Public Key Cryptography¶

Public key cryptography is a collection of techniques allowing for encryption, signatures, and key agreement.

Key Objects¶

Public and private keys are represented by classes Public_Key and Private_Key. Both derive from Asymmetric_Key.

Currently there is an inheritance relationship between Private_Key and Public_Key, so that a private key can also be used as the corresponding public key. It is best to avoid relying on this, as this inheritance will be removed in a future major release.

class Asymmetric_Key¶

std::string algo_name()¶: Return a short string identifying the algorithm of this key, eg “RSA” or “ML-DSA”.

size_t estimated_strength() const¶: Return an estimate of the strength of this key, in terms of brute force key search. For example if this function returns 128, then it is is estimated to be roughly as difficult to crack as AES-128.

OID object_identifier() const¶: Return an object identifier which can be used to identify this type of key.

bool supports_operation(PublicKeyOperation op) const¶: Check if this key could be used for the queried operation type.

class Public_Key¶

size_t key_length() const = 0;¶: Return an integer value that most accurately captures for the security level of the key. For example for RSA this returns the length of the public modules, while for ECDSA keys it returns the size of the elliptic curve group.

bool check_key(RandomNumberGenerator &rng, bool strong) const = 0;¶: Check if the key seems to be valid. If strong is set to true then more expensive tests are performed.

AlgorithmIdentifier algorithm_identifier() const = 0;¶: Return an X.509 algorithm identifier that can be used to identify the key.

std::vector<uint8_t> public_key_bits() const = 0;¶

Returns a binary representation of the public key. Typically this is a BER encoded structure that includes metadata like the algorithm and parameter set used to generate the key.

Note that pre-standard post-quantum algorithms of the NIST competition (e.g. Kyber, Dilithium, FrodoKEM, etc) do not have a standardized BER encoding, yet. For the time being, the raw public key bits are returned for these algorithms. That might change as the standards evolve.

std::vector<uint8_t> raw_public_key_bits() const = 0;¶

Returns a binary representation of the public key’s canonical structure. Typically, this does not include any metadata like an algorithm identifier or parameter set. Note that some schemes (e.g. RSA) do not know such “raw” canonical structure and therefore throw Not_Implemented. For key agreement algorithms, this is the canonical public value of the scheme.

Decoding the resulting raw bytes typically requires knowledge of the algorithm and parameters used to generate the key.

std::vector<uint8_t> subject_public_key() const;¶: Return the X.509 SubjectPublicKeyInfo encoding of this key

std::string fingerprint_public(const std::string &alg = "SHA-256") const;¶: Return a hashed fingerprint of this public key.

Public Key Algorithms¶

Botan includes a number of public key algorithms, some of which are in common use, others only used in specialized or niche applications.

RSA¶

Based on the difficulty of factoring. Usable for encryption, signatures, and key encapsulation.

ECDSA¶

Fast signature scheme based on elliptic curves.

ECDH, DH, X25519 and X448¶

Key agreement schemes. DH uses arithmetic over finite fields and is slower and with larger keys. ECDH, X25519 and X448 use elliptic curves instead.

ML-DSA (FIPS 204)¶

Post-quantum secure signature scheme based on (structured) lattices. This algorithm is standardized in FIPS 204. Signing keys are always stored and expanded from the 32-byte private random seed (xi), loading the expanded key format specified in FIPS 204 is explicitly not supported.

Support for ML-DSA is implemented in the module ml_dsa

Additionally, support for the pre-standardized version “Dilithium” is retained for the time being. The implemented specification is commonly referred to as version 3.1 of the CRYSTALS-Dilithium submission to NIST’s third round of the PQC competition. This is not compatible to the “Initial Public Draft” version of FIPS 204 for which Botan does not offer an implementation.

Currently two flavors of Dilithium are implemented in separate Botan modules:

dilithium, that uses Keccak (SHAKE), and that saw some public usage by early adopters.

dilithium_aes, that uses AES instead of Keccak-based primitives. This mode is deprecated and will be removed in a future release.

ML-KEM (FIPS 203)¶

Post-quantum key encapsulation scheme based on (structured) lattices. This algorithm is standardized in FIPS 203. Decapsulation keys are always stored and expanded from the 64-byte private random seeds (d || z), loading the expanded key format specified in FIPS 203 is explicitly not supported.

Support for ML-KEM is implemented in the module ml_kem.

Additionally, support for the pre-standardized version “Kyber” is retained for the time being. The implemented specification is commonly referred to as version 3.01 of the CRYSTALS-Kyber submission to NIST’s third round of the PQC competition. This is not compatible to the “Initial Public Draft” version of FIPS 203 for which Botan does not offer an implementation.

Currently two flavors of Kyber are implemented in separate Botan modules:

kyber, that uses Keccak (SHAKE and SHA-3), and that saw some public usage by early adopters.

kyber_90s, that uses AES/SHA-2 instead of Keccak-based primitives. This mode is deprecated and will be removed in a future release.

Ed25519 and Ed448¶

Signature schemes based on a specific elliptic curve.

XMSS¶

A post-quantum secure signature scheme whose security is based (only) on the security of a hash function. Unfortunately XMSS is stateful, meaning the private key changes with each signature, and only a certain pre-specified number of signatures can be created. If the same state is ever used to generate two signatures, then the whole scheme becomes insecure, and signatures can be forged.

HSS-LMS¶

A post-quantum secure hash-based signature scheme similar to XMSS. Contains support for multitrees. It is stateful, meaning the private key changes after each signature.

SLH-DSA (FIPS 205)¶

The Stateless Hash-Based Digital Signature Standard (SLH-DSA) is the FIPS 205 post-quantum secure signature scheme whose security is solely based on the security of a hash function. Unlike XMSS, it is a stateless signature scheme, meaning that the private key does not change with each signature. It has high security but very long signatures and high runtime.

Support for SLH-DSA is implemented in the modules slh_dsa_sha2 and slh_dsa_shake.

Additionally, support for the pre-standardized version “SPHINCS+” is retained for the time being. The implemented specification is commonly referred to as version 3.1 of the SPHINCS+ submission to NIST’s third round of the PQC competition. This is not compatible with the “Initial Public Draft” version of FIPS 205 for which Botan does not offer an implementation. Also, Botan does not support the Haraka hash function.

Currently, two flavors of SPHINCS+ are implemented in separate Botan modules:

sphincsplus_shake, that uses Keccak (SHAKE) hash functions

sphincsplus_sha2, that uses SHA-256

FrodoKEM¶

A post-quantum secure key encapsulation scheme based on (unstructured) lattices.

McEliece¶

Post-quantum secure key encapsulation scheme based on the hardness of certain decoding problems.

ElGamal¶

Encryption scheme based on the discrete logarithm problem. Generally unused except in PGP.

DSA¶

Finite field based signature scheme. A NIST standard but now quite obsolete.

ECGDSA, ECKCDSA, SM2, GOST-34.10¶

A set of signature schemes based on elliptic curves. All are national standards in their respective countries (Germany, South Korea, China, and Russia, resp), and are completely obscure and unused outside of that context.

Creating New Private Keys¶

Creating a new private key requires two things: a source of random numbers (see Random Number Generators) and some algorithm specific parameters that define the security level of the resulting key. For instance, the security level of an RSA key is (at least in part) defined by the length of the public key modulus in bits. So to create a new RSA private key, you would call

RSA_PrivateKey::RSA_PrivateKey(RandomNumberGenerator &rng, size_t bits)¶

A constructor that creates a new random RSA private key with a modulus of length bits.

RSA key generation is relatively slow, and can take an unpredictable amount of time. Generating a 2048 bit RSA key might take 5 to 10 seconds on a slow machine like a Raspberry Pi 2. Even on a fast desktop it might take up to half a second. In a GUI blocking for that long can be a problem. The usual approach is to perform key generation in a new thread, with a animated modal UI element so the user knows the application is still alive. If you wish to provide a progress estimate things get a bit complicated but some library users documented their approach in a blog post.

Algorithms based on the discrete-logarithm problem use what is called a group; a group can safely be used with many keys, and for some operations, like key agreement, the two keys must use the same group. There are currently two kinds of discrete logarithm groups supported in botan: the integers modulo a prime, represented by DL_Group, and elliptic curves in GF(p), represented by EC_Group. A rough generalization is that the larger the group is, the more secure the algorithm is, but correspondingly the slower the operations will be.

Given a DL_Group, you can create new DSA, Diffie-Hellman and ElGamal key pairs with

DSA_PrivateKey::DSA_PrivateKey(RandomNumberGenerator &rng, const DL_Group &group, const BigInt &x = 0)¶

DH_PrivateKey::DH_PrivateKey(RandomNumberGenerator &rng, const DL_Group &group, const BigInt &x = 0)¶

ElGamal_PrivateKey::ElGamal_PrivateKey(RandomNumberGenerator &rng, const DL_Group &group, const BigInt &x = 0)¶: The optional x parameter to each of these constructors is a private key value. This allows you to create keys where the private key is formed by some special technique; for instance you can use the hash of a password (see Password Based Key Derivation for how to do that) as a private key value. Normally, you would leave the value as zero, letting the class generate a new random key.

Finally, given an EC_Group object, you can create a new ECDSA, ECKCDSA, ECGDSA, ECDH, or GOST 34.10-2001 private key with

ECDSA_PrivateKey::ECDSA_PrivateKey(RandomNumberGenerator &rng, const EC_Group &domain, const BigInt &x = 0)¶

ECKCDSA_PrivateKey::ECKCDSA_PrivateKey(RandomNumberGenerator &rng, const EC_Group &domain, const BigInt &x = 0)¶

ECGDSA_PrivateKey::ECGDSA_PrivateKey(RandomNumberGenerator &rng, const EC_Group &domain, const BigInt &x = 0)¶

ECDH_PrivateKey::ECDH_PrivateKey(RandomNumberGenerator &rng, const EC_Group &domain, const BigInt &x = 0)¶

GOST_3410_PrivateKey::GOST_3410_PrivateKey(RandomNumberGenerator &rng, const EC_Group &domain, const BigInt &x = 0)¶

Serializing Private Keys Using PKCS #8¶

The standard format for serializing a private key is PKCS #8, the operations for which are defined in pkcs8.h. It supports both unencrypted and encrypted storage.

secure_vector<uint8_t> PKCS8::BER_encode(const Private_Key &key, RandomNumberGenerator &rng, const std::string &password, const std::string &pbe_algo = "")¶

Takes any private key object, serializes it, encrypts it using password, and returns a binary structure representing the private key.

The final (optional) argument, pbe_algo, specifies a particular password based encryption (or PBE) algorithm. If you don’t specify a PBE, a sensible default will be used.

The currently supported PBE is PBES2 from PKCS5. Format is as follows: PBE-PKCS5v20(CIPHER,PBKDF) or PBES2(CIPHER,PBKDF).

Cipher can be any block cipher using CBC or GCM modes, for example “AES-128/CBC” or “Camellia-256/GCM”. For best interop with other systems, use AES in CBC mode. The PBKDF can be either the name of a hash function (in which case PBKDF2 is used with that hash) or “Scrypt”, which causes the scrypt memory hard password hashing function to be used. Scrypt is supported since version 2.7.0.

Use PBE-PKCS5v20(AES-256/CBC,SHA-256) if you want to ensure the keys can be imported by different software packages. Use PBE-PKCS5v20(AES-256/GCM,Scrypt) for best security assuming you do not care about interop.

For ciphers you can use anything which has an OID defined for CBC, GCM or SIV modes. Currently this includes AES, Camellia, Serpent, Twofish, and SM4. Most other libraries only support CBC mode for private key encryption. GCM has been supported in PBES2 since 2.0. SIV has been supported since 2.8.

std::string PKCS8::PEM_encode(const Private_Key &key, RandomNumberGenerator &rng, const std::string &pass, const std::string &pbe_algo = "")¶: This formats the key in the same manner as BER_encode, but additionally encodes it into a text format with identifying headers. Using PEM encoding is highly recommended for many reasons, including compatibility with other software, for transmission over 8-bit unclean channels, because it can be identified by a human without special tools, and because it sometimes allows more sane behavior of tools that process the data.

Unencrypted serialization is also supported.

Warning

In most situations, using unencrypted private key storage is a bad idea, because anyone can come along and grab the private key without having to know any passwords or other secrets. Unless you have very particular security requirements, always use the versions that encrypt the key based on a passphrase, described above.

secure_vector<uint8_t> PKCS8::BER_encode(const Private_Key &key)¶: Serializes the private key and returns the result.

std::string PKCS8::PEM_encode(const Private_Key &key)¶: Serializes the private key, base64 encodes it, and returns the result.

Last but not least, there are some functions that will load (and decrypt, if necessary) a PKCS #8 private key:

std::unique_ptr<Private_Key> load_key(DataSource &source, std::function<std::string()> get_passphrase)¶

std::unique_ptr<Private_Key> load_key(DataSource &source, const std::string &pass)¶

std::unique_ptr<Private_Key> load_key(DataSource &source)¶

These functions will return an object allocated key object based on the data from whatever source it is using (assuming, of course, the source is in fact storing a representation of a private key, and the decryption was successful). The encoding used (PEM or BER) need not be specified; the format will be detected automatically. The DataSource is usually a DataSource_Stream to read from a file or DataSource_Memory for an in-memory buffer.

The versions taking a std::string attempt to decrypt using the password given (if the key is encrypted; if it is not, the passphase value will be ignored). If the passphrase does not decrypt the key, an exception will be thrown.

Serializing Public Keys¶

To import and export public keys, use:

std::vector<uint8_t> X509::BER_encode(const Public_Key &key)¶

std::string X509::PEM_encode(const Public_Key &key)¶

std::unique_ptr<Public_Key> X509::load_key(DataSource &in)¶

std::unique_ptr<Public_Key> X509::load_key(const secure_vector<uint8_t> &buffer)¶

std::unique_ptr<Public_Key> X509::load_key(const std::string &filename)¶: These functions operate in the same way as the ones described in Serializing Private Keys Using PKCS #8, except that no encryption option is available.

Note

In versions prior to 3.0, these functions returned a raw pointer instead of a unique_ptr.

DL_Group¶

As described in Creating New Private Keys, a discrete logarithm group can be shared among many keys, even keys created by users who do not trust each other. However, it is necessary to trust the entity who created the group; that is why organization like NIST use algorithms which generate groups in a deterministic way such that creating a bogus group would require breaking some trusted cryptographic primitive like SHA-2.

Instantiating a DL_Group simply requires calling

DL_Group::DL_Group(const std::string &name)¶

The name parameter is a specially formatted string that consists of three things, the type of the group (“modp” or “dsa”), the creator of the group, and the size of the group in bits, all delimited by ‘/’ characters.

Currently all “modp” groups included in botan are ones defined by the Internet Engineering Task Force, so the provider is “ietf”, and the strings look like “modp/ietf/N” where N can be any of 1024, 1536, 2048, 3072, 4096, 6144, or 8192. This group type is used for Diffie-Hellman and ElGamal algorithms.

The other type, “dsa” is used for DSA keys. They can also be used with Diffie-Hellman and ElGamal, but this is less common. The currently available groups are “dsa/jce/1024” and “dsa/botan/N” with N being 2048 or 3072. The “jce” groups are the standard DSA groups used in the Java Cryptography Extensions, while the “botan” groups were randomly generated using the FIPS 186-3 algorithm by the library maintainers.

You can generate a new random group using

DL_Group::DL_Group(RandomNumberGenerator &rng, PrimeType type, size_t pbits, size_t qbits = 0)¶: The type can be either Strong, Prime_Subgroup, or DSA_Kosherizer. pbits specifies the size of the prime in bits. If the type is Prime_Subgroup or DSA_Kosherizer, then qbits specifies the size of the subgroup.

You can serialize a DL_Group using

secure_vector<uint8_t> DL_Group::DER_Encode(Format format)¶

std::string DL_Group::PEM_encode(Format format)¶

where format is any of

ANSI_X9_42 (or DH_PARAMETERS) for modp groups
ANSI_X9_57 (or DSA_PARAMETERS) for DSA-style groups
PKCS_3 is an older format for modp groups; it should only be used for backwards compatibility.

You can reload a serialized group using

void DL_Group::BER_decode(DataSource &source, Format format)¶

void DL_Group::PEM_decode(DataSource &source)¶

Code Example: DL_Group¶

The example below creates a new 2048 bit DL_Group, prints the generated parameters and ANSI_X9_42 encodes the created group for further usage with DH.

#include <botan/auto_rng.h>
#include <botan/dl_group.h>
#include <botan/rng.h>

#include <iostream>

int main() {
   Botan::AutoSeeded_RNG rng;
   auto group = std::make_unique<Botan::DL_Group>(rng, Botan::DL_Group::Strong, 2048);

   std::cout << "P = " << group->get_p().to_hex_string() << "\n"
             << "Q = " << group->get_q().to_hex_string() << "\n"
             << "G = " << group->get_g().to_hex_string() << "\n";

   std::cout << "\nPEM:\n" << group->PEM_encode(Botan::DL_Group_Format::ANSI_X9_42) << "\n";

   return 0;
}

EC_Group¶

An EC_Group is initialized by passing the name of the group to be used to the constructor. These groups have semi-standardized names like “secp256r1” and “brainpool512r1”.

Key Checking¶

Most public key algorithms have limitations or restrictions on their parameters. For example RSA requires an odd exponent, and algorithms based on the discrete logarithm problem need a generator > 1.

Each public key type has a function

bool Public_Key::check_key(RandomNumberGenerator &rng, bool strong)¶

This function performs a number of algorithm-specific tests that the key seems to be mathematically valid and consistent, and returns true if all of the tests pass.

It does not have anything to do with the validity of the key for any particular use, nor does it have anything to do with certificates that link a key (which, after all, is just some numbers) with a user or other entity. If strong is true, then it does “strong” checking, which includes expensive operations like primality checking.

As key checks are not automatically performed they must be called manually after loading keys from untrusted sources. If a key from an untrusted source is not checked, the implementation might be vulnerable to algorithm specific attacks.

The following example loads the Subject Public Key from the x509 certificate cert.pem and checks the loaded key. If the key check fails a respective error is thrown.

#include <botan/auto_rng.h>
#include <botan/pk_keys.h>
#include <botan/rng.h>
#include <botan/x509cert.h>
#include <iostream>

int main() {
   Botan::X509_Certificate cert("cert.pem");
   Botan::AutoSeeded_RNG rng;
   auto key = cert.subject_public_key();
   if(!key->check_key(rng, false)) {
      std::cerr << "Loaded key is invalid";
      return 1;
   }

   return 0;
}

Public Key Encryption/Decryption¶

Safe public key encryption requires the use of a padding scheme which hides the underlying mathematical properties of the algorithm. Additionally, they will add randomness, so encrypting the same plaintext twice produces two different ciphertexts.

The primary interface for encryption is

class PK_Encryptor¶

std::vector<uint8_t> encrypt(const uint8_t in[], size_t length, RandomNumberGenerator &rng) const¶

std::vector<uint8_t> encrypt(std::span<const uint8_t> in, RandomNumberGenerator &rng) const¶: These encrypt a message, returning the ciphertext.

size_t maximum_input_size() const¶: Returns the maximum size of the message that can be processed, in bytes. If you call PK_Encryptor::encrypt with a value larger than this the operation will fail with an exception.

size_t ciphertext_length(size_t ctext_len) const¶: Return an upper bound on the returned size of a ciphertext, if this particular key/padding scheme is used to encrypt a message of the provided length.

PK_Encryptor is only an interface - to actually encrypt you have to create an implementation, of which there are currently three available in the library, PK_Encryptor_EME, DLIES_Encryptor and ECIES_Encryptor. DLIES is a hybrid encryption scheme (from IEEE 1363) that uses Diffie-Hellman key agreement technique in combination with a KDF, a MAC and a symmetric encryption algorithm to perform message encryption. ECIES is similar to DLIES, but uses ECDH for the key agreement. Normally, public key encryption is done using algorithms which support it directly, such as RSA or ElGamal; these use the EME class:

class PK_Encryptor_EME¶

PK_Encryptor_EME(const Public_Key &key, std::string padding)¶

With key being the key you want to encrypt messages to. The padding method to use is specified in padding.

If you are not sure what padding to use, use “OAEP(SHA-256)”. If you need compatibility with protocols using the PKCS #1 v1.5 standard, you can also use “EME-PKCS1-v1_5”.

class DLIES_Encryptor¶

Available in the header dlies.h

DLIES_Encryptor(const DH_PrivateKey &own_priv_key, RandomNumberGenerator &rng, std::unique_ptr<KDF> kdf, std::unique_ptr<MessageAuthenticationCode> mac, size_t mac_key_len = 20)¶: Where kdf is a key derivation function (see Key Derivation Functions (KDF)) and mac is a MessageAuthenticationCode. The encryption is performed by XORing the message with a stream of bytes provided by the KDF.

DLIES_Encryptor(const DH_PrivateKey &own_priv_key, RandomNumberGenerator &rng, std::unique_ptr<KDF> kdf, std::unique_ptr<Cipher_Mode> cipher, size_t cipher_key_len, std::unique_ptr<MessageAuthenticationCode> mac, size_t mac_key_len = 20)¶: Instead of XORing the message with KDF output, a cipher mode can be used

class ECIES_Encryptor¶

Available in the header ecies.h.

Parameters for encryption and decryption are set by the ECIES_System_Params class which stores the EC domain parameters, the KDF (see Key Derivation Functions (KDF)), the cipher (see Cipher Modes) and the MAC.

ECIES_Encryptor(const PK_Key_Agreement_Key &private_key, const ECIES_System_Params &ecies_params, RandomNumberGenerator &rng)¶: Where private_key is the key to use for the key agreement. The system parameters are specified in ecies_params and the RNG to use is passed in rng.

ECIES_Encryptor(RandomNumberGenerator &rng, const ECIES_System_Params &ecies_params)¶: Creates an ephemeral private key which is used for the key agreement.

The decryption classes are named PK_Decryptor, PK_Decryptor_EME, DLIES_Decryptor and ECIES_Decryptor. They are created in the exact same way, except they take the private key, and the processing function is named decrypt.

Botan implements the following encryption algorithms:

RSA. Requires a padding scheme as parameter.
DLIES
ECIES
SM2. Takes an optional HashFunction as parameter which defaults to SM3.
ElGamal. Requires a padding scheme as parameter.

Code Example: RSA Encryption¶

The following code sample reads a PKCS #8 keypair from the passed location and subsequently encrypts a fixed plaintext with the included public key, using OAEP with SHA-256. For the sake of completeness, the ciphertext is then decrypted using the private key.

#include <botan/auto_rng.h>
#include <botan/hex.h>
#include <botan/pk_keys.h>
#include <botan/pkcs8.h>
#include <botan/pubkey.h>
#include <botan/rng.h>

#include <iostream>

int main(int argc, char* argv[]) {
   if(argc != 2) {
      return 1;
   }
   std::string plaintext(
      "Your great-grandfather gave this watch to your granddad for good luck. "
      "Unfortunately, Dane's luck wasn't as good as his old man's.");
   std::vector<uint8_t> pt(plaintext.data(), plaintext.data() + plaintext.length());
   Botan::AutoSeeded_RNG rng;

   // load keypair
   Botan::DataSource_Stream in(argv[1]);
   auto kp = Botan::PKCS8::load_key(in);

   // encrypt with pk
   Botan::PK_Encryptor_EME enc(*kp, rng, "OAEP(SHA-256)");
   std::vector<uint8_t> ct = enc.encrypt(pt, rng);

   // decrypt with sk
   Botan::PK_Decryptor_EME dec(*kp, rng, "OAEP(SHA-256)");
   Botan::secure_vector<uint8_t> pt2 = dec.decrypt(ct);

   std::cout << "\nenc: " << Botan::hex_encode(ct) << "\ndec: " << Botan::hex_encode(pt2);

   return 0;
}

Available encryption padding schemes¶

Note

Padding schemes in the context of encryption are sometimes also called Encoding Method for Encryption (EME).

OAEP¶

OAEP (called EME1 in IEEE 1363 and in earlier versions of the library) as specified in PKCS#1 v2.0 (RFC 2437) or PKCS#1 v2.1 (RFC 3447).

Name: OAEP,
Deprecated aliases: EME-OAEP, EME1
Parameters specification:
- (<HashFunction>)
- (<HashFunction>,MGF1)
- (<HashFunction>,MGF1(<HashFunction>))
- (<HashFunction>,MGF1(<HashFunction>),<optional label>)
The only Mask generation function available is MGF1, which is also the default.
By default the same hash function will be used for the label and MGF1.
By default the OAEP label is the empty string
Examples: OAEP(SHA-256), OAEP(SHA-256,MGF1), OAEP(SHA-256,MGF1(SHA-512)), OAEP(SHA-512,MGF1(SHA-512),TCPA)

PKCS #1 v1.5 Type 2 (encryption)¶

PKCS #1 v1.5 Type 2 (encryption) padding.

Name: PKCS1v15 Deprecated alias: EME-PKCS1-v1_5

Raw EME¶

Does not change the input during padding. Don’t use this unless you know what you are doing. Un-padding will strip leading zeros.

Name: Raw

Public Key Signature Schemes¶

Signature generation is performed using

class PK_Signer¶

PK_Signer(const Private_Key &key, const std::string &padding, Signature_Format format = Siganture_Format::Standard)¶

Constructs a new signer object for the private key key using the hash/padding specified in padding. The key must support signature operations. In the current version of the library, this includes e.g. RSA, ECDSA, Dilithium, ECKCDSA, ECGDSA, GOST 34.10-2001, and SM2.

Note

Botan both supports non-deterministic and deterministic (as per RFC 6979) DSA and ECDSA signatures. Either type of signature can be verified by any other (EC)DSA library, regardless of which mode it prefers. If the rfc6979 module is enabled at build time, deterministic DSA and ECDSA signatures will be created.

The proper value of padding depends on the algorithm. For many signature schemes including ECDSA and DSA, simply naming a hash function like “SHA-256” is all that is required.

For RSA, more complicated padding is required. The two most common schemes for RSA signature padding are PSS and PKCS1v1.5, so you must specify both the padding mechanism as well as a hash, for example “PSS(SHA-256)” or “PKCS1v15(SHA-256)”.

Certain newer signature schemes, especially post-quantum based ones, hardcode the hash function associated with their signatures, and no configuration is possible. There padding should be left blank, or may possibly be used to identify some algorithm-specific option. For instance Dilithium may be parameterized with “Randomized” or “Deterministic” to choose if the generated signature is randomized or not. If left blank, a default is chosen.

Another available option, usable in certain specialized scenarios, is using padding scheme “Raw”, where the provided input is treated as if it was already hashed, and directly signed with no other processing.

The format defaults to Standard which is either the usual, or the only, available formatting method, depending on the algorithm. For certain signature schemes including ECDSA, DSA, ECGDSA and ECKCDSA you can also use DerSequence, which will format the signature as an ASN.1 SEQUENCE value. This formatting is used in protocols such as TLS and Bitcoin.

void update(const uint8_t *in, size_t length)¶

void update(std::span<const uint8_t> in)¶

void update(uint8_t in)¶: These add more data to be included in the signature computation. Typically, the input will be provided directly to a hash function.

std::vector<uint8_t> signature(RandomNumberGenerator &rng)¶: Creates the signature and returns it. The rng may or may not be used, depending on the scheme.

std::vector<uint8_t> sign_message(const uint8_t *in, size_t length, RandomNumberGenerator &rng)¶

std::vector<uint8_t> sign_message(std::span<const uint8_t> in, RandomNumberGenerator &rng)¶: These functions are equivalent to calling PK_Signer::update and then PK_Signer::signature. Any data previously provided using update will also be included in the signature.

size_t signature_length() const¶: Return an upper bound on the length of the signatures returned by this object.

AlgorithmIdentifier algorithm_identifier() const¶: Return an algorithm identifier appropriate to identify signatures generated by this object in an X.509 structure.

std::string hash_function() const¶: Return the hash function which is being used

Signatures are verified using

class PK_Verifier¶

PK_Verifier(const Public_Key &pub_key, const std::string &padding, Signature_Format format = Signature_Format::Standard)¶: Construct a new verifier for signatures associated with public key pub_key. The padding and format should be the same as that used by the signer.

void update(const uint8_t *in, size_t length)¶

void update(std::span<const uint8_t> in)¶

void update(uint8_t in)¶: Add further message data that is purportedly associated with the signature that will be checked.

bool check_signature(const uint8_t *sig, size_t length)¶

bool check_signature(std::span<const uint8_t> sig)¶: Check to see if sig is a valid signature for the message data that was written in. Return true if so. This function clears the internal message state, so after this call you can call PK_Verifier::update to start verifying another message.

bool verify_message(const uint8_t *msg, size_t msg_length, const uint8_t *sig, size_t sig_length)¶

bool verify_message(std::span<const uint8_t> msg, std::span<const uint8_t> sig)¶: These are equivalent to calling PK_Verifier::update on msg and then calling PK_Verifier::check_signature on sig. Any data previously provided to PK_Verifier::update will also be included.

Botan implements the following signature algorithms:

RSA. Requires a padding scheme as parameter.
DSA. Requires a hash function as parameter.
ECDSA. Requires a hash function as parameter.
ECGDSA. Requires a hash function as parameter.
ECKDSA. Requires a hash function as parameter, not supporting Raw.
GOST 34.10-2001. Requires a hash function as parameter.
Ed25519 and Ed448. See Ed25519 and Ed448 Variants for parameters.
SM2. Takes one of the following as parameter:
- <user ID> (uses SM3)
- <user ID>,<HashFunction>
ML-DSA (Dilithium). Takes the optional parameter Deterministic (default) or Randomized.
SLH-DSA. Takes the optional parameter Deterministic (default) or Randomized.
XMSS. Takes no parameter.
HSS-LMS. Takes no parameter.

Code Example: ECDSA Signature¶

The following sample program below demonstrates the generation of a new ECDSA keypair over the curve secp512r1 and a ECDSA signature using SHA-256. Subsequently the computed signature is validated.

#include <botan/auto_rng.h>
#include <botan/ec_group.h>
#include <botan/ecdsa.h>
#include <botan/hex.h>
#include <botan/pubkey.h>

#include <iostream>

int main() {
   Botan::AutoSeeded_RNG rng;
   // Generate ECDSA keypair
   const auto group = Botan::EC_Group::from_name("secp521r1");
   Botan::ECDSA_PrivateKey key(rng, group);

   const std::string message("This is a tasty burger!");

   // sign data
   Botan::PK_Signer signer(key, rng, "SHA-256");
   signer.update(message);
   std::vector<uint8_t> signature = signer.signature(rng);
   std::cout << "Signature:\n" << Botan::hex_encode(signature);

   // now verify the signature
   Botan::PK_Verifier verifier(key, "SHA-256");
   verifier.update(message);
   std::cout << "\nis " << (verifier.check_signature(signature) ? "valid" : "invalid");
   return 0;
}

Available signature padding schemes¶

Note

Padding schemes in the context of signatures are sometimes also called Encoding methods for signatures with appendix (EMSA).

PKCS #1 v1.5 Type 1 (signature)¶

PKCS #1 v1.5 Type 1 (signature) padding, aka EMSA3 in IEEE 1363.

Name: PKCS1v15
Deprecated aliases: EMSA_PKCS1, EMSA-PKCS1-v1_5, EMSA3
Parameters specification:
- (<HashFunction>)
- (Raw,<optional HashFunction>)
The raw variant encodes a precomputed hash, optionally with the digest ID of the given hash.
Examples: PKCS1v15(SHA-256), PKCS1v15(Raw), PKCS1v15(Raw,MD5),

EMSA-PSS¶

Probabilistic signature scheme (PSS) (called EMSA4 in IEEE 1363).

Name: PSS
Deprecated aliases: EMSA-PSS, PSSR, PSS-MGF1, EMSA4
Parameters specification:
- (<HashFunction>)
- (<HashFunction>,MGF1,<optional salt size>)
Examples: PSS(SHA-256), PSS(SHA-256,MGF1,32),

There also exists a raw version, which accepts a pre-hashed buffer instead of the message. Don’t use this unless you know what you are doing.

Name: PSS_Raw
Deprecated alias: PSSR_Raw
Parameters specification:
- (<HashFunction>)
- (<HashFunction>,MGF1,<optional salt size>)

ISO-9796-2¶

ISO-9796-2 - Digital signature scheme 2 (probabilistic).

Name: ISO_9796_DS2
Parameters specification:
- (<HashFunction>)
- (<HashFunction>,<exp|imp>,<optional salt size>)
Defaults to the explicit mode.
Examples: ISO_9796_DS2(RIPEMD-160), ISO_9796_DS2(RIPEMD-160,imp)

ISO-9796-2 - Digital signature scheme 3 (deterministic), i.e. DS2 without a salt.

Name: ISO_9796_DS3
Parameters specification:
- (<HashFunction>)
- (<HashFunction>,<exp|imp>
Defaults to the explicit mode.
Examples: ISO_9796_DS3(RIPEMD-160), ISO_9796_DS3(RIPEMD-160,imp),

X9.31¶

EMSA from X9.31 (EMSA2 in IEEE 1363).

Name: X9.31
Deprecated aliases: EMSA2, EMSA_X931
Parameters specification: (<HashFunction>)
Example: X9.31(SHA-256)

Raw EMSA¶

Sign inputs directly. Don’t use this unless you know what you are doing.

Name: Raw
Parameters specification: (<optional HashFunction>)
Examples: Raw, Raw(SHA-256)

Signature with Hash¶

For many signature schemes including ECDSA and DSA, simply naming a hash function like SHA-256 is all that is required.

Previous versions of Botan required using a hash specifier like EMSA1(SHA-256) when generating or verifying ECDSA/DSA signatures, with the specified hash. The EMSA1 was a reference to a now obsolete IEEE standard.

Parameters specification:

<HashFunction>
EMSA1(<HashFunction>)

There also exists a raw mode, which accepts a pre-hashed buffer instead of the message. Don’t use this unless you know what you are doing.

Parameters specification:

Raw
Raw(<HashFunction>)

Ed25519 and Ed448 Variants¶

Most signature schemes in Botan follow a hash-then-sign paradigm. That is, the entire message is digested to a fixed length representative using a collision resistant hash function, and then the digest is signed. Ed25519 and Ed448 instead sign the message directly. This is beneficial, in that the design should remain secure even in the (extremely unlikely) event that a collision attack on SHA-512 is found. However it means the entire message must be buffered in memory, which can be a problem for many applications which might need to sign large inputs. To use this variety of Ed25519/Ed448, use a padding name of “Pure”.

This is the default mode if no padding name is given.

Parameter specification: Pure / Identity

Ed25519ph (or Ed448) (pre-hashed) instead hashes the message with SHA-512 (or SHAKE256(512)) and then signs the digest plus a special prefix specified in RFC 8032. To use it, specify padding name “Ed25519ph” (or “Ed448ph”).

Parameter specification: Ed25519ph

Another variant of pre-hashing is used by GnuPG. There the message is digested with any hash function, then the digest is signed. To use it, specify any valid hash function. Even if SHA-512 is used, this variant is not compatible with Ed25519ph.

Parameter specification: <HashFunction>

For best interop with other systems, prefer “Ed25519ph”.

Key Agreement¶

Key agreement is a scheme where two parties exchange public keys, after which it is possible for them to derive a secret key which is known only to the two of them.

There are different approaches possible for key agreement. In many protocols, both parties generate a new key, exchange public keys, and derive a secret, after which they throw away their private keys, using them only the once. However this requires the parties to both be online and able to communicate with each other.

In other protocols, one of the parties publishes their public key online in some way, and then it is possible for someone to send encrypted messages to that recipient by generating a new keypair, performing key exchange with the published public key, and then sending both the message along with their ephemeral public key. Then the recipient uses the provided public key along with their private key to complete the key exchange, recover the shared secret, and decrypt the message.

Typically the raw output of the key agreement function is not uniformly distributed, and may not be of an appropriate length to use as a key. To resolve these problems, key agreement will use a Key Derivation Functions (KDF) on the shared secret to produce an output of the desired length.

ECDH over GF(p) Weierstrass curves
ECDH over x25519
DH over prime fields

class PK_Key_Agreement¶

PK_Key_Agreement(const Private_Key &key, RandomNumberGenerator &rng, const std::string &kdf, const std::string &provider = "")¶: Set up to perform key derivation using the given private key and specified KDF.

SymmetricKey derive_key(size_t key_len, const uint8_t in[], size_t in_len, const uint8_t params[], size_t params_len) const¶

SymmetricKey derive_key(size_t key_len, std::span<const uint8_t> in, const uint8_t params[], size_t params_len) const¶

SymmetricKey derive_key(size_t key_len, const uint8_t in[], size_t in_len, const std::string &params = "") const¶

SymmetricKey derive_key(size_t key_len, const std::span<const uint8_t> in, const std::string &params = "") const¶

Return a shared key. The params will be hashed along with the shared secret by the KDF; this can be useful to bind the shared secret to a specific usage.

The in parameter must be the public key associated with the other party.

Code Example: ECDH Key Agreement¶

The code below performs an unauthenticated ECDH key agreement using the secp521r elliptic curve and applies the key derivation function KDF2(SHA-256) with 256 bit output length to the computed shared secret.

#include <botan/auto_rng.h>
#include <botan/ec_group.h>
#include <botan/ecdh.h>
#include <botan/hex.h>
#include <botan/pubkey.h>

#include <iostream>

int main() {
   Botan::AutoSeeded_RNG rng;

   // ec domain and KDF
   const auto domain = Botan::EC_Group::from_name("secp521r1");
   const std::string kdf = "KDF2(SHA-256)";

   // the two parties generate ECDH keys
   Botan::ECDH_PrivateKey key_a(rng, domain);
   Botan::ECDH_PrivateKey key_b(rng, domain);

   // now they exchange their public values
   const auto key_apub = key_a.public_value();
   const auto key_bpub = key_b.public_value();

   // Construct key agreements and agree on a shared secret
   Botan::PK_Key_Agreement ka_a(key_a, rng, kdf);
   const auto sA = ka_a.derive_key(32, key_bpub).bits_of();

   Botan::PK_Key_Agreement ka_b(key_b, rng, kdf);
   const auto sB = ka_b.derive_key(32, key_apub).bits_of();

   if(sA != sB) {
      return 1;
   }

   std::cout << "agreed key:\n" << Botan::hex_encode(sA);
   return 0;
}

Key Encapsulation¶

Key encapsulation (KEM) is a variation on public key encryption which is commonly used by post-quantum secure schemes. Instead of choosing a random secret and encrypting it, as in typical public key encryption, a KEM encryption takes no inputs and produces two values, the shared secret and the encapsulated key. The decryption operation takes in the encapsulated key and returns the shared secret.

class PK_KEM_Encryptor¶

PK_KEM_Encryptor(const Public_Key &key, const std::string &kdf = "", const std::string &provider = "")¶: Create a KEM encryptor

size_t shared_key_length(size_t desired_shared_key_len) const¶: Size in bytes of the shared key being produced by this PK_KEM_Encryptor.

size_t encapsulated_key_length() const¶: Size in bytes of the encapsulated key being produced by this PK_KEM_Encryptor.

KEM_Encapsulation encrypt(RandomNumberGenerator &rng, size_t desired_shared_key_len = 32, std::span<const uint8_t> salt = {})¶: Perform a key encapsulation operation with the result being returned as a convenient struct.

void encrypt(std::span<uint8_t> out_encapsulated_key, std::span<uint8_t> out_shared_key, RandomNumberGenerator &rng, size_t desired_shared_key_len = 32, std::span<const uint8_t> salt = {})¶: Perform a key encapsulation operation by passing in out-buffers of the correct output length. Use encapsulated_key_length() and shared_key_length() to pre-allocate the output buffers.

void encrypt(secure_vector<uint8_t> &out_encapsulated_key, secure_vector<uint8_t> &out_shared_key, size_t desired_shared_key_len, RandomNumberGenerator &rng, std::span<const uint8_t> salt)¶: Perform a key encapsulation operation by passing in out-vectors that will be re-allocated to the correct output size.

class KEM_Encapsulation¶

std::vector<uint8_t> encapsulated_shared_key() const¶

secure_vector<uint8_t> shared_key() const¶

class PK_KEM_Decryptor¶

PK_KEM_Decryptor(const Public_Key &key, const std::string &kdf = "", const std::string &provider = "")¶: Create a KEM decryptor

size_t encapsulated_key_length() const¶: Size in bytes of the encapsulated key expected by this PK_KEM_Decryptor.

size_t shared_key_length(size_t desired_shared_key_len) const¶: Size in bytes of the shared key being produced by this PK_KEM_Encryptor.

secure_vector<uint8> decrypt(std::span<const uint8> encapsulated_key, size_t desired_shared_key_len, std::span<const uint8_t> salt)¶: Perform a key decapsulation operation

void decrypt(std::span<uint8_t> out_shared_key, std::span<const uint8_t> encap_key, size_t desired_shared_key_len = 32, std::span<const uint8_t> salt = {})¶: Perform a key decapsulation operation by passing in a pre-allocated out-buffer. Use shared_key_length() to determine the byte-length required.

Botan implements the following KEM schemes:

RSA
ML-KEM (Kyber)
FrodoKEM
McEliece

Code Example: ML-KEM¶

The code below demonstrates key encapsulation using ML-KEM (FIPS 203), formerly known as Kyber.

#include <botan/ml_kem.h>
#include <botan/pubkey.h>
#include <botan/system_rng.h>
#include <array>
#include <iostream>

int main() {
   const size_t shared_key_len = 32;
   const std::string kdf = "HKDF(SHA-512)";

   Botan::System_RNG rng;

   std::array<uint8_t, 16> salt;
   rng.randomize(salt);

   Botan::ML_KEM_PrivateKey priv_key(rng, Botan::ML_KEM_Mode::ML_KEM_768);
   auto pub_key = priv_key.public_key();

   Botan::PK_KEM_Encryptor enc(*pub_key, kdf);

   const auto kem_result = enc.encrypt(rng, shared_key_len, salt);

   Botan::PK_KEM_Decryptor dec(priv_key, rng, kdf);

   auto dec_shared_key = dec.decrypt(kem_result.encapsulated_shared_key(), shared_key_len, salt);

   if(dec_shared_key != kem_result.shared_key()) {
      std::cerr << "Shared keys differ\n";
      return 1;
   }

   return 0;
}

McEliece cryptosystem¶

McEliece is a cryptographic scheme based on error correcting codes which is thought to be resistant to quantum computers. First proposed in 1978, it is fast and patent-free. Variants have been proposed and broken, but with suitable parameters the original scheme remains secure. However the public keys are quite large, which has hindered deployment in the past.

The implementation of McEliece in Botan was contributed by cryptosource GmbH. It is based on the implementation HyMES, with the kind permission of Nicolas Sendrier and INRIA to release a C++ adaption of their original C code under the Botan license. It was then modified by Falko Strenzke to add side channel and fault attack countermeasures. You can read more about the implementation at http://www.cryptosource.de/docs/mceliece_in_botan.pdf

Encryption in the McEliece scheme consists of choosing a message block of size n, encoding it in the error correcting code which is the public key, then adding t bit errors. The code is created such that knowing only the public key, decoding t errors is intractable, but with the additional knowledge of the secret structure of the code a fast decoding technique exists.

The McEliece implementation in HyMES, and also in Botan, uses an optimization to reduce the public key size, by converting the public key into a systemic code. This means a portion of the public key is a identity matrix, and can be excluded from the published public key. However it also means that in McEliece the plaintext is represented directly in the ciphertext, with only a small number of bit errors. Thus it is absolutely essential to only use McEliece with a CCA2 secure scheme.

For a given security level (SL) a McEliece key would use parameters n and t, and have the corresponding key sizes listed:

SL	n	t	public key KB	private key KB
80	1632	33	59	140
107	2280	45	128	300
128	2960	57	195	459
147	3408	67	265	622
191	4624	95	516	1234
256	6624	115	942	2184

You can check the speed of McEliece with the suggested parameters above using botan speed McEliece

eXtended Merkle Signature Scheme (XMSS)¶

Botan implements the single tree version of the eXtended Merkle Signature Scheme (XMSS) using Winternitz One Time Signatures+ (WOTS+). The implementation is based on RFC 8391 “XMSS: eXtended Merkle Signature Scheme”.

Warning

XMSS is stateful, meaning the private key updates after each signature creation. Applications are responsible for updating their persistent secret with the new output of Private_Key::private_key_bits() after each signature creation. If the same private key is ever used to generate two different signatures, then the scheme becomes insecure. For this reason, it can be challenging to use XMSS securely.

XMSS uses the Botan interfaces for public key cryptography. The following algorithms are implemented:

XMSS-SHA2_10_256
XMSS-SHA2_16_256
XMSS-SHA2_20_256
XMSS-SHA2_10_512
XMSS-SHA2_16_512
XMSS-SHA2_20_512
XMSS-SHAKE_10_256
XMSS-SHAKE_16_256
XMSS-SHAKE_20_256
XMSS-SHAKE_10_512
XMSS-SHAKE_16_512
XMSS-SHAKE_20_512

The algorithm name contains the hash function name, tree height and digest width defined by the corresponding parameter set. Choosing XMSS-SHA2_10_256 for instance will use the SHA2-256 hash function to generate a tree of height ten.

Code Example: XMSS¶

The following code snippet shows a minimum example on how to create an XMSS public/private key pair and how to use these keys to create and verify a signature:

#include <botan/auto_rng.h>
#include <botan/pubkey.h>
#include <botan/secmem.h>
#include <botan/xmss.h>

#include <iostream>
#include <vector>

int main() {
   // Create a random number generator used for key generation.
   Botan::AutoSeeded_RNG rng;

   // create a new public/private key pair using SHA2 256 as hash
   // function and a tree height of 10.
   Botan::XMSS_PrivateKey private_key(Botan::XMSS_Parameters::xmss_algorithm_t::XMSS_SHA2_10_256, rng);
   const Botan::XMSS_PublicKey& public_key(private_key);

   // create Public Key Signer using the private key.
   Botan::PK_Signer signer(private_key, rng, "");

   // create and sign a message using the Public Key Signer.
   Botan::secure_vector<uint8_t> msg{0x01, 0x02, 0x03, 0x04};
   signer.update(msg.data(), msg.size());
   std::vector<uint8_t> sig = signer.signature(rng);

   // create Public Key Verifier using the public key
   Botan::PK_Verifier verifier(public_key, "");

   // verify the signature for the previously generated message.
   verifier.update(msg.data(), msg.size());
   if(verifier.check_signature(sig.data(), sig.size())) {
      std::cout << "Success.\n";
      return 0;
   } else {
      std::cout << "Error.\n";
      return 1;
   }
}

Hierarchical Signature System with Leighton-Micali Hash-Based Signatures (HSS-LMS)¶

HSS-LMS is a stateful hash-based signature scheme which is defined in RFC 8554 “Leighton-Micali Hash-Based Signatures”.

It is a multitree scheme, which is highly configurable. Multitree means, it consists of multiple layers of Merkle trees, which can be defined individually. Moreover, the used hash function and the Winternitz Parameter of the underlying one-time signature can be chosen for each tree layer. For a sensible selection of parameters refer to RFC 8554 Section 6.4..

Warning

HSS-LMS is stateful, meaning the private key updates after each signature creation. Applications are responsible for updating their persistent secret with the new output of Private_Key::private_key_bits() after each signature creation. If the same private key is ever used to generate two different signatures, then the scheme becomes insecure. For this reason, it can be challenging to use HSS-LMS securely.

HSS-LMS uses the Botan interfaces for public key cryptography. The params argument of the HSS-LMS private key is used to define the parameter set. The syntax of this argument must be the following:

HSS-LMS(<hash>,HW(<h>,<w>),HW(<h>,<w>),...)

e.g. HSS-LMS(SHA-256,HW(5,1),HW(5,1)) to use SHA-256 in a two-layer HSS instance with LMS tree height 5 and Winternitz parameter 1. This results in a private key that can be used to create up to 2^(5+5)=1024 signatures.

The following parameters are allowed (which are specified in RFC 8554 and and draft-fluhrer-lms-more-parm-sets-11):

hash: SHA-256, Truncated(SHA-256,192), SHAKE-256(256), SHAKE-256(192)
h: 5, 10, 15, 20, 25
w: 1, 2, 4, 8

Botan

Public Key Cryptography¶

Key Objects¶

Public Key Algorithms¶

RSA¶

ECDSA¶

ECDH, DH, X25519 and X448¶

ML-DSA (FIPS 204)¶

ML-KEM (FIPS 203)¶

Ed25519 and Ed448¶

XMSS¶

HSS-LMS¶

SLH-DSA (FIPS 205)¶

FrodoKEM¶

McEliece¶

ElGamal¶

DSA¶

ECGDSA, ECKCDSA, SM2, GOST-34.10¶

Creating New Private Keys¶

Serializing Private Keys Using PKCS #8¶

Serializing Public Keys¶

DL_Group¶

Code Example: DL_Group¶

EC_Group¶

Key Checking¶

Public Key Encryption/Decryption¶

Code Example: RSA Encryption¶

Available encryption padding schemes¶

OAEP¶

PKCS #1 v1.5 Type 2 (encryption)¶

Raw EME¶

Public Key Signature Schemes¶

Code Example: ECDSA Signature¶

Available signature padding schemes¶

PKCS #1 v1.5 Type 1 (signature)¶

EMSA-PSS¶

ISO-9796-2¶

X9.31¶

Raw EMSA¶

Signature with Hash¶

Ed25519 and Ed448 Variants¶

Key Agreement¶

Code Example: ECDH Key Agreement¶

Key Encapsulation¶

Code Example: ML-KEM¶

McEliece cryptosystem¶

eXtended Merkle Signature Scheme (XMSS)¶

Code Example: XMSS¶

Hierarchical Signature System with Leighton-Micali Hash-Based Signatures (HSS-LMS)¶

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