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Parent page: Internet and Networking >> SSH

Public key authentication is more secure than password authentication. This is particularly important if the computer is visible on the internet. If you don't think it's important, try logging the login attempts you get for the next week. My computer - a perfectly ordinary desktop PC - had over 4,000 attempts to guess my password and almost 2,500 break-in attempts in the last week alone.

With public key authentication, the authenticating entity has a public key and a private key. Each key is a large number with special mathematical properties. The private key is kept on the computer you log in from, while the public key is stored on the .ssh/authorized_keys file on all the computers you want to log in to. When you log in to a computer, the SSH server uses the public key to 'lock' messages in a way that can only be 'unlocked' by your private key - this means that even the most resourceful attacker can't snoop on, or interfere with, your session. As an extra security measure, most SSH programs store the private key in a passphrase-protected format, so that if your computer is stolen or broken in to, you should have enough time to disable your old public key before they break the passphrase and start using your key. Wikipedia has a more detailed explanation of how keys work.

Public key authentication is a much better solution than passwords for most people. In fact, if you don't mind leaving a private key unprotected on your hard disk, you can even use keys to do secure automatic log-ins - as part of a network backup, for example. Different SSH programs generate public keys in different ways, but they all generate public keys in a similar format:

Key-based authentication is the most secure of several modes of authentication usable with OpenSSH, such as plain password and Kerberos tickets. Key-based authentication has several advantages over password authentication, for example the key values are significantly more difficult to brute-force, or guess than plain passwords, provided an ample key length. Other authentication methods are only used in very specific situations.

SSH can use either 'RSA' (Rivest-Shamir-Adleman) or 'DSA' ('Digital Signature Algorithm') keys. Both of these were considered state-of-the-art algorithms when SSH was invented, but DSA has come to be seen as less secure in recent years. RSA is the only recommended choice for new keys, so this guide uses 'RSA key' and 'SSH key' interchangeably.

Key-based authentication uses two keys, one 'public' key that anyone is allowed to see, and another 'private' key that only the owner is allowed to see. To securely communicate using key-based authentication, one needs to create a key pair, securely store the private key on the computer one wants to log in from, and store the public key on the computer one wants to log in to.

Using key based logins with ssh is generally considered more secure than using plain password logins. This section of the guide will explain the process of generating a set of public/private RSA keys, and using them for logging into your Ubuntu computer(s) via OpenSSH.

The first step involves creating a set of RSA keys for use in authentication.

This should be done on the client.

To create your public and private SSH keys on the command-line:

You will be prompted for a location to save the keys, and a passphrase for the keys. This passphrase will protect your private key while it's stored on the hard drive:

Your public key is now available as .ssh/id_rsa.pub in your home folder.

Congratulations! You now have a set of keys. Now it's time to make your systems allow you to login with them

Choosing a good passphrase

You need to change all your locks if your RSA key is stolen. Otherwise the thief could impersonate you wherever you authenticate with that key.

An SSH key passphrase is a secondary form of security that gives you a little time when your keys are stolen. If your RSA key has a strong passphrase, it might take your attacker a few hours to guess by brute force. That extra time should be enough to log in to any computers you have an account on, delete your old key from the .ssh/authorized_keys file, and add a new key.

Your SSH key passphrase is only used to protect your private key from thieves. It's never transmitted over the Internet, and the strength of your key has nothing to do with the strength of your passphrase.

The decision to protect your key with a passphrase involves convenience x security. Note that if you protect your key with a passphrase, then when you type the passphrase to unlock it, your local computer will generally leave the key unlocked for a time. So if you use the key multiple times without logging out of your local account in the meantime, you will probably only have to type the passphrase once.

If you do adopt a passphrase, pick a strong one and store it securely in a password manager. You may also write it down on a piece of paper and keep it in a secure place. If you choose not to protect the key with a passphrase, then just press the return when ssh-keygen asks.

Key Encryption Level

Note: The default is a 2048 bit key. You can increase this to 4096 bits with the -b flag (Increasing the bits makes it harder to crack the key by brute force methods).

Password Authentication

The main problem with public key authentication is that you need a secure way of getting the public key onto a computer before you can log in with it. If you will only ever use an SSH key to log in to your own computer from a few other computers (such as logging in to your PC from your laptop), you should copy your SSH keys over on a memory stick, and disable password authentication altogether. If you would like to log in from other computers from time to time (such as a friend's PC), make sure you have a strong password.

The key you need to transfer to the host is the public one. If you can log in to a computer over SSH using a password, you can transfer your RSA key by doing the following from your own computer:

Where <username> and <host> should be replaced by your username and the name of the computer you're transferring your key to.

Due to this bug, you cannot specify a port other than the standard port 22. You can work around this by issuing the command like this: ssh-copy-id '<username>@<host> -p <port_nr>'. If you are using the standard port 22, you can ignore this tip.

Another alternative is to copy the public key file to the server and concatenate it onto the authorized_keys file manually. It is wise to back that up first:

You can make sure this worked by doing:

You should be prompted for the passphrase for your key:

Enter your passphrase, and provided host is configured to allow key-based logins, you should then be logged in as usual.

Encrypted Home Directory

If you have an encrypted home directory, SSH cannot access your authorized_keys file because it is inside your encrypted home directory and won't be available until after you are authenticated. Therefore, SSH will default to password authentication.

To solve this, create a folder outside your home named /etc/ssh/<username> (replace '<username>' with your actual username). This directory should have 755 permissions and be owned by the user. Move the authorized_keys file into it. The authorized_keys file should have 644 permissions and be owned by the user.

Then edit your /etc/ssh/sshd_config and add:

Rsa Public Private Key Explained

Finally, restart ssh with:

The next time you connect with SSH you should not have to enter your password.

username@host's password:

If you are not prompted for the passphrase, and instead get just the

prompt as usual with password logins, then read on. There are a few things which could prevent this from working as easily as demonstrated above. On default Ubuntu installs however, the above examples should work. If not, then check the following condition, as it is the most frequent cause:

On the host computer, ensure that the /etc/ssh/sshd_config contains the following lines, and that they are uncommented;

If not, add them, or uncomment them, restart OpenSSH, and try logging in again. If you get the passphrase prompt now, then congratulations, you're logging in with a key!

Permission denied (publickey)

If you're sure you've correctly configured sshd_config, copied your ID, and have your private key in the .ssh directory, and still getting this error:

Create Rsa Public Private Key

Chances are, your /home/<user> or ~/.ssh/authorized_keys permissions are too open by OpenSSH standards. You can get rid of this problem by issuing the following commands:

Error: Agent admitted failure to sign using the key.

This error occurs when the ssh-agent on the client is not yet managing the key. Issue the following commands to fix:

This command should be entered after you have copied your public key to the host computer.

Debugging and sorting out further problems

The permissions of files and folders is crucial to this working. You can get debugging information from both the client and server.

if you think you have set it up correctly , yet still get asked for the password, try starting the server with debugging output to the terminal.

To connect and send information to the client terminal

No matter how your public key was generated, you can add it to your Ubuntu system by opening the file .ssh/authorized_keys in your favourite text editor and adding the key to the bottom of the file. You can also limit the SSH features that the key can use, such as disallowing port-forwarding or only allowing a specific command to be run. This is done by adding 'options' before the SSH key, on the same line in the authorized_keys file. For example, if you maintain a CVS repository, you could add a line like this:

When the user with the specified key logged in, the server would automatically run /usr/bin/cvs server, ignoring any requests from the client to run another command such as a shell. For more information, see the sshd man page. /755

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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.

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• 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.

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• 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

Public And Private Keys Rsa Key Generation 10

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.

Public And Private Keys Rsa Key Generation Linux

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.