July 13, by Mark Hughes You might love math. But do you love math enough to learn a Slavic language, travel behind the Iron Curtain during the height of the Cold War, and visit the only other Mathematics Center outside the US? Neal Koblitz loves math.

Neal Koblitz and Dr. Victor Miller independently co-discovered elliptic-curve cryptography ECC , part of the mathematics that allows encrypted communication on the internet today. Neal Koblitz. NK : I have been interested in pure mathematics since I was a young child. When I went to college, I took courses in pure mathematics and a few courses in physics, but no engineering or other courses in practical application.

My undergraduate work was at Harvard, and the department courses were very theoretical in nature. I never applied the math I was learning to real-life situations.

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## Fabrice Mouhartem − Privacy-preserving cryptography from pairings and lattices

I even learned calculus in a very abstract way. I had to teach the course in a very different way from the way I learned it. In graduate school, I also studied an abstract area of mathematics: algebraic geometry in relation to number theory. The same was true for the other fellow who came up with the ideas at the same time that I did. Victor Miller received his Ph. But my background, and my early work into the s, was in very theoretical mathematics. I had no grounding in applications. AAC: When did your mathematics background shift from the theoretical realm to security applications?

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That was the first important application of number theory to computer security. This really captured the imagination of a lot of people. Many people who had previously thought of number theory as a purely theoretical area now saw that there might be some interesting work to be done in applications. Author's note: Neither Dr. Koblitz nor Dr. Miller obtained patents for their work. AAC: I understand that you also spent some time in Russia.

What compelled you to do that? NK : When I was very young, during the Cold War between the United States and the Soviet Union, both countries were going all out to develop their scientific and technical capacity in a very competitive way.

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There was the race to the moon and other things like that. The Soviet Union had at this point, in the s, an extremely high level of theoretical mathematics. They were the only mathematical superpower besides the US. I realized at a young age that it would be really nice to be able to go over there and work with Soviet mathematicians, to learn how they do things.

For this, I studied Russian intensively. In fact, as an undergraduate, I double-majored in Mathematics and Slavic Languages and Literature, which included Russian.

## Dr. Neal Koblitz: Independent Co-creator of Elliptic Curve Cryptography

I was very serious about my studies and I started applying to study-abroad programs. In those days, it was much less common than now. I first went on a six-week summer language study program.

My first trip to the Soviet Union was in After that, I went again as an independent tourist. Then, after I was married, my wife became very interested in Russian history, so we went again once I completed my PhD. We went several times in the s and s.

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So there were reasonable ways to get over there to spend a fair amount of time collaborating with Soviet specialists in theoretical mathematics. In , I took my last long trip to the Soviet Union. That trip coincided with the starting of my work in cryptography. The idea of elliptic-curve cryptography came in I, along with several other people received a pre-print, a rather preliminary-version, of an algorithm that Hendrik Lenstra developed to factor large integer numbers.

If this algorithm was sufficiently fast, it could be a threat to RSA cryptography. It turned out that this algorithm was not that much more efficient than other algorithms that existed at the time. The fact that he was able to, in a clever way, use the geometry and number theory of elliptic curves to factor integers, was really intriguing. This was the first practical application of that sort of number theory, and I had that fresh in my mind when I went to the Soviet Union in While I was in the Soviet Union, I thought that elliptic curves might be a really good basis for constructing a cryptosystem.

Of the mathematicians I knew at the time, he was the person who most bridged the gap between pure and applied mathematics. He was knowledgeable about cryptography and had major work in pure number theory, so I mailed him this idea. There was no email then, so replies were slow to come in through the mail. He encouraged me, and I suspect Victor Miller, to pursue it. He saw no fallacies in basing a cryptosystem on what Victor and I suggested.

That trip was my last long visit—six months in No one in the Soviet Union worked openly on cryptography. There were attempts by the NSA to restrict and create a [government] monopoly on cryptography research. Ultimately, the people in the government that wanted to [restrict cryptography research] lost the battle. There was no way I could collaborate with anyone in the Soviet Union on cryptography, and in any case, I was just a beginner at that time.

One of them, named Yuri Manin , was one of the top Soviet specialists in number theory. In addition, since nobody was working in cryptography, my visit in was not as productive as my previous visits. There, working in Moscow, it was really a unique experience because it was the biggest concentration of mathematicians anywhere in the world at the time. The US is much more decentralized. For over 30 years we have done research in Secure Multiparty Computation. Aarhus University and The Alexandra Institute are world leaders in this technology. Have a look at our recent projects or see selected examples on some of the thesis written in the group.

We all realize the importance of protecting systems from attacks by viruses and hackers, but the applications of information security go far beyond that: In a modern information-driven society, we need to do business and exchange information between people who may have conflicting interests, and do not necessarily trust each other a priori.

As examples, one may think of systems for e-commerce, e-voting or electronic auctions. In such scenarios, security is a complex notion. In an e-commerce system, for instance, the need to prevent misuse of the system seems to necessitate identification and surveillance of users, whereas the need to protect privacy pulls in the opposite direction. It follows that we need to understand what security really means, which goals can be achieved in which scenarios and which cannot, and finally how we can build efficient systems and solutions achieving these goals.

If you want to know more about the group and our history visit this site. If you are interested in joining our group, as master's thesis student, PhD student, or postdoc, feel free to contact the faculty members. Focus points.