are generated together but cannot be directly calculated from each other.
Usually, a public / private key pair is randomly generated in a secure environment (e.g. in a hardware wallet) and the public key is revealed, while the private key is securely stored in a crypto-wallet and is protected by a password or by multi-factor authentication.
Example of 256-bit private key and its corresponding 256-bit public key (based on secp256k1 curve):
privKey: 648fc1fa828c7f185d825c04a5b21af9e473b867eeee1acea4dbab938433e158pubKey: 02c324648931b89e3e8a0fc42c96e8e3be2e42812986573a40d46563bceaf75110
Message encryption and signing is done by a private key. The private keys are always kept secret by their owner, just like passwords. In the server infrastructure, private key usually stay in an encrypted and protected keystore. In the blockchain systems the private keys usually stay in specific software or hardware apps or devices called "crypto wallets", which store securely a set of private keys.
Example of 256-bit private key:
Message decryption and signature verification is done by the public key. Public keys are by design public information (not a secret). It is mathematically infeasible to calculate the private key from its corresponding public key.
In many systems the public key is encapsulated in a digital certificate, which binds certain identity (e.g. person or Internet domain name) to certain public key. In blockchain systems public keys are usually published as parts of the blockchain transactions to help identify who has signed each transaction. In systems like PGP and SSH the public key is downloaded from the server once (after manual user verification) and is remembered for further use.
Example of 256-bit public key:
In most blockchain systems the blockchain address is derived from the public key (by hashing and other transformations), so if you have someone's public key, you are assumed to have his blockchain address as well.
A certain public key can be connected to certain person or organization or is used anonymously. You can never know who is the owner of the private key, corresponding to certain public key, unless you have additional proof, e.g. a digital certificate.
Public key cryptosystems provide mathematical framework and algorithms to generate public + private key pairs, to sign, verify, encrypt and decrypt messages and exchange keys, in a cryptographically secure way.
Well-known public-key cryptosystems are: RSA, ECC and ElGamal. Many crypto algorithms are based on the primitives from these cryptosystems like RSA sign, RSA encrypt / decrypt, ECDH key exchange and ECDSA and EdDSA signatures.
The RSA public-key cryptosystem is based on the math of modular exponentiations (numbers raised to a power by modulus) and some additional assumptions, together with the computational difficulty of the integer factorization problem. We shall discuss it later in details, along with examples.
The elliptic-curve cryptography (ECC) public-key cryptosystem is based on the math of the on the algebraic structure of the elliptic curves over finite fields and the difficulty of the elliptic curve discrete logarithm problem (ECDLP). The ECC usually comes together with the ECDSA algorithm (elliptic-curve digital signature algorithm). We shall discuss ECC and ECDSA in details, along with examples.
ECC uses smaller keys, ciphertexts and signatures than RSA and is recommended for most applications. It is mathematically proven that a 3072-bit RSA key has similar cryptographic strength to a 256-bit ECC key. Key generation is in ECC is significantly faster than with RSA.
Due to the above reasons most blockchain networks (like Bitcoin and Ethereum) use elliptic-curve-based cryptography (ECC) to secure the transactions.
Note that both RSA and ECC cryptosystems are not quantum-safe, which means that if someone has enough powerful quantum computer, he will be able to derive the private key from given public key in just few seconds.
Asymmetric encryption works for small messages only (limited by the public / private key length). To encrypt larger messages key encapsulation mechanisms or other techniques can be used, which encrypt asymmetrically a random secret key, then use it to symmetrically encrypt the larger messages. In practice, modern asymmetric encryption schemes involve using a symmetric encryption algorithm together with a public-key cryptosystem, key encapsulation and message authentication.
Popular asymmetric encryption schemes are: RSA-OAEP (based on RSA and OAEP padding), RSAES-PKCS1-v1_5 (based on RSA and PKCS#1 v1.5 padding), DLIES (based on discrete logarithms and symmetric encryption) and ECIES (based on elliptic curve cryptography and symmetric encryption).
In order to better understand the idea behind the asymmetric encryption, you can play with some online public key encryption tool to encrypt / decrypt a sample message by sample RSA private / public key. You can play a bit with this site: http://travistidwell.com/jsencrypt/demo/.
In the above online demo you can generate RSA public / private key pairs and encrypt / decrypt text messages. Note that the message size is limited by the key length, so you can't encrypt long text. You will get an error if you try. Internally, the above Web site uses the RSAES-PKCS1-v1_5 public key encryption scheme as specified in RFC3447.
We shall discuss the RSA and ECC cryptosystems in details in the next few chapters. Now, it is important to learn that symmetric and asymmetric cryptosystems work differently and are used in different scenarios.