Monday, November 14, 2016

Mathematics of the Blind

How does a blind know the other person is blind or not? One of the toughest explorations of humankind is the exploration for the truth. Human search for food ended with finding food. In some way, it is parallel to what other living beings also do. Hunger for food brings human beings, animal and plants on the same platform. They work towards getting good, sometimes good food. As a rational animal, human beings went further ahead. They searched for the truth behind it. They found that there is much more in the truth.

Among the three types of living beings, human beings and animals are superior to plants because of their capacity to change their positions. They find better positions, in the face of threats and sometimes for comfort. Hence we may say that plants live in the first dimension.  Human beings are superior to animals because of their capacity to change their positions in thinking. Though animals think, their thinking is a very static thinking. Hence, we may say that animals live in the second dimension. They live according their instinct. Human beings live in the third dimension. They have life. They can move. They can remember the past and dream about the future. They can imagine.

Among the many capacities human beings have, the most important one is the capacity to reason. Reason is the reason for human beings to search for truth. Human beings, in general, are in an incessant search for truth. Some search for the truth of life. Some others search for the truth of the nature. There are some others who search for the truth of the truth itself. This search for truth is the basis of recording and documentation. Documentation is the backbone of all the developments in the world. As human beings change their styles and modes of living, a continuous progress is visible in the world. It is by encountering and polishing the truths of the nature, the truths of the truths, human beings progress towards better life.  In this pursuit of progress, the most important and precise recording is in the form of Mathematics.

Many say, “Mathematics is the study of anything that has reason.” Mathematics is about recognizing patterns in the world and contributing suggestions for its repetition and improving.  In establishing the claims of what is right and what is good, mathematicians use a technique called “Proof.” In fact, everyone with rationality does proving, in many different ways. Hence, in general everyone is a mathematician. However, academic mathematicians do more documented proving. There are various methods of Proof.

One without the capacity to see things using one’s own eyes is termed as a blind. Of course, blindness is not a hindrance to reasoning. Hence, a blind person also does proving. But, can a blind person use a visual proof? To be precise, we come back to the initial question, “how does a blind know the other person is blind or not?” One can always bluff the blind by pretending to be blind or not. Here comes a beautiful mathematical proof method, named as “Zero Knowledge Proof.” It is illustrated in the case a blind person (B) who tests the blindness of another (A).

B has two identical marbles of different colours.
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B knows that the marbles are of different colours. However, B cannot distinguish them by their colours. B holds the marbles in either of the palms. B asks A to identify the colours of the marbles. There is a possibility that A might come with the two colours correctly. But, B is still not sure if A is truthful. Hence, B keeps swapping the marbles quickly for some time between the palms with the perfect knowledge of the movement of the marbles. If A is blind, A has no clue whether the marble is exchanged or not. B then asks A to identify the location of the different colours. This procedure is repeated a couple of times more. If A is not blind, then A can never deviate from the initial identification of colours of the marbles.

In modern cryptography, Zero Knowledge Proof is efficiently utilized. It was initially conceived and published by the trio Shafi Goldwasser, Silvio Micali and Charles Rackoff.* The power of Zero Knowledge Proof is that the prover never passes the proof but a verifier can always test the prover of the knowledge of the proof and be satisfied.

*S. Goldwasser et al. “The knowledge complexity of interactive proof systems,” SIAM J. Comput. vol. 18, no. 1, pp. 186-208, Feb. 1989.