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I am taking a course in linear algebra which was to be credited towards a program in applied math. My interests have moved towards matters of "security" (information, network, cryptographic, etc.). It seems that the math required for security is more towards discrete math and finite automata, probability theory, and number theory. Is there any motivation to study linear algebra? What are its applications to "security"?

For example, there's the concept of a "protection matrix", but I'm not sure if this can be related to the "matrix" in the linear algebra sense; a system of equations (would eigenvectors have any meaning?).

Andrei Botalov
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T. Webster
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There is no opposition between "linear algebra" and "discrete math". For instance, a Linear Feedback Shift Register is "linear" in the sense of linear algebra, but also totally discrete. What is seldom encountered in computers is not the "linear" part, but the use of real or complex numbers as base field -- because computers are not good at storing numbers with an infinite number of digits (an infinite storage space is a bit expensive). Computers use approximations (all the "floating point types" like double), or use finite fields. In the case of a LFSR, the finite field is GF(2), the field with two elements (0 and 1).

Thomas Pornin
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    Would you know of security careeer paths in which one might encounter applications of linera algebra outside of research and academia? – T. Webster Nov 24 '12 at 05:41
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Yes, not only is linear algebra required in many computer science programs it also has security implications.

Linear Algebra can be used to detect doctored photographs.

Vector Clocks are important in distributed systems and time can play a role in security.

Linear algebra is also used in GPS and Missie Guidance. GPS Spoofing requires Linear Algebra and this can be used to attack gullible drones.

rook
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Linear algebra is not especially relevant to practical security. There are some applications, but they are scattered and minor, and not particularly more prevalent than many other areas of mathematics. It pains me to say it, as I loved linear algebra in college and would love to tell you that it is super-useful in computer security -- but sadly, it ain't so.

Linear algebra does have some relevance in some aspects of cryptography (but that's better answered on Crypto.SE).

D.W.
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While it is true that number theory is the basis of most current public key cryptography, there are other cryptographical schemes that are based on linear algebra. For example, Multivariate Cryptography - the following is from the Wikipedia article:

Multivariate cryptography is the generic term for asymmetric cryptographic primitives based on multivariate polynomials over finite fields. In certain cases those polynomials could be defined over both a ground and an extension field. If the polynomials have the degree two, we talk about multivariate quadratics. Solving systems of multivariate polynomial equations is proven to be NP-Hard or NP-Complete. That's why those schemes are often considered to be good candidates for post-quantum cryptography, once quantum computers can break the current schemes.

So if you want to be prepared for doing security in a post quantum computing (if and when), linear algebra may be even more relevant to cryptography than number theory.

Community
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David Wachtfogel
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