Application of Linear Algebra in Cryptography: RSA Encryption and Decryption of Text Data

Linear algebra used in cryography encryption and decryption of text data with rsa using Ill start with a practical answer, but what I believe is the true answer is down at the bottom. One main point of linear algebra is that linear equations are easy to solve. Let me explain. Equations arise naturally all the time, for example when we try to model something in the world mathematically. Sometimes the equations will involve special functions of the parameters trigonometric, exponential, logarithm, etc. But more often than not the equations are polynomials they involve only performing addition, subtraction, multiplication, and division operations upon the parameters, together with scaling various quantities by co.nstants. This is quite reasonable, as a big part of why mathematics is a useful tool for describing the world is that the basic operations of arithmetic already yield enormous versatility. But there is some bad news polynomial equations, while in some sense simple, can have exceptionally complicated behavior. An entire field of mathematical research algebraic geometry is dedicated to describing the shapes of the solution sets of systems of polynomial equations in many variables. The fact that a lot of effort is expended upon understanding these objects is at least heuristic evidence that understanding them can be nontrivial. In fact, another illustration of the complexity of polynomial equations is given by an amazing theorem roughly speaking, any computation that you can perform on a computer can be encoded into the problem of finding integer solutions to a very complicated system of polynomial equations. More practically, when we have some equations that we care about, we often want to solve them. That is, we want to know if there is at least one solution, and if so, we would like to parametrize the set of all solutions in a co.nvenient way. The unfortunate fact is that beyond the case of linear equations the domain of linear algebra solving systems of polynomial equations very quickly becomes intractable, even using a computer. To explain, let us co.nsider the case of equations in one variable x. Solving a linear equation Ax B 0 is something most elementary school students have the capacity for even if they dont lea.rn the algebraic language until a little later. Solving a quadratic equation Ax^2 Bx C 0 is part of the standard high school curriculum in fact there exists a closed formula for the two solutions in the set of complex numbers to this equation in terms of the coefficients A, B, C. A similar formula exists for cubic degree 3 equations. But one of the great triumphs of early 19th century mathematics was the proof that as soon as the equation involves x^5 or any higher power, there can be no closed formula for its solutions that is, there is no formula involving only extracting square roots, cube roots, etc., of various combinations of the coefficients. So solving non linear equations is in general not a trivial task. Now when there is just one equation in just one variable, theres not really a problem its relatively easy to approximate the solutions to any desired degree of precision in an efficient way. For example, the Newton Raphson iteration we teach calculus students can do the trick. The real issue is when you have a complicated system involving many equations in many variables. It turns out that when your equations are linear you can still solve them efficiently even when there are tons of them. To be precise, the time it takes to solve a linear system of equations is polynomial in the number of equations or variables. When you remove the linearity co.ndition and allow polynomials of higher degree, the be known algorithms for solving such systems take time that is exponential in the number of variables and equations. In other words, solving nonlinear equations on a computer is possible but often impractical in complicated situations. To sum up, linear algebra is important partly because it is the science of the only types of equations that we know how to solve easily. Fortunately, many non linear situations can be studied using linear methods. The be known example of this is calculus, which is entirely about understanding curvy functions and objects using straight, linear approximations to them. Like any good tool, linear algebra can be put to an incredible variety of uses. An I think elegant example is principal component analysis http //en.wikipedia/wiki/Principalcomponentanalysis , but examples abound. Thus there is no shortage of tangible reasons for a student to lea.rn linear algebra in fact, like basic facts about chemistry, physics, and biology, I think one should be embarrassed if one does not know linear algebra. To compare with physics, Newtonian mechanics notions of inertia, momentum, mass, etc. is a starting point for modern physics, a basic tool in u