SSH/OpenSSH/Keys on Ubuntu

Public and Private Keys

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:

<ssh-rsa or ssh-dss> <really long string of nonsense> <username>@<host>

Key-Based SSH Logins

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.

Generating RSA Keys

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:

mkdir ~/.ssh
chmod 700 ~/.ssh
ssh-keygen -t rsa

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:

Generating public/private rsa key pair.
Enter file in which to save the key (/home/b/.ssh/id_rsa):
Enter passphrase (empty for no passphrase):
Enter same passphrase again:
Your identification has been saved in /home/b/.ssh/id_rsa.
Your public key has been saved in /home/b/.ssh/id_rsa.pub.

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).

ssh-keygen -t rsa -b 4096

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.

Transfer Client Key to Host

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:

ssh-copy-id <username>@<host>

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

(i) 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:

cp authorized_keys authorized_keys_Backup
cat id_rsa.pub >> authorized_keys

You can make sure this worked by doing:

ssh <username>@<host>

You should be prompted for the passphrase for your key:

Enter passphrase for key ‘/home/<user>/.ssh/id_rsa’:

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

Troubleshooting

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:

AuthorizedKeysFile    /etc/ssh/%u/authorized_keys

Finally, restart ssh with:

sudo service ssh restart

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

username@host’s password:

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;

PubkeyAuthentication yes
RSAAuthentication yes

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:

Permission denied (publickey).

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:

chmod go-w ~/
chmod 700 ~/.ssh
chmod 600 ~/.ssh/authorized_keys

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:

ssh-add

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.

sudo /usr/sbin/sshd -d

To connect and send information to the client terminal

ssh -v ( or -vv) username@host's

Where to From Here?

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:

command="/usr/bin/cvs server",no-agent-forwarding,no-port-forwarding,no-X11-forwarding,no-user-rc ssh-dss <string of nonsense>...

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.

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How RSA public key encryption works

RSA is an algorithm used by modern computers to encrypt and decrypt messages. It is an asymmetric cryptographic algorithm. Asymmetric means that there are two different keys. This is also called public key cryptography, because one of them can be given to everyone. The other key must be kept private. It is based on the fact that finding the factors of an integer is hard (the factoring problem). RSA stands for Ron Rivest, Adi Shamir and Leonard Adleman, who first publicly described it in 1978. A user of RSA creates and then publishes the product of two large prime numbers, along with an auxiliary value, as their public key. The prime factors must be kept secret. Anyone can use the public key to encrypt a message, but with currently published methods, if the public key is large enough, only someone with knowledge of the prime factors can feasibly decode the message.

Operation

RSA involves a public key and private key. The public key can be known to everyone, it is used to encrypt messages. Messages encrypted using the public key can only be decrypted with the private key. The keys for the RSA algorithm are generated the following way:

  1. Choose two different large random prime numbers  {\displaystyle p\,} and  {\displaystyle q\,}
  2. Calculate {\displaystyle n=pq\,}
    •   {\displaystyle n\,} is the modulus for the public key and the private keys
  3. Calculate the totient: {\displaystyle \phi (n)=(p-1)(q-1)\,}.
  4. Choose an integer {\displaystyle e\,} such that 1 <  {\displaystyle e\,} < {\displaystyle \phi (n)\,}, and {\displaystyle e\,} is coprime to {\displaystyle \phi (n)\,} ie: {\displaystyle e\,} and {\displaystyle \phi (n)\,} share no factors other than 1; gcd(  {\displaystyle e\,}, {\displaystyle \phi (n)\,}) = 1.
    •   {\displaystyle e\,} is released as the public key exponent
  5. Compute {\displaystyle d\,} to satisfy the congruence relation {\displaystyle de\equiv 1{\pmod {\phi (n)}}\,} ie: {\displaystyle de=1+k\phi (n)\,} for some integer  {\displaystyle k\,}.
    •   {\displaystyle d\,} is kept as the private key exponent

Notes on the above steps:

  • Step 1: Numbers can be probabilistically tested for primality.
  • Step 2: changed in PKCS#1 v2.0 to {\displaystyle \lambda (n)={\rm {lcm}}(p-1,q-1)\,} instead of  {\displaystyle \phi (n)=(p-1)(q-1)\,}.
  • Step 3: A popular choice for the public exponents is  {\displaystyle e\,} = 216 + 1 = 65537. Some applications choose smaller values such as {\displaystyle e\,} = 3, 5, or 35 instead. This is done to make encryption and signature verification faster on small devices like smart cards but small public exponents may lead to greater security risks.
  • Steps 4 and 5 can be performed with the extended Euclidean algorithm; see modular arithmetic.

The public key is made of the modulus  {\displaystyle n\,} and the public (or encryption) exponent  {\displaystyle e\,}.
The private key is made of the modulus  {\displaystyle n\,} and the private (or decryption) exponent  {\displaystyle d\,} which must be kept secret.

  • For efficiency a different form of the private key can be stored:
    •   {\displaystyle p\,} and  {\displaystyle q\,}: the primes from the key generation,
    •   {\displaystyle d\mod (p-1)\,} and {\displaystyle d\mod (q-1)\,}: often called dmp1 and dmq1.
    •   {\displaystyle q^{-1}\mod (p)\,}: often called iqmp
  • All parts of the private key must be kept secret in this form.  {\displaystyle p\,} and {\displaystyle q\,} are sensitive since they are the factors of  {\displaystyle n\,}, and allow computation of {\displaystyle d\,} given  {\displaystyle e\,}. If  {\displaystyle p\,} and  {\displaystyle q\,} are not stored in this form of the private key then they are securely deleted along with other intermediate values from key generation.
  • Although this form allows faster decryption and signing by using the Chinese Remainder Theorem (CRT) it is considerably less secure since it enables side channel attacks. This is a particular problem if implemented on smart cards, which benefit most from the improved efficiency. (Start with {\displaystyle y=x^{e}{\pmod {n}}} and let the card decrypt that. So it computes  {\displaystyle y^{d}{\pmod {p}}} or  {\displaystyle y^{d}{\pmod {q}}} whose results give some value  {\displaystyle z}. Now, induce an error in one of the computations. Then  {\displaystyle \gcd(z-x,n)} will reveal  {\displaystyle p} or q.)

Encrypting messages

Alice gives her public key (  {\displaystyle n\,} {\displaystyle e\,}) to Bob and keeps her private key secret. Bob wants to send message M to Alice.

First he turns M into a number  m smaller than  n by using an agreed-upon reversible protocol known as a padding scheme. He then computes the ciphertext {\displaystyle c\,} corresponding to:

  {\displaystyle c=m^{e}\mod {n}}

This can be done quickly using the method of exponentiation by squaring. Bob then sends {\displaystyle c\,} to Alice.

Decrypting messages

Alice can recover  {\displaystyle m\,} from  {\displaystyle c\,} by using her private key  {\displaystyle d\,} in the following procedure:

  {\displaystyle m=c^{d}\mod {n}}

Given  {\displaystyle m\,}, she can recover the original message M.

The decryption procedure works because first

  {\displaystyle c^{d}\equiv (m^{e})^{d}\equiv m^{ed}{\pmod {n}}}.

Now, since

  {\displaystyle ed\equiv 1{\pmod {p-1}}\,} and
  {\displaystyle ed\equiv 1{\pmod {q-1}}\,}

Fermat’s little theorem yields

  {\displaystyle m^{ed}\equiv m{\pmod {p}}} and
  {\displaystyle m^{ed}\equiv m{\pmod {q}}}.

Since  {\displaystyle p\,} and  {\displaystyle q\,} are distinct prime numbers, applying the Chinese remainder theorem to these two congruences yields

{\displaystyle m^{ed}\equiv m{\pmod {pq}}}.

Thus,

{\displaystyle c^{d}\equiv m{\pmod {n}}}.

A working example

Here is an example of RSA encryption and decryption. The parameters used here are artificially small, but you can also use OpenSSL to generate and examine a real keypair.

  1. Choose two random prime numbers
  2.  :  {\displaystyle p=61} and  {\displaystyle q=53}
  3. Compute  {\displaystyle n=pq\,}
  4.  :  {\displaystyle n=61*53=3233}
  5. Compute the totient  {\displaystyle \phi (n)=(p-1)(q-1)\,}
  6.  : {\displaystyle \phi (n)=(61-1)(53-1)=3120}
  7. Choose  {\displaystyle e>1} coprime to 3120
  8.  :  {\displaystyle e=17}
  9. Choose  {\displaystyle d\,} to satisfy {\displaystyle de\equiv 1{\pmod {\phi (n)}}\,}
  10.  :  {\displaystyle d=2753}
  11.  :  {\displaystyle 17*2753=46801=1+15*3120}.

The public key is (  {\displaystyle n=3233}, {\displaystyle e=17}). For a padded message  {\displaystyle m\,} the encryption function is:

  {\displaystyle c=m^{e}\mod {n}=m^{17}\mod 3233\,}.

The private key is (  {\displaystyle n=3233} {\displaystyle d=2753}). The decryption function is:

  {\displaystyle m=c^{d}\mod {n}=c^{2753}\mod 3233\,}.

For example, to encrypt  {\displaystyle m=123}, we calculate

  {\displaystyle c=123^{17}\mod 3233=855}

To decrypt  {\displaystyle c=855}, we calculate

  {\displaystyle m=855^{2753}\mod 3233=123}.

Both of these calculations can be computed efficiently using the square-and-multiply algorithm for modular exponentiation.

Padding schemes

When used in practice, RSA must be combined with some form of padding scheme, so that no values of M result in insecure ciphertexts. RSA used without padding may have some problems:

  • The values m = 0 or m = 1 always produce ciphertexts equal to 0 or 1 respectively, due to the properties of exponentiation.
  • When encrypting with small encryption exponents (e.g., e = 3) and small values of the m, the (non-modular) result of m e {\displaystyle m^{e}} {\displaystyle m^{e}} may be strictly less than the modulus n. In this case, ciphertexts may be easily decrypted by taking the eth root of the ciphertext with no regard to the modulus.
  • RSA encryption is a deterministic encryption algorithm. It has no random component. Therefore, an attacker can successfully launch a chosen plaintext attack against the cryptosystem. They can make a dictionary by encrypting likely plaintexts under the public key, and storing the resulting ciphertexts. The attacker can then observe the communication channel. As soon as they see ciphertexts that match the ones in their dictionary, the attackers can then use this dictionary in order to learn the content of the message.

In practice, the first two problems can arise when short ASCII messages are sent. In such messages, m might be the concatenation of one or more ASCII-encoded character(s). A message consisting of a single ASCII NUL character (whose numeric value is 0) would be encoded as m = 0, which produces a ciphertext of 0 no matter which values of e and N are used. Likewise, a single ASCII SOH (whose numeric value is 1) would always produce a ciphertext of 1. For systems which conventionally use small values of e, such as 3, all single character ASCII messages encoded using this scheme would be insecure, since the largest m would have a value of 255, and 2553 is less than any reasonable modulus. Such plaintexts could be recovered by simply taking the cube root of the ciphertext.

To avoid these problems, practical RSA implementations typically embed some form of structured, randomized padding into the value m before encrypting it. This padding ensures that m does not fall into the range of insecure plaintexts, and that a given probe, once padded, will encrypt to one of a large number of different possible ciphertexts. The latter property can increase the cost of a dictionary attack beyond the capabilities of a reasonable attacker.

Standards such as PKCS have been carefully designed to securely pad messages prior to RSA encryption. Because these schemes pad the plaintext m with some number of additional bits, the size of the un-padded message M must be somewhat smaller. RSA padding schemes must be carefully designed so as to prevent sophisticated attacks. This may be made easier by a predictable message structure. Early versions of the PKCS standard used ad-hoc constructions, which were later found vulnerable to a practical adaptive chosen ciphertext attack. Modern constructions use secure techniques such as Optimal Asymmetric Encryption Padding (OAEP) to protect messages while preventing these attacks. The PKCS standard also has processing schemes designed to provide additional security for RSA signatures, e.g., the Probabilistic Signature Scheme for RSA (RSA-PSS).

Signing messages

Suppose Alice uses Bob’s public key to send him an encrypted message. In the message, she can claim to be Alice but Bob has no way of verifying that the message was actually from Alice since anyone can use Bob’s public key to send him encrypted messages. So, in order to verify the origin of a messages, RSA can also be used to sign a message.

Suppose Alice wishes to send a signed message to Bob. She produces a hash value of the message, raises it to the power of d mod n (just like when decrypting a message), and attaches it as a “signature” to the message. When Bob receives the signed message, he raises the signature to the power of e mod n (just like encrypting a message), and compares the resulting hash value with the message’s actual hash value. If the two agree, he knows that the author of the message was in possession of Alice’s secret key, and that the message has not been tampered with since.

Note that secure padding schemes such as RSA-PSS are as essential for the security of message signing as they are for message encryption, and that the same key should never be used for both encryption and signing purposes.