Digital Signature - Online Article

A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a signature in digital, rather than written, form. Digital signature schemes normally give two algorithms, one for signing which involves the user's secret or private key, and one for verifying signatures which involves the user's public key. The output of the signature process is called the "digital signature."

Digital signatures, like written signatures, are used to provide authentication of the associated input, usually called a "message." Messages may be anything, from electronic mail to a contract, or even a message sent in a more complicated cryptographic protocol. Digital signatures are used to create public key infrastructure (PKI) schemes in which a user's public key (whether for public-key encryption, digital signatures, or any other purpose) is tied to a user by a digital identity certificate issued by a certificate authority. PKI schemes attempt to unbreakably bind user information (name, address, phone number, etc.) to a public key, so that public keys can be used as a form of identification.

Digital signatures are often used to implement electronic signatures, a broader term that refers to any electronic data that carries the intent of a signature, but not all electronic signatures use digital signatures.In some countries, including the United States, and in the European Union, electronic signatures have legal significance. However, laws concerning electronic signatures do not always make clear their applicability towards cryptographic digital signatures, leaving their legal importance somewhat unspecified.


A digital signature scheme typically consists of three algorithms:

  • A key generation algorithm G that randomly produces a "key pair" (PK, SK) for the signer. PK is the verifying key, which is to be public, and SK is the signing key, to be kept private.
  • A signing algorithm S, that on input of a message m and a signing key SK, produces a signature σ.
  • A signature verifying algorithm V, that on input a message m, a verifying key PK, and a signature σ, either accepts or rejects.

Two main properties are required. First, signatures computed honestly should always verify. That is, V should accept (m, PK, S (m, SK)) where SK is the secret key related to PK, for any message m. Secondly, it should be hard for any adversary, knowing only PK, to create valid signature(s).


In the famous paper "New Directions in Cryptography", Whitfield Diffie and Martin Hellman first described the notion of a digital signature scheme, although they only conjectured that such schemes existed. Soon afterwards, Ronald Rivest, Adi Shamir, and Len Adleman invented the RSA algorithm that could be used for primitive digital signatures. (Note that this just serves as a proof-of-concept, and "plain" RSA signatures are not secure.) The first widely marketed software package to offer digital signature was Lotus Notes 1.0, released in 1989, which used the RSA algorithm.

Basic RSA signatures are computed as follows. To generate RSA signature keys, one simply generates an RSA key pair containing a modulus N that is the product of two large primes, along with integers e and d such that e d = 1 mod φ(N), where φ is the Euler phi-function. The signer's public key consists of N and e, and the signer's secret key contains d.

To sign a message m, the signer computes σ=md mod N. To verify, the receiver checks that σe = m mod N.

As noted earlier, this basic scheme is not very secure. To prevent attacks, one can first apply a cryptographic hash function to the message m and then apply the RSA algorithm described above to the result. This approach can be proven secure in the so-called random oracle model.

Other digital signature schemes were soon developed after RSA, the earliest being Lamport signatures, Merkle signatures (also known as "Merkle trees" or simply "Hash trees"), and Rabin signatures.

In 1984, Shafi Goldwasser, Silvio Micali, and Ronald Rivest became the first to rigorously define the security requirements of digital signature schemes. They described a hierarchy of attack models:

  • In a key-only attack, the attacker is only given the public verification key.
  • In a known message attack, the attacker is given valid signatures for a variety of messages known by the attacker but not chosen by the attacker.
  • In a chosen message attack, the attacker first learns signatures on arbitrary messages of the attacker's choice.

They also describe a hierarchy of attack results:

  • A total break results in the recovery of the signing key.
  • A universal forgery attack results in the ability to forge signatures for any message.
  • A selective forgery attack results in a signature on a message of the adversary's choice.
  • An existential forgery merely results in some valid message/signature pair not already known to the adversary.

They also present the GMR signature scheme, the first that can be proven to prevent even an existential forgeries against even a chosen message attack.

Most early signature schemes were of a similar type: they involve the use of a trapdoor permutation, such as the RSA function, or in the case of the Rabin signature scheme, computing square modulo composite n. A trapdoor permutation family is a family of permutations, specified by a parameter, that is easy to compute in the forward direction, but is difficult to compute in the reverse direction. However, for every parameter there is a "trapdoor" that enables easy computation of the reverse direction. Trapdoor permutations can be viewed as public-key encryption systems, where the parameter is the public key and the trapdoor is the secret key, and where encrypting corresponds to computing the forward direction of the permutation, while decrypting corresponds to the reverse direction. Trapdoor permutations can also be viewed as digital signature schemes, where computing the reverse direction with the secret key is thought of as signing, and computing the forward direction is done to verify signatures. Because of this correspondence, digital signatures are often described as based on public-key cryptosystems, where signing is equivalent to decryption and verification is equivalent to encryption, but this is not the only way digital signatures are computed.

Used directly, this type of signature scheme is vulnerable to a key-only existential forgery attack. To create a forgery, the attacker picks a random signature σ and uses the verification procedure to determine the message m corresponding to that signature. In practice, however, this type of signature is not used directly, but rather, the message to be signed is first hashed to produce a short digest that is then signed. This forgery attack, then, only produces the hash function output that corresponds to σ, but not a message that leads to that value, which does not lead to an attack. In the random oracle model, this hash-and-decrypt form of signature is existentially unforgivable, even against a chosen-message attack.

There are several reasons to sign such a hash (or message digest) instead of the whole document.

  • For efficiency: The signature will be much shorter and thus save time since hashing is generally much faster than signing in practice.
  • For compatibility: Messages are typically bit strings, but some signature schemes operate on other domains (such as, in the case of RSA, numbers modulo a composite number N). A hash function can be used to convert an arbitrary input into the proper format.
  • For integrity: Without the hash function, the text "to be signed" may have to be split (separated) in blocks small enough for the signature scheme to act on them directly. However, the receiver of the signed blocks is not able to recognize if all the blocks are present and in the appropriate order.

Benefits of Digital Signatures

These are common reasons for applying a digital signature to communications:


Although messages may often include information about the entity sending a message, that information may not be accurate. Digital signatures can be used to authenticate the source of messages. When ownership of a digital signature secret key is bound to a specific user, a valid signature shows that the message was sent by that user. The importance of high confidence in sender authenticity is especially obvious in a financial context. For example, suppose a bank's branch office sends instructions to the central office requesting a change in the balance of an account. If the central office is not convinced that such a message is truly sent from an authorized source, acting on such a request could be a grave mistake.


In many scenarios, the sender and receiver of a message may have a need for confidence that the message has not been altered during transmission. Although encryption hides the contents of a message, it may be possible to change an encrypted message without understanding it. (Some encryption algorithms, known as nonmalleable ones, prevent this, but others do not.) However, if a message is digitally signed, any change in the message will invalidate the signature. Furthermore, there is no efficient way to modify a message and its signature to produce a new message with a valid signature, because this is still considered to be computationally infeasible by most cryptographic hash functions (see collision resistance).

Drawbacks of Digital Signatures

Despite their usefulness, digital signatures do not alone solve all the problems we might wish them to.

Association of digital signatures and trusted time stamping

Digital signature algorithms and protocols do not inherently provide certainty about the date and time at which the underlying document was signed. The signer might, or might not, have included a time stamp with the signature, or the document itself might have a date mentioned on it, but a later reader cannot be certain the signer did not, for instance, backdate the date or time of the signature. Such misuse can be made impracticable by using trusted time stamping in addition to digital signatures.


In a cryptographic context, the word repudiation refers to any act of disclaiming responsibility for a message. A message's recipient may insist the sender attach a signature in order to make later repudiation more difficult, since the recipient can show the signed message to a third party (eg, a court) to reinforce a claim as to its signatories and integrity. However, loss of control over a user's private key will mean that all digital signatures using that key, and so ostensibly 'from' that user, are suspect. Nonetheless, a user cannot repudiate a signed message without repudiating their signature key. It is aggravated by the fact there is no trusted time stamp, so new documents (after the key compromise) cannot be separated from old ones, further complicating signature key invalidation. Certificate Authorities usually maintain a public repository of public-key so the association user-key is certified and signatures cannot be repudiated. Expired certificates are normally removed from the directory. It is a matter for the security policy and the responsibility of the authority to keep old certificates for a period of time if a non-repudiation of data service is provide.

Additional Security Precautions

Putting the private key on a smart card

All public key / private key cryptosystems depend entirely on keeping the private key secret. A private key can be stored on a user's computer, and protected by, for instance, a local password, but this has two disadvantages:

  • the user can only sign documents on that particular computer and
  • the security of the private key completely depends on the security of the computer, which is notoriously unreliable for many PCs and operating systems.

A more secure alternative is to store the private key on a smart card. Many smart cards are deliberately designed to be tamper resistant (however, quite a few designs have been broken, notably by Ross Anderson and his students). In a typical implementation, the hash calculated from the document is sent to the smart card, whose CPU encrypts the hash using the stored private key of the user and returns it. Typically, a user must activate his smart card by entering a personal identification number or PIN code (thus providing a two-factor authentication). Note that it can be sensibly arranged (but is not always done) that the private key never leaves the smart card. If the smart card is stolen, the thief will still need the PIN code to generate a digital signature. This reduces the security of the scheme to that of the PIN system, but is nevertheless more secure than are many PCs.

Using smart card readers with a separate keyboard

Entering a PIN code to activate the smart card, commonly requires a numeric keypad. Some card readers have their own numeric keypad. This is safer than using a card reader integrated into a PC, and then entering the PIN using that computer's keyboard. The computer might be running a keystroke logger (by its owner/operators intention or otherwise -- due to a virus, for instance) so that the PIN code becomes compromised. Specialized card readers are less vulnerable, though not invulnerable, against tampering with their software or hardware. And, of course, eavesdropping attacks against all such equipment are possible.

Other smart card designs

Smart card design is an active field, and there are smart card schemes which are intended to avoid these particular problems, though so far with little security proofs.

Using digital signatures only with trusted applications

One of the main differences between a digital signature and a written signature is that the user does not "see" what he signs. It's the application that presents a hash code to be encrypted with the private key, but in the case of a malicious application a hash code of another document might be presented so that the users thinks he is signing the document he sees on the screen but is actually unwillingly signing another (probably less favourable).

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