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27 February 2022

Reflections on Signatures

For thousands of years, signatures and more broadly stamps and seals had been used in literally every society to bind a person with an entity. Even in the Stone Age, before the emergence of written languages, human carved symbolic shapes to signify their unique cultural identity. Signatures, nowadays in more refined forms, are still widely employed, though in some cases they could also fail to fulfil their intended goals.

Then one may ask: what is the machinery in their effectiveness and sometimes ineffectiveness? How can one design and implement a more effective signature? These lead to a quest for the nature of signatures.

Back to the primitive tribes in prehistoric times, signatures on an appliance conceivably served to display possession of superior power or culture. The tradition continues all the way to modern days, for example when an artist signs her painting to proudly claim the ownership of the aesthetic value she created, or when a food company brands trademark on its product to associate the taste with the enterprise itself. Conversely, a craftsman would be unmotivated to sign a clumsy experimental work due to lack of merit. We could thus put it this way:

Via signature, a signer can actively tie himself to the inherent quality of an entity.

Hence signatures are meant to be proprietary from a signer’s perpective. He intends to establish via signature a permanent, immutable bond to certain quality. His foremost demand is unforgeability, namely no one else could change his signature on the signed entity (thus jeopardising the bond), or duplicate his signature onto a new entity (thus announcing fake bond and potentially harming his fame).

Now we observe some accidental and dramatic effects from the persepective of other parties. Signatures (which are assumed unforgeable) allow a third party to quickly connect a person with a quality – may it be good or bad. In other words, a signature is capable of witnessing an undeniable announcement made by the signer.

This way of thinking feeds back and reshapes how signatures are used. A person could now request others to sign a given entity in order to create an undeniable bond in his favour. He may later show this piece on court to end disputes. In this setting, the person who signs does not actively sign for merit, but rather passively signs for reassurance to the requester.

Via signature, a signer passively ties himself to the quality set by a requester.

Such twofold, bidirectional functions rendered (unforgeable) signatures communally desirable. The crux is, then, the enforcement of unforgeability. Rephrased: how can one tell a real signature from a forgery?

Asymmetry turns out crucial, since symmetric persons can sign indistinguishably. Towards unforgeability, the signer must incorporate his hidden distinctive characteristics, or asymmetry, into his signature. Signing is an occasion where the signer reveals his secret, so to speak.

Another necessity is the inseparable marriage of the signature and the signed entity. Any attempt of removing the signature from the entity shall shatter their completeness; transferring the signature as is to another entity would never fit.

Hence:

The action of signing (with unforgeability as its goal) should instantiate signer’s secret on an entity, so that the two are deeply entangled.

Various methods were designed and employed in history to meet the requirement. For instance,

• A classical signature instantiates one’s handwriting on paper documents. The patterns of his hand motion, ink selection, and pressing on the nib are all tangled with the fibre of paper, both structurally and visually.
• A seal in ancient China reveals the physical characteristics of the seal itself (which is exlusively owned by the signer) on paper. Secrets such as the stone material, surface flatness, cracks, the carvings of the characters, are incorporated into paper as in handwritten signatures.

But a serious concern hides underneath. If one’s signature pattern is accessible to the public, then everyone including malicious parties could study it in detail. After sufficient exposures of concrete (instantiated) signatures, experts might manage to immitate it in close proximity. He the signer is now facing a perpetual paradox: On the one hand he must give up some secret to implement asymmetry and to allow public verification; on the other hand the revealed secret can be learnt by others, thus turning him and the forger symmetric again.

Modern cryptography maintains that it is not an issue if we adds the dimension of resources into consideration:

• The signer reveals some secret to the public with little effort.
• The public learns the secret and could verify it with little effort. But to really understand and replicate it, the public must invest substantial amount of effort.

Hence, the asymmetry of information entails a lasting asymmetry in resource demand, and it is exactly the latter asymmetry that empowers the unforgeability. This revolutionary viewpoint is the pearl of modern cryptography, in light of which we could say:

Signature is an art of controlling secret exposure. Forgery is always possible, but if it takes unreasonably many resources to complete, then the signature is practically unforgeable, or secure.