IEEE Solid-States Circuits Magazine - Winter 2023 - 40

Lattice-based cryptographic schemes use
computationally infeasible problems from
the lattice theory.
each mathematical class has its own
beauty and advantages (and some
limitations). Lattice-based cryptographic
schemes use computationally
infeasible problems from the
lattice theory. One such problem is
the learning with errors (LWE) one,
which has seen applications in constructing
many postquantum signatures
and key agreement schemes
as well as advanced protocols such as
fully homomorphic encryption, functional
encryption, zero-knowledge
proofs, and so on [3]. So, what is this
LWE problem?
The LWE Problem
In the following, we present a simple
introduction to the LWE problem and
then discuss its application to the
postquantum key agreement. Let's
consider a system of linear equations
using a commonly used matrix-vector
notation, e.g., [2X + 3Y; 3X + 5Y] =
[8; 13]. The solution to this system
of linear equations will be X = 1 and
Y = 2. The system has a solution as
the number of unknowns (i.e., X and
Y) is two and the number of independent
equations is also two. As long as
the number of unknown variables is
equal to or smaller than the number of
equations, we can compute the solution
by hand calculations for small
dimensions or by using a Gaussian
elimination program for larger dimensions.
So, solving such a well-defined
system of linear equations is easy.
Now consider the system of equations
[2X + 3Y + e1; 3X + 5Y + e2] = [9; 12] mod
15 where an unknown small error is
added to each linear equation. You are
still able to find a solution, e.g., X = 1
and Y = 2 with e1 = 1 and e2 = −1, but
it requires more effort. Computing a
solution becomes infeasibly hard
when the number of unknown variables
X, Y, Z, ... excluding the error
terms is approximately 512 or larger.
Even a very large quantum computer
40 WINTER 2023
cannot find a solution! This mathematical
problem of finding the secret
vector [s] from a given matrix [A] and
vector [b] where [b] = [A]·[s] + [e] is
known as the LWE as we have to learn
or solve the unknown variables X, Y,
Z, ... in the presence of unknown
error vector [e]. Solving an LWE

IEEE Solid-States Circuits Magazine - Winter 2023

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IEEE Solid-States Circuits Magazine - Winter 2023 - Cover1
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