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- Elliptic Curve Cryptography Generate Public Key From Private Key
- Elliptic Curve Cryptography Vs Rsa
- Elliptic Curve Cryptography Generate Public Keys
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Public Key Cryptography
Feb 22, 2012 The fig 3 show are simple elliptic curve. Key Generation. Key generation is an important part where we have to generate both public key and private key. The sender will be encrypting the message with receiver’s public key and the receiver will decrypt its private key. Now, we have to select a number ‘d’ within the range of ‘n’. Welcome to part four in our series on Elliptic Curve Cryptography. In this episode we dive into the development of the public key. In just 44 lines of code, with no special functions or imports, we produce the elliptic curve public key for use in Bitcoin. Better still, we walk you through it line by line, constant by constant. Nothing makes the process clearer and easier to understand than. Jun 08, 2014 In just 44 lines of code, with no special functions or imports, we produce the elliptic curve public key for use in Bitcoin. Better still, we walk you through it line by line, constant by constant.
Unlike symmetric key cryptography, we do not find historical use of public-key cryptography. It is a relatively new concept.
Symmetric cryptography was well suited for organizations such as governments, military, and big financial corporations were involved in the classified communication.
With the spread of more unsecure computer networks in last few decades, a genuine need was felt to use cryptography at larger scale. The symmetric key was found to be non-practical due to challenges it faced for key management. This gave rise to the public key cryptosystems.
The process of encryption and decryption is depicted in the following illustration −
The most important properties of public key encryption scheme are −
- Different keys are used for encryption and decryption. This is a property which set this scheme different than symmetric encryption scheme.
- Each receiver possesses a unique decryption key, generally referred to as his private key.
- Receiver needs to publish an encryption key, referred to as his public key.
- Some assurance of the authenticity of a public key is needed in this scheme to avoid spoofing by adversary as the receiver. Generally, this type of cryptosystem involves trusted third party which certifies that a particular public key belongs to a specific person or entity only.
- Encryption algorithm is complex enough to prohibit attacker from deducing the plaintext from the ciphertext and the encryption (public) key.
- Though private and public keys are related mathematically, it is not be feasible to calculate the private key from the public key. In fact, intelligent part of any public-key cryptosystem is in designing a relationship between two keys.
There are three types of Public Key Encryption schemes. We discuss them in following sections −
RSA Cryptosystem
This cryptosystem is one the initial system. It remains most employed cryptosystem even today. The system was invented by three scholars Ron Rivest, Adi Shamir, and Len Adleman and hence, it is termed as RSA cryptosystem.
We will see two aspects of the RSA cryptosystem, firstly generation of key pair and secondly encryption-decryption algorithms.
Generation of RSA Key Pair
Each person or a party who desires to participate in communication using encryption needs to generate a pair of keys, namely public key and private key. The process followed in the generation of keys is described below −
- Generate the RSA modulus (n)
- https://buyersskyey321.weebly.com/fallout-3-pc-product-key-generator.html. Select two large primes, p and q.
- Calculate n=p*q. For strong unbreakable encryption, let n be a large number, typically a minimum of 512 bits.
- Find Derived Number (e)
- Number e must be greater than 1 and less than (p − 1)(q − 1).
- There must be no common factor for e and (p − 1)(q − 1) except for 1. In other words two numbers e and (p – 1)(q – 1) are coprime.
- Form the public key
- The pair of numbers (n, e) form the RSA public key and is made public.
- Interestingly, though n is part of the public key, difficulty in factorizing a large prime number ensures that attacker cannot find in finite time the two primes (p & q) used to obtain n. This is strength of RSA.
- Generate the private key
- Private Key d is calculated from p, q, and e. For given n and e, there is unique number d.
- Number d is the inverse of e modulo (p - 1)(q – 1). This means that d is the number less than (p - 1)(q - 1) such that when multiplied by e, it is equal to 1 modulo (p - 1)(q - 1).
- This relationship is written mathematically as follows −
The Extended Euclidean Algorithm takes p, q, and e as input and gives d as output.
![Elliptic Curve Cryptography Generate Public Key Elliptic Curve Cryptography Generate Public Key](/uploads/1/2/6/4/126448585/453755602.png)
Example
An example of generating RSA Key pair is given below. (For ease of understanding, the primes p & q taken here are small values. Practically, these values are very high).
- Let two primes be p = 7 and q = 13. Thus, modulus n = pq = 7 x 13 = 91.
- Select e = 5, which is a valid choice since there is no number that is common factor of 5 and (p − 1)(q − 1) = 6 × 12 = 72, except for 1.
- The pair of numbers (n, e) = (91, 5) forms the public key and can be made available to anyone whom we wish to be able to send us encrypted messages.
- Input p = 7, q = 13, and e = 5 to the Extended Euclidean Algorithm. The output will be d = 29.
- Check that the d calculated is correct by computing −
- Hence, public key is (91, 5) and private keys is (91, 29).
Encryption and Decryption
Once the key pair has been generated, the process of encryption and decryption are relatively straightforward and computationally easy.
Interestingly, RSA does not directly operate on strings of bits as in case of symmetric key encryption. It operates on numbers modulo n. Hence, it is necessary to represent the plaintext as a series of numbers less than n.
RSA Encryption
- Suppose the sender wish to send some text message to someone whose public key is (n, e).
- The sender then represents the plaintext as a series of numbers less than n.
- To encrypt the first plaintext P, which is a number modulo n. The encryption process is simple mathematical step as −
- In other words, the ciphertext C is equal to the plaintext P multiplied by itself e times and then reduced modulo n. This means that C is also a number less than n.
- Returning to our Key Generation example with plaintext P = 10, we get ciphertext C −
RSA Decryption
- The decryption process for RSA is also very straightforward. Suppose that the receiver of public-key pair (n, e) has received a ciphertext C.
- Receiver raises C to the power of his private key d. The result modulo n will be the plaintext P.
- Returning again to our numerical example, the ciphertext C = 82 would get decrypted to number 10 using private key 29 −
RSA Analysis
The security of RSA depends on the strengths of two separate functions. The RSA cryptosystem is most popular public-key cryptosystem strength of which is based on the practical difficulty of factoring the very large numbers.
- Encryption Function − It is considered as a one-way function of converting plaintext into ciphertext and it can be reversed only with the knowledge of private key d.
- Key Generation − The difficulty of determining a private key from an RSA public key is equivalent to factoring the modulus n. An attacker thus cannot use knowledge of an RSA public key to determine an RSA private key unless he can factor n. It is also a one way function, going from p & q values to modulus n is easy but reverse is not possible.
If either of these two functions are proved non one-way, then RSA will be broken. In fact, if a technique for factoring efficiently is developed then RSA will no longer be safe.
The strength of RSA encryption drastically goes down against attacks if the number p and q are not large primes and/ or chosen public key e is a small number.
ElGamal Cryptosystem
Along with RSA, there are other public-key cryptosystems proposed. Many of them are based on different versions of the Discrete Logarithm Problem.
ElGamal cryptosystem, called Elliptic Curve Variant, is based on the Discrete Logarithm Problem. It derives the strength from the assumption that the discrete logarithms cannot be found in practical time frame for a given number, while the inverse operation of the power can be computed efficiently.
Let us go through a simple version of ElGamal that works with numbers modulo p. In the case of elliptic curve variants, it is based on quite different number systems.
Generation of ElGamal Key Pair
Each user of ElGamal cryptosystem generates the key pair through as follows −
- Choosing a large prime p. Generally a prime number of 1024 to 2048 bits length is chosen.
- Choosing a generator element g.
- This number must be between 1 and p − 1, but cannot be any number.
- It is a generator of the multiplicative group of integers modulo p. This means for every integer m co-prime to p, there is an integer k such that gk=a mod n.For example, 3 is generator of group 5 (Z5 = {1, 2, 3, 4}).
N | 3n | 3n mod 5 |
---|---|---|
1 | 3 | 3 |
2 | 9 | 4 |
3 | 27 | 2 |
4 | 81 | 1 |
Elliptic Curve Cryptography Problems
- Choosing the private key. The private key x is any number bigger than 1 and smaller than p−1.
- Computing part of the public key. The value y is computed from the parameters p, g and the private key x as follows −
- Obtaining Public key. The ElGamal public key consists of the three parameters (p, g, y).For example, suppose that p = 17 and that g = 6 (It can be confirmed that 6 is a generator of group Z17). The private key x can be any number bigger than 1 and smaller than 71, so we choose x = 5. The value y is then computed as follows −
- Thus the private key is 62 and the public key is (17, 6, 7).
Elliptic Curve Cryptography Generate Public Key In Linux
Encryption and Decryption
The generation of an ElGamal key pair is comparatively simpler than the equivalent process for RSA. But the encryption and decryption are slightly more complex than RSA.
ElGamal Encryption
Suppose sender wishes to send a plaintext to someone whose ElGamal public key is (p, g, y), then −
- Sender represents the plaintext as a series of numbers modulo p.
- To encrypt the first plaintext P, which is represented as a number modulo p. The encryption process to obtain the ciphertext C is as follows −
- Randomly generate a number k;
- Compute two values C1 and C2, where −
- Send the ciphertext C, consisting of the two separate values (C1, C2), sent together.
- Referring to our ElGamal key generation example given above, the plaintext P = 13 is encrypted as follows −
- Randomly generate a number, say k = 10
- Compute the two values C1 and C2, where −
- Send the ciphertext C = (C1, C2) = (15, 9).
ElGamal Decryption
- To decrypt the ciphertext (C1, C2) using private key x, the following two steps are taken −
- Compute the modular inverse of (C1)x modulo p, which is (C1)-x , generally referred to as decryption factor.
- Obtain the plaintext by using the following formula −
- In our example, to decrypt the ciphertext C = (C1, C2) = (15, 9) using private key x = 5, the decryption factor is
- Extract plaintext P = (9 × 9) mod 17 = 13.
ElGamal Analysis
In ElGamal system, each user has a private key x. and has three components of public key − prime modulus p, generator g, and public Y = gx mod p. The strength of the ElGamal is based on the difficulty of discrete logarithm problem.
The secure key size is generally > 1024 bits. Today even 2048 bits long key are used. On the processing speed front, Elgamal is quite slow, it is used mainly for key authentication protocols. Due to higher processing efficiency, Elliptic Curve variants of ElGamal are becoming increasingly popular.
Elliptic Curve Cryptography (ECC)
Elliptic Curve Cryptography (ECC) is a term used to describe a suite of cryptographic tools and protocols whose security is based on special versions of the discrete logarithm problem. It does not use numbers modulo p.
ECC is based on sets of numbers that are associated with mathematical objects called elliptic curves. There are rules for adding and computing multiples of these numbers, just as there are for numbers modulo p.
Elliptic Curve Cryptography Generate Public Key Git
ECC includes a variants of many cryptographic schemes that were initially designed for modular numbers such as ElGamal encryption and Digital Signature Algorithm.
It is believed that the discrete logarithm problem is much harder when applied to points on an elliptic curve. This prompts switching from numbers modulo p to points on an elliptic curve. Also an equivalent security level can be obtained with shorter keys if we use elliptic curve-based variants.
Elliptic Curve Cryptography Generate Public Key From Private Key
The shorter keys result in two benefits −
- Ease of key management
- Efficient computation
These benefits make elliptic-curve-based variants of encryption scheme highly attractive for application where computing resources are constrained.
RSA and ElGamal Schemes – A Comparison
Let us briefly compare the RSA and ElGamal schemes on the various aspects.
RSA | ElGamal |
---|---|
It is more efficient for encryption. | It is more efficient for decryption. |
It is less efficient for decryption. | It is more efficient for decryption. |
For a particular security level, lengthy keys are required in RSA. | For the same level of security, very short keys are required. |
It is widely accepted and used. | It is new and not very popular in market. |
Definition
Provides a Cryptography Next Generation (CNG) implementation of the Elliptic Curve Diffie-Hellman (ECDH) algorithm. This class is used to perform cryptographic operations.
ECDiffieHellmanCng
Examples
The following example shows how to use the ECDiffieHellmanCng class to establish a key exchange and how to use that key to encrypt a message that can be sent over a public channel and decrypted by the receiver.
Remarks
The ECDiffieHellmanCng class enables two parties to exchange private key material even if they are communicating through a public channel. Both parties can calculate the same secret value, which is referred to as the secret agreement in the managed Diffie-Hellman classes. The secret agreement can then be used for a variety of purposes, including as a symmetric key. However, instead of exposing the secret agreement directly, the ECDiffieHellmanCng class does some post-processing on the agreement before providing the value. This post processing is referred to as the key derivation function (KDF); you can select which KDF you want to use and set its parameters through a set of properties on the instance of the Diffie-Hellman object.
Key derivation function | Properties |
---|---|
Hash | HashAlgorithm - The hash algorithm that is used to process the secret agreement. SecretPrepend - An optional byte array to prepend to the secret agreement before hashing it. SecretAppend - An optional byte array to append to the secret agreement before hashing it. |
Hmac | HashAlgorithm - The hash algorithm that is used to process the secret agreement. SecretPrepend- An optional byte array to prepend to the secret agreement before hashing it. SecretAppend - An optional byte array to append to the secret agreement before hashing it. |
Tls | Label - The label for key derivation. Seed - The seed for key derivation. |
The result of passing the secret agreement through the key derivation function is a byte array that may be used as key material for your application. The number of bytes of key material generated is dependent on the key derivation function; for example, SHA-256 will generate 256 bits of key material, whereas SHA-512 will generate 512 bits of key material.The basic flow of an ECDH key exchange is as follows:
- Alice and Bob create a key pair to use for the Diffie-Hellman key exchange operation
- Alice and Bob configure the KDF using parameters the agree on.
- Alice sends Bob her public key.
- Bob sends Alice his public key.
- Alice and Bob use each other's public keys to generate the secret agreement, and apply the KDF to the secret agreement to generate key material.
Constructors
ECDiffieHellmanCng() | Initializes a new instance of the ECDiffieHellmanCng class with a random key pair. |
ECDiffieHellmanCng(CngKey) | Initializes a new instance of the ECDiffieHellmanCng class by using the specified CngKey object. |
ECDiffieHellmanCng(ECCurve) | Creates a new instance of the ECDiffieHellmanCng class whose public/private key pair is generated over the specified curve. |
ECDiffieHellmanCng(Int32) | Initializes a new instance of the ECDiffieHellmanCng class with a random key pair, using the specified key size. |
Fields
KeySizeValue | Represents the size, in bits, of the key modulus used by the asymmetric algorithm. (Inherited from AsymmetricAlgorithm) |
LegalKeySizesValue | Specifies the key sizes that are supported by the asymmetric algorithm. (Inherited from AsymmetricAlgorithm) |
Properties
HashAlgorithm | Gets or sets the hash algorithm to use when generating key material. |
HmacKey | Gets or sets the Hash-based Message Authentication Code (HMAC) key to use when deriving key material. |
Key | Specifies the CngKey that is used by the current object for cryptographic operations. |
KeyDerivationFunction | Gets or sets the key derivation function for the ECDiffieHellmanCng class. |
KeyExchangeAlgorithm | Gets the name of the key exchange algorithm. (Inherited from ECDiffieHellman) |
KeySize | Gets or sets the size, in bits, of the key modulus used by the asymmetric algorithm. |
Label | Gets or sets the label value that is used for key derivation. |
LegalKeySizes | Gets the key sizes that are supported by the asymmetric algorithm. (Inherited from AsymmetricAlgorithm) |
PublicKey | Gets the public key that can be used by another ECDiffieHellmanCng object to generate a shared secret agreement. |
SecretAppend | Gets or sets a value that will be appended to the secret agreement when generating key material. |
SecretPrepend | Gets or sets a value that will be added to the beginning of the secret agreement when deriving key material. |
Seed | Gets or sets the seed value that will be used when deriving key material. |
SignatureAlgorithm | Gets the name of the signature algorithm. (Inherited from ECDiffieHellman) |
UseSecretAgreementAsHmacKey | Gets a value that indicates whether the secret agreement is used as a Hash-based Message Authentication Code (HMAC) key to derive key material. |
Elliptic Curve Cryptography Vs Rsa
Methods
Clear() | Releases all resources used by the AsymmetricAlgorithm class. (Inherited from AsymmetricAlgorithm) |
DeriveKeyFromHash(ECDiffieHellmanPublicKey, HashAlgorithmName) | Performs key derivation using a specified hash algorithm. (Inherited from ECDiffieHellman) |
DeriveKeyFromHash(ECDiffieHellmanPublicKey, HashAlgorithmName, Byte[], Byte[]) | Performs key derivation using a specified hash algorithm with optional prepended or appended data. |
DeriveKeyFromHmac(ECDiffieHellmanPublicKey, HashAlgorithmName, Byte[]) | Performs key derivation using a specified HMAC (Hash-based Message Authentication Code) algorithm. (Inherited from ECDiffieHellman) |
DeriveKeyFromHmac(ECDiffieHellmanPublicKey, HashAlgorithmName, Byte[], Byte[], Byte[]) | Performs key derivation using a specified HMAC (Hash-based Message Authentication Code) algorithm with optional prepended or appended data. |
DeriveKeyMaterial(CngKey) | Derives the key material that is generated from the secret agreement between two parties, given a CngKey object that contains the second party's public key. |
DeriveKeyMaterial(ECDiffieHellmanPublicKey) | Derives the key material that is generated from the secret agreement between two parties, given an ECDiffieHellmanPublicKey object that contains the second party's public key. |
DeriveKeyTls(ECDiffieHellmanPublicKey, Byte[], Byte[]) | Performs key derivation using the TLS (Transport Layer Security) 1.1 PRF (Pseudo-Random Function). |
DeriveSecretAgreementHandle(CngKey) | Gets a handle to the secret agreement generated between two parties, given a CngKey object that contains the second party's public key. |
DeriveSecretAgreementHandle(ECDiffieHellmanPublicKey) | Gets a handle to the secret agreement generated between two parties, given an ECDiffieHellmanPublicKey object that contains the second party's public key. |
Dispose() | Releases all resources used by the current instance of the AsymmetricAlgorithm class. (Inherited from AsymmetricAlgorithm) |
Dispose(Boolean) | Releases the unmanaged resources used by the AsymmetricAlgorithm class and optionally releases the managed resources. (Inherited from AsymmetricAlgorithm) |
Equals(Object) | Determines whether the specified object is equal to the current object. (Inherited from Object) |
ExportECPrivateKey() | Exports the current key in the ECPrivateKey format. (Inherited from ECDiffieHellman) |
ExportEncryptedPkcs8PrivateKey(ReadOnlySpan<Byte>, PbeParameters) | Exports the current key in the PKCS#8 EncryptedPrivateKeyInfo format with a byte-based password. (Inherited from AsymmetricAlgorithm) |
ExportEncryptedPkcs8PrivateKey(ReadOnlySpan<Char>, PbeParameters) | Exports the current key in the PKCS#8 EncryptedPrivateKeyInfo format with a char-based password. Paretologic pc health advisor license key generator. (Inherited from AsymmetricAlgorithm) |
ExportExplicitParameters(Boolean) | Exports the key and explicit curve parameters used by the ECCurve object into an ECParameters object. |
ExportParameters(Boolean) | Exports the key used by the ECCurve object into an ECParameters object. |
ExportPkcs8PrivateKey() | Exports the current key in the PKCS#8 PrivateKeyInfo format. (Inherited from AsymmetricAlgorithm) |
ExportSubjectPublicKeyInfo() | Exports the public-key portion of the current key in the X.509 SubjectPublicKeyInfo format. (Inherited from AsymmetricAlgorithm) |
FromXmlString(String) | This method is not implemented. |
FromXmlString(String, ECKeyXmlFormat) | Deserializes the key information from an XML string by using the specified format. |
GenerateKey(ECCurve) | Generates a new ephemeral public/private key pair for the specified curve. |
GetHashCode() | Serves as the default hash function. (Inherited from Object) |
GetType() | Gets the Type of the current instance. (Inherited from Object) |
ImportECPrivateKey(ReadOnlySpan<Byte>, Int32) | Imports the public/private keypair from an ECPrivateKey structure, replacing the keys for this object. (Inherited from ECDiffieHellman) |
ImportEncryptedPkcs8PrivateKey(ReadOnlySpan<Byte>, ReadOnlySpan<Byte>, Int32) | Imports the public/private keypair from a PKCS#8 EncryptedPrivateKeyInfo structure after decrypting with a byte-based password, replacing the keys for this object. (Inherited from ECDiffieHellman) |
ImportEncryptedPkcs8PrivateKey(ReadOnlySpan<Char>, ReadOnlySpan<Byte>, Int32) | Imports the public/private keypair from a PKCS#8 EncryptedPrivateKeyInfo structure after decrypting with a char-based password, replacing the keys for this object. (Inherited from ECDiffieHellman) |
ImportParameters(ECParameters) | Imports the specified parameters for an ECCurve object as a key into the current instance. |
ImportPkcs8PrivateKey(ReadOnlySpan<Byte>, Int32) | Imports the public/private keypair from a PKCS#8 PrivateKeyInfo structure after decryption, replacing the keys for this object. (Inherited from ECDiffieHellman) |
ImportSubjectPublicKeyInfo(ReadOnlySpan<Byte>, Int32) | Imports the public key from an X.509 SubjectPublicKeyInfo structure after decryption, replacing the keys for this object. (Inherited from ECDiffieHellman) |
MemberwiseClone() | Creates a shallow copy of the current Object. (Inherited from Object) |
ToString() | Returns a string that represents the current object. (Inherited from Object) |
ToXmlString(Boolean) | This method is not implemented. |
ToXmlString(ECKeyXmlFormat) | Serializes the key information to an XML string by using the specified format. |
TryExportECPrivateKey(Span<Byte>, Int32) | Microsoft word 7product key generator. Attempts to export the current key in the ECPrivateKey format into a provided buffer. (Inherited from ECDiffieHellman) |
TryExportEncryptedPkcs8PrivateKey(ReadOnlySpan<Byte>, PbeParameters, Span<Byte>, Int32) | Attempts to export the current key in the PKCS#8 EncryptedPrivateKeyInfo format into a provided buffer, using a byte-based password. (Inherited from ECDiffieHellman) |
TryExportEncryptedPkcs8PrivateKey(ReadOnlySpan<Char>, PbeParameters, Span<Byte>, Int32) | Attempts to export the current key in the PKCS#8 EncryptedPrivateKeyInfo format into a provided buffer, using a char-based password. (Inherited from ECDiffieHellman) |
TryExportPkcs8PrivateKey(Span<Byte>, Int32) | Attempts to export the current key in the PKCS#8 PrivateKeyInfo format into a provided buffer. (Inherited from ECDiffieHellman) |
TryExportSubjectPublicKeyInfo(Span<Byte>, Int32) | Attempts to export the current key in the X.509 SubjectPublicKeyInfo format into a provided buffer. (Inherited from ECDiffieHellman) |
Elliptic Curve Cryptography Generate Public Keys
Explicit Interface Implementations
IDisposable.Dispose() | For a description of this member, see Dispose(). (Inherited from AsymmetricAlgorithm) |