Address at the International Conference on "Number Theory for Secure Communications"

Ramanujan: A continuing source of inspiration

I am indeed delighted to participate in the International Conference on Number Theory for Secure Communications. This conference coincides with the 116th Birth Anniversary of our mathematical genius Shri Srinivasa Ramanujan who with his intimate familiarity with numbers and excellence especially in number theory and modular function theory made the country proud. I greet the Vice Chancellor of Shanmugha Arts, Science, Technology and Research Academy (SASTRA), Faculty Members, delegates of the Conference from India and abroad, students and other dignitaries on this important occasion.

A genius well ahead of time

Srinivasa Ramanujan was one of the greatest geniuses known and acknowledged of his time. He lived only for 33 years and did not have formal higher education or means of living. Yet, his inexhaustible spirit and love for his subject made him contribute to the treasure houses of mathematical research - some of which are still under serious study and engaging all-available world mathematicians' efforts to establish formal proofs. Ramanujan was a unique Indian genius who could melt the heart of the most hardened and outstanding Cambridge mathematician Prof G H Hardy. In fact, it is not an exaggeration to say that it was Prof. Hardy who discovered Ramanujan for the world. Professor Hardy rated various geniuses on a scale of 100. While most of the mathematicians got a rating of around 30 with rare exceptions reaching to 60, Ramanujan got a rating of 100. There cannot be any better tribute to either Ramanujan or to Indian heritage. His works cover vast areas including Prime Numbers, Hyper geometric Series, Modular Functions, Elliptic Functions, Mock Theta Functions, even magic squares, apart from serious side works on geometry of ellipses, squaring the circle etc. One of the tributes to Ramanujan says that, 'every Integer is a personal friend of Ramanujan'.

Number theory and spiritual connectivity

Ramanujan used to say "An equation means nothing to me unless it expresses a thought of God".

For him the understanding of numbers was a process of spiritual revelation and connection. In his investigations into pure mathematics, he drew extraordinary conclusions that mystified his colleagues, but were usually proven, eventually, to be right. He opened a universe of theory that still today is reaping applications. The landscape of the infinite was to Ramanujan a reality of both mathematics and spirit.

He would talk for hours on the relationship he saw between God, zero and infinity. He spoke of the quantity two to the power 'n' minus one ("2n -1"), explaining that it stood for "the primordial God and several divinities. When n is zero, the expression denotes zero, there is nothing; when n is "1", the expression denotes God; when n is "2", the expression denotes Trinity; when n is "3", the expression denotes "7", the Saptha Rishis". (A group of seven stars called the "Great Bear"). And he continued with the idea that "Zero represents Absolute Reality. Infinity is the myriad manifestations of that Reality.

Their mathematical product, Infinity x 0 is not one number, but all numbers each of which corresponds to individual acts of creation". For Ramanujan, numbers and their mathematical relationship were the measure of how the universe fits together. Each new theorem he explored was one more piece of the infinite to fathom.

Communication models

One of the important applications of Number Theory is in designing error correcting codes which are robust against noise introduced in communication channels. The idealistic communication models can be described as follows:

Source to Encoder to Channel (added with noise) to Decoder to Receiver.

The problems of defining a suitable measure of information and of efficiency of coding have been satisfactorily solved. The second problem of coding is concerned with finding a method whereby for each message received we can identify the message transmitted with the least amount of error, even when the transmitted message is corrupted by noise. The fundamental theorem of information theory assures us that under certain conditions this can be done. The construction of error correcting codes have been a difficult and fascinating mathematical problem and its more or less successful solution has made it possible for us to think of channels or great reliability to work with computers and automation equipment.

In the area of analogue signal processing one uses a mathematical technique called Fourier Transform. When one enters the digital world a different tool called Discrete Fourier Transform is used. Whereas, if one has to analyse noise signals, engineers have recently come to the conclusion that an efficient mathematical tool would be the Ramanujan Fourier Transformation or in short RFT . This once again demonstrates that though Ramanujan did the work on RFT purely to satisfy his urge to explore the beauty of mathematics, it had come to be of use in day-to-applications like communications- almost six decades later.

Early recognition of talents

Friends, the genius in Ramanujan had to be discovered by Prof. Hardy. This has been cryptically remarked at that time by Poondi Namasivaya Mudaliar with anguish "It is the destiny of our nation that an Indian brain requires an acknowledgement from a foreigner. Why our people are hesitant to appreciate such a personality". I would like to narrate another incident which took place a few years back.

A young man, Loveligen, from a remote area of Kerala, who could not complete PUC, wrote to me saying that he has discovered a new mathematical theory and he would like to talk to me. I saw in the letter that the boy was very sincere. Since he has written to me, I thought our specialist team can study his work and direct him to the right type of researchers. I called this boy to Delhi for a few days. What surprised us was that he had arrived at part of the equations of the Ramanujan's number theory, which this boy was not at all aware of. He had discovered something and added some new points to it and the result is new. To a great extent the achievements in the field of mathematics generally seem to come out of a desire to look into the beautiful aspects of nature, including natural phenomena such as the star studded skies, which have always interested the astronomers from time immemorial. An additional contributory factor seems to be an inherent drive towards recognition of patterns even if it be in the sense of mathematical sequences or series.

It is interesting to note that Loveligen has currently delved into the equally exciting topic of power sequences and series. What I felt was that he needed a good mathematical education or a patronage of a good mathematics teacher. It is like having Prof Hardy for Ramanujan, the mathematics genius to come. I asked this boy, why he didn't meet a mathematics teacher. He said, meeting a mathematics teacher is an expedition. He says, it is below their dignity to meet somebody who is not even a graduate. How do we promote this kind of young and enthusiastic minds? Can our teachers and philanthropists or the social activists spot these buds to blossom? Those who spot such talents and make them blossom will themselves be a different kind of a flower as described in the Bhagwad Gita: "See the flower, how generously it distributes its perfume and its honey. It gives to all, gives freely of its love. When its work is done, it falls away quietly. Try to be like the flower unassuming despite all its qualities". What a beautiful message for all generations of this Nation.

Recently, I have asked Dr. R. Chidambaram, Dr. R.A.Mashelkar and Prof.N Balakrishnan to evolve a system that will provide an outlet for innovativeness in the budding mathematicians. One of the suggestions that I have is that, to bring out Indian talents, like the way Hardy did for Ramanujam, many of you must agree to host these talents at least for a few months in your establishments. You are the ones who could become tomorrow's Hardy by helping to shape the uncut diamond to a best jewel that we could be proud of.

Let me now share with you the latest development in the field of Information Communication Technology and Information Security.

Cryptography and Information Security

Number Theory once epitomized pure mathematics. Today the applied Number Theory usually refers to Cryptography, which enables Secure Communications. Very simple mathematics, cleverly used, occasionally produces spectacular practical results and indeed the first public key crypto systems needed only the most rudimentary number theory. But the modern elaborations of the number theory use all the number theoretic tools one can reasonably expect from an undergraduate. Thus, cryptography motivates a new generation of students to study number theory. Cryptography aside, applied number theory might also mean communication networks. Expander graphs are basic building blocks in the design of networks and have vast number of applications in areas of computer and communication sciences. In the last two decades the theory of Ramanujan Graphs has gained prominence primarily for two reasons. First, from a practical point of view these graphs resolved an external problem in communication network theory. Second, from a more aesthetic point of view.

In the digital world, bits and bytes represent knowledge which in turn represents the nation's wealth. For example, the source codes of the software that a company sells, the long term strategies of companies, bank accounts, the purchase through e-Commerce and even our land records are in digital form today. Movement of bits across the network creates further wealth in the digital era. More than 25% of most nations soon will be directly or indirectly connected with Information and Communication products. In this new world, the economic and physical securities are tightly intertwined by Information Security. Nations that are capable of generating and managing information in a secure way will become world leaders and world super powers.

Cryptography - the art of hiding information is central to information security. There are two major ways by which information can be cryptographically secured in computer mediated communication. One is the symmetric cipher where the same key is used for encryption as well as decryption. The other is the Asymmetric key cryptography, wherein the encryption key and the decryption key are different. The latter can be also used for Public-Key Encryption. The Asymmetric key cryptography like the widely used RSA (Ron Rivest, Adi Shamir and Len Adleman) is dependent on the complexity of calculations needed for finding out the two prime numbers p and q, given their product n =p x q.

This is where the Number Theory comes in - a field wherein Ramanujan made long lasting contributions. Had Ramanujan lived for a few more years, had we continued to maintain the lead that Ramanujan gave us in the field of Number Theory; today we would have become the "Leaders in Secure communication" and possibly made many innovations that would have made our nation proud and also rich. Ramanujan, in making contributions to Number Theory was well ahead of his time- blissfully unaware of the potential of his work either in securing information or in creating wealth. He did what he did simply for the beauty and purity of all he did. What Ramanujan said is so much, what he implied is even more, what he left behind is the legacy that many generations that would never forget and perpetuate his mission.

Information & Communication Technology (ICT)

ICT has established that the data transformed into information has a business proposition which has given a competitive advantage. I am sure by the end of this decade countries like India will have IT enabled services in the fields of human resources, customer interaction, finance and accounting, data search and integration, e-education, tele-medicine and e-governance.

Core competencies that can be exploited in addition to the above include Information Security, Scientific Software development that can spearhead a strong domestic market, Entertainment, Education, Hardware and chip design and Wireless. In India the software industry is exploring these areas to create a wealth of $80 billion by the year 2010. We strongly believe by a proper planning and the ability to move up the value chain we may even touch a target of $150 billion by 2010.

National mission for nurturing mathematicians

When we all talk about the nation's strides in IT, Space, Defence, Agriculture and academic institutions we have not yet fully recognised the importance of mathematics. It is becoming even more difficult to get bright students taking to Mathematics - the purest of the sciences as a career when they are young. This will in the next few years would stifle innovations and make the role of Science & Technology in societal transformation a saturated ground. On this historic day, can we launch a National Mission to generate Mathematicians in large numbers and also create suitable employment potential for them, so that we will enrich our scientific work and enrich our nation which had a tradition of mathematics right from Aryabhatta. This is eminently possible since India for several centuries had been the home of some of the best talents in mathematics - a tradition that should be nurtured for the world to benefit.

There may be hundreds of such minds spread over in the country. They are looking for many of the gifted senior scientists and professors to encourage their thinking and lead to best contribution in the field. We have to shed the 'minimum educational qualification' syndrome for the sake of discovering and encouraging the young minds' dreams. Throughout the country, the educated community, blessed with higher educational potential, should spot and encourage the creative minds wherever they are, irrespective of their regions.

Conclusion

I suggest the following action plan to promote mathematics in the nation as a fitting tribute to the contributions of Ramanujan:

1. We need to form a Ramanujan Foundation with a corpus amount Rs. 100 crores.

2. Foundation should conduct bi-annual international meets on advancement of mathematics and also spot young talents and encourage them to pursue mathematics.

3. DAE, ADA, CSIR, DRDO, DST, ISRO and other scientific institutions may set apart certain fund annually on a long term basis for the promising young mathematicians to pursue mathematics for mathematics sake at the place of their choice.

4. Create a mechanism to locate and tap the best minds from rural areas, which are goldmines of hidden talents.

5. Institute Young Mathematician awards of Rs. 10 lakh to be given every year to the best young mathematician (below 35 years) for path-breaking contribution to mathematics.

Now, I dedicate the house, where the great Srinivasa Ramanujan had stayed for over 15 years before he left for Trinity College, UK, as an International Monument. This is the place where vibrations of the thoughts of Ramanujan is being felt and will always be felt for generations to come.

I am happy to dedicate the Srinivasa Ramanujan Mathematics centre. Co-located within this complex are state of the art scanning machines which form part of the Universal Digital Library project. I believe that it is likely to grow into a mega scanning centre very soon. I also inaugurate the Srinivasa Ramanujan Museum which will ignite the minds of our younger generation and create number of Ramanujan's in the years to come.

I wish you all success.