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1. Rsa Public Key Generation Example For Kids
2. Rsa Public Key Generation Example Definition
3. Rsa Public Key Algorithm
4. Rsa Public Key Generation Example 2016
5. Generate Rsa Public Key From Private
6. Rsa Public Key Generation Examples

The flowcharts above shows how to generate a public and private key using RSA. Example of RSA: Here is an example of RSA encryption and decryption with generation of the public and private key. Generate public and private key. Encryption and Decryption. How to use the RSA Algorithm in a C# Windows Forms application. The -t option specifies the key generation algorithm (RSA in this case), while the -b option specifies the length of the key in bits. The -f option sets the name of the output file. If not present, ssh-keygen will ask the name of the file, offering to save it to the default file /.ssh/idrsa. Start Writing. Start Writing ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ ‌ Help; About; Start Writing; Sponsor: Brand-as-Author; Sitewide Billboard.

The RSA algorithm involves four steps: key generation, key distribution, encryption and decryption. A basic principle behind RSA is the observation that it is practical to find three very large positive integers e, d and n such that with modular exponentiation for all integers m (with 0 ≤ m. Generate RSA keys with SSH by using PuTTYgen. Last updated on: 2016-06-23; Authored by: Rackspace Support; One effective way of securing SSH access to your cloud server is to use a public-private key pair. This means that a public key is placed on the server and a private key is placed on your local workstation. Using a key pair makes it. A separate public key file is not created at the same step though. To extract public key from the private key file into separate public key file you use your openssl rsa -in private.pem -pubout -out public.pem command. When you produce a public key this way, it is extracted from the private key file, not calculated.

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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).

  • 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

• 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 −

• 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.

to export the private key, select Export OpenSSH key from the Conversions menu. After this, the raw contents of the public key will be displayed alongside its fingerprint and a timestamp comment.Two important fields, Key passphrase and Confirm passphrase, allow you to enter a passphrase to protect the private key.Finally, you will need to export both the private and public keys separately:. Generate public key from private key ubuntu windows 7. Consequently, you won’t be asked for a passphrase when using your key.When the process has finished, the private key and the public key can be found in the /.ssh directory accessible from the Ubuntu terminal, or the following folder from Windows file manager: C:Users%USERNAME%AppDataLocalPackagesCanonicalGroupLimited.UbuntuonWindows79rhkp1fndgscLocalStaterootfsThe AppData and directory is hidden from the default view and will need to be entered manually. Key generation with PuTTYTo generate a key pair with the PuTTY key generator, simply run puttygen.exe and click the Generate button in the window that appears.You will be asked to move the mouse and press keys to improve the random number generation at the heart of SSH security.

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.

Contents

• 3. Saving the Keys in Binary Format
• Source Code

1. Introduction

Let us learn the basics of generating and using RSA keys in Java.

Java provides classes for the generation of RSA public and private key pairs with the package java.security. You can use RSA keys pairs in public key cryptography.

Rsa Public Key Generation Example For Kids

Public key cryptography uses a pair of keys for encryption. Distribute the public key to whoever needs it but safely secure the private key.

Public key cryptography can be used in two modes:

Encryption: Only the private key can decrypt the data encrypted with the public key.

Authentication: Data encrypted with the private key can only be decrypted with the public key thus proving who the data came from.

2. Generating a Key Pair

First step in creating an RSA Key Pair is to create a KeyPairGeneratorfrom a factory method by specifying the algorithm (“RSA” in this instance):

Initialize the KeyPairGenerator with the key size. Use a key size of 1024 or 2048. Currently recommended key size for SSL certificates used in e-commerce is 2048 so that is what we use here.

From the KeyPair object, get the public key using getPublic() and the private key using getPrivate().

Rsa Public Key Generation Example Definition

3. Saving the Keys in Binary Format

Save the keys to hard disk once they are obtained. This allows re-using the keys for encryption, decryption and authentication.

What is the format of the saved files? The key information is encoded in different formats for different types of keys. Here is how you can find what format the key was saved in. On my machine, the private key was saved in PKCS#8 format and the public key in X.509 format. We need this information below to load the keys.

3.1. Load Private Key from File

After saving the private key to a file (or a database), you might need to load it at a later time. You can do that using the following code. Note that you need to know what format the data was saved in: PKCS#8 in our case.

Rsa Public Key Algorithm

3.2 Load Public Key from File

Load the public key from a file as follows. The public key has been saved in X.509 format so we use the X509EncodedKeySpec class to convert it.

4. Use Base64 for Saving Keys as Text

Save the keys in text format by encoding the data in Base64. Java 8 provides a Base64 class which can be used for the purpose. Save the private key with a comment as follows:

And the public key too (with a comment):

5. Generating a Digital Signature

As mentioned above, one of the purposes of public key cryptography is digital signature i.e. you generate a digital signature from a file contents, sign it with your private key and send the signature along with the file. The recipient can then use your public key to verify that the signature matches the file contents.

Here is how you can do it. Use the signature algorithm “SHA256withRSA” which is guaranteed to be supported on all JVMs. Use the private key (either generated or load from file as shown above) to initialize the Signatureobject for signing. It is then updated with contents from the data file and the signature is generated and written to the output file. This output file contains the digital signature and must be sent to the recipient for verification.

6. Verifying the Digital Signature

Rsa Public Key Generation Example 2016

The recipient uses the digital signature sent with a data file to verify that the data file has not been tampered with. It requires access to the sender’s public key and can be loaded from a file if necessary as presented above.

Generate Rsa Public Key From Private

The code below updates the Signature object with data from the data file. It then loads the signature from file and uses Signature.verify() to check if the signature is valid.

Rsa Public Key Generation Examples

And that in a nutshell is how you can use RSA public and private keys for digital signature and verification.

Source Code

Go here for the source code.