Communicating on the Internet is a bit like having a private conversation with a friend across a crowded room filled with people keen on hearing what you have to say. Lots of individuals (and organizations) have the opportunity to eavesdrop on or even manipulate messages you send and receive online. Yet, somehow, most of us communicate over the Internet without thinking about these risks. We use our credit card for online purchases, communicate with our doctor about our medical records, and access our bank accounts.

How is such privacy and security possible? Rooted in both new and very old mathematics, a tree of technologies (*cryptosystems* and *authentication schemes*) has grown and enables us to communicate from anywhere without fear of our data being manipulated, publicized, or used maliciously in other ways.

These technologies help to establish Internet security, and all rely in a fundamental way on mathematics, some of which is very old. For example, RSA relies on *Euler's theorem*, which is more than 250 years old! It is astounding—but not unusual—that a piece of mathematics from 1763 can have its broadest and most impactful application today. As new mathematics is discovered every day, it builds upon existing mathematics and expands upon what is already known. Unlike technology, new mathematics does not replace old mathematics—the old mathematics remains true.

In addition, while new technologies are often built using mathematics, the demand for new technologies also stimulates the development of new mathematics.

Since around 2010, excitement over cryptocurrencies and related **blockchains** has exploded. Blockchains are a kind of tamper-proof public distributed database of digital information (blocks), typically consisting of records of transactions. People are excited to discover or invent new things to be done with blockchains and to find new and more efficient ways to engineer them. This has stimulated new applications of mathematics.

As one example, researchers used to study *zero-knowledge proofs* as an abstract topic with seemingly distant potential practical applications. These proofs provide a way for you to prove to someone else that you know a specific thing (such as an account number), without giving them any useful information about the thing itself (that’s why they are called “zero-knowledge”). This concept is subtle, and the whole subject used to be of largely academic interest, but now research in this area has become truly practical: zero-knowledge proofs are being put to real use as parts of some blockchains.

Internet security in an era of quantum computing is also motivating new mathematics. Quantum computers use the laws of quantum mechanics to process information and have the potential to solve certain computational problems faster than classical computers. In 1994, Peter Shor discovered a way that a quantum computer (if one existed) could break the encryption algorithms used in public key technologies. Today, it appears more and more possible that scientists will succeed in building a quantum computer that is powerful enough to render these technologies obsolete. Consequently, a global effort is under way to find strong and efficient *post-quantum cryptosystems*: replacements for RSA and the other vulnerable public key technologies. Such a new technology has to be based on the difficulty of solving a fundamentally different mathematical problem, something different from factoring whole numbers or finding discrete logarithms. Some of the hard problems being used in the design of cryptosystems include *decoding random linear codes*, *finding an isogeny between given elliptic curves*, and *finding the short vector in a lattice*.

Threats to safe use of the Internet will undoubtedly evolve, but mathematics will continue to support and enable new security technologies. Today, the combination of new and old mathematics allows us to feel confident that we are indeed communicating with the secure website that we think we are. It lets us do our banking online without fear that someone will eavesdrop and steal all of our money. And it lets us download apps on our cell phone without fear that some adversary will replace them with malware in route. While it is unclear what challenges to Internet security will emerge in years to come, we can be sure that many solutions will be rooted in mathematics.

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