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2. Generate Encryption Key With Diffie Hellman Key
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I need to support Diffie Hellman encryption. DH is key exchange (or key agreement) protocol, not encryption. DH is used to securely generate a common key between two parties, other algorithms are used for encryption itself. I need to create a certificate with DH key parameters eg. Key-length - 2048 etc.

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• 2019-10-10  I need to support Diffie Hellman encryption. DH is key exchange (or key agreement) protocol, not encryption. DH is used to securely generate a common key between two parties, other algorithms are used for encryption itself. I need to create a certificate with DH key parameters eg. Key-length - 2048 etc.
• What if we could dynamically generate a key? What if there was some way that both a client and a server could agree upon an encryption key without ever having to tell each other the actual key? Enter the Diffie-Hellman key exchange. Diffie-Hellman Explained.
• 2019-9-12  Oakley算法具有五个重要特征： 它采用称为cookie程序的机制来对抗阻塞攻击.
• Wikipedia: 'The Diffie–Hellman key exchange method allows two parties that have no prior knowledge of each other to jointly establish a shared secret key over an insecure communications channel. This key can then be used to encrypt subsequent communications using a symmetric key cipher.'
• Mar 25, 2008 This is not an encryption application: it demonstrates how Diffie-Hellman works and is not intended to be used by end-users. DiffieHellman.cs can be used to create an application based on this key exchange mechanism.

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Public Key Cryptography

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)

  • 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).

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  • This relationship is written mathematically as follows −

The Extended Euclidean Algorithm takes p, q, and e as input and gives d as output.

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

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• 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 g^k=a mod n.

    For example, 3 is generator of group 5 (Z₅ = {1, 2, 3, 4}).

• 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).

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 −

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• 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 = g^x 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.

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.

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.

The Diffie-Hellman algorithm provides the capability for two communicating parties to agree upon a shared secret between them. Its an agreement scheme because both parties add material used to derive the key (as opposed to transport, where one party selects the key). The shared secret can then be used as the basis for some encryption key to be used for further communication.

If Alice and Bob wish to communicate with each other, they first agree between them a large prime number p, and a generator (or base) g (where 0 < g < p).

Alice chooses a secret integer a (her private key) and then calculates g^a mod p (which is her public key).Bob chooses his private key b, and calculates his public key in the same way.

Alice and Bob then send each other their public keys. Alice now knows a and Bob's public key g^b mod p. She is not able to calculate the value b from Bob's public key as this is a hard mathematical problem (known as the discrete logarithm problem). She can however calculate (g^b)^a mod p = g^ab mod p.

Bob knows b and g^a, so he can calculate (g^a)^b mod p = g^ab mod p. Therefore both Alice and Bob know a shared secret g^ab mod p. Eve who was listening in on the communication knows p, g, Alice's public key (g^a mod p) and Bob's public key (g^b mod p). She is unable to calculate the shared secret from these values.

In static-static mode both Alice and Bob retain their private/public keys over multiple communications. Therefore the resulting shared secret will be the same every time. In ephemeral-static mode one party will generate a new private/public key every time, thus a new shared secret will be generated.

Diffie-Hellman Standards[edit]

There are a number of standards relevant to Diffie-Hellman key agreement. Some of the key ones are:

• PKCS 3 defines the basic algorithm and data formats to be used.
• ANSI X9.42 is a later standard than PKCS 3 and provides further guidance on its use (note OpenSSL does not support ANSI X9.42 in the released versions - support is available in the as yet unreleased 1.0.2 and 1.1.0)
• RFC 2631 summarizes the key points of ANSI X9.42
• RFC 5114 defines 3 standard sets of parameters for use with Diffie-Hellman (OpenSSL will have built-in support for these parameters from OpenSSL 1.0.2 - not yet released)
• NIST SP 800-56A is a NIST publication that provides recommendations on the implementation of X9.42

Generate Encryption Key With Diffie Hellman Key

Diffie-Hellman in SSL/TLS[edit]

There are three versions of Diffie-Hellman used in SSL/TLS.

• Anonymous Diffie-Hellman
• Fixed Diffie-Hellman
• Ephemeral Diffie-Hellman

Anonymous Diffie-Hellman uses Diffie-Hellman, but without authentication. Because the keys used in the exchange are not authenticated, the protocol is susceptible to Man-in-the-Middle attacks. Note: if you use this scheme, a call to SSL_get_peer_certificate will return NULL because you have selected an anonymous protocol. This is the only time SSL_get_peer_certificate is allowed to return NULL under normal circumstances.

You should not use Anonymous Diffie-Hellman. You can prohibit its use in your code by using '!ADH' in your call to SSL_set_cipher_list.

Fixed Diffie-Hellman embeds the server's public parameter in the certificate, and the CA then signs the certificate. That is, the certificate contains the Diffie-Hellman public-key parameters, and those parameters never change.

Ephemeral Diffie-Hellman uses temporary, public keys. Each instance or run of the protocol uses a different public key. The authenticity of the server's temporary key can be verified by checking the signature on the key. Because the public keys are temporary, a compromise of the server's long term signing key does not jeopardize the privacy of past sessions. This is known as Perfect Forward Secrecy (PFS).

You should always use Ephemeral Diffie-Hellman because it provides PFS. You can specify ephemeral methods by providing 'kEECDH:kEDH' in your call to SSL_set_cipher_list.

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Working with Parameters and Generating Keys[edit]

The first step with the Diffie-Hellman algorithm is to ensure that both parties are using the same set of parameters (i.e. the same values for p and g). Since parameter generation can be an expensive process this is normally done once in advance and then the same set of parameters are used over many key exchanges. A new set of parameters can be generated by OpenSSL, or alternatively there is support for built-in standard sets of parameters.

To generate your own parameters refer to EVP Key and Parameter Generation.

Note: The function DH_get_2048_256 is scheduled for release in OpenSSL 1.0.2; it is not available in 1.0.1e or earlier.

Generating a Shared Secret[edit]

Having obtained a private/public key pair you need to also obtain the public key of the other communicating party. Refer to EVP Key Agreement for information on how to agree a shared secret.

Using the Low Level APIs[edit]

Users of the OpenSSL library are expected to normally use the EVP method for working with Diffie Hellman as described above and on the EVP Key Agreement page. The EVP api is implemented by a lower level Diffie Hellman API. In some circumstances, expert users may need to use the low level api. This is not recommended for most users. However, if you need to use this then an example of use is shown below. The manual page for the low level API is available here: Manual:dh(3)

There are (currently) no DH_ level routines to read and writea public OR private key, but the generic PUBKEY andPrivateKey routines do so as an X.509 SubjectPublickKeyInfo structure (aka SPKI or PKCS#8). This includes the parameters plus the public key (and the private key for the PrivateKey routines) (see Manual:Pem(3)).

There are three possible cases:

• ephemeral parameters: A must send new parameters AND the public key to the peer (B), who needs to send back only their public key (although it may be convenient to embed it in an SPKI structure)
• static but undistributed parameters: effectively the same
• pre-distributed parameters: A only needs to send their public key, but may embed in an SPKI structure; B doesn't need to wait for A to get parameters but may wait anyway, and only needs to send B's public key but may embed it in an SPKI structure.

See also[edit]