Yet the mathematics underpinning these new standards remains locked behind graduate-level abstraction, and a team of researchers at the Universidad de La Laguna in Spain argues that this is a problem education must solve now. In a study published in the Journal of New Approaches in Educational Research, Édgar Pérez-Ramos, Candelaria Hernández-Goya, and Pino Caballero-Gil present an interactive GeoGebra-based activity designed to introduce high school students to the core ideas of lattice-based post-quantum cryptography. The work is notable not only for its subject matter but for its results: after a single session, students with no prior cryptography experience showed dramatic gains in both motivation and conceptual understanding, with some test scores improving by more than 40 percentage points.

The mathematical heart of the activity is the lattice, an object the researchers describe as a multidimensional infinite grid. In its simplest two-dimensional form, a lattice is the collection of all points reachable by adding and subtracting whole-number multiples of a finite set of vectors. This definition deliberately builds on concepts students already meet in secondary school: the dot product of two vectors, orthogonality, Euclidean distance, and the so-called taxi or Manhattan distance, which measures travel along a rectangular grid rather than a straight line. The dot product links the lattice-generating vectors to the cosine of the angle between them, creating a natural bridge to trigonometry, while the taxi distance gives students a tangible way to think about how lattice points relate to one another in space.

Why do these geometric ideas matter for security? Modern lattice-based schemes rest on computational problems that are believed to be intractable even for quantum computers, including the Shortest Vector Problem, the Closest Vector Problem, and the Module Learning With Errors Problem. Finding the shortest vector from a lattice point to an arbitrary target looks trivial in the flat Cartesian plane, but the real cryptographic schemes operate in dimensions up to 1024, with coefficients drawn modulo large prime numbers such as q = 3329 in CRYSTALS-Kyber. Crucially, the less orthogonal the vectors defining a lattice are, the harder the associated problem becomes. A student who can see this relationship in two dimensions has grasped the seed of an idea that protects encrypted communications at planetary scale.

The researchers chose GeoGebra, an open-source mathematics platform combining dynamic geometry, algebra, and data visualization, as the vehicle for the lesson. The publicly available activity, titled A Brief Introduction to Lattice Theory, lets students drag vectors on screen and watch the resulting lattice reshape itself in real time. They can verify that the only null lattice consists entirely of null vectors, discover that non-null vectors with infinite coefficients generate an infinite lattice, and explore what happens when generating vectors are perpendicular versus skewed. Short-answer and open-ended questions are woven throughout, pushing students beyond passive observation toward constructing their own lattices and articulating the rules that govern them.

The design is grounded in three established frameworks from mathematics education research. The Theory of Didactical Situations structures the session into action, formulation, validation, and institutionalization phases: students first manipulate vectors independently, then formulate hypotheses about how vector choices shape the lattice, test those conjectures through GeoGebra’s immediate feedback, and finally formalize their discoveries with the teacher using concepts like basis, orthogonality, and scalar products. APOS Theory describes the cognitive progression from concrete actions to internalized processes, reified objects, and integrated schemas, mapping how a lattice transforms from something students click on into a mathematical object with intrinsic properties. The Theory of Instrumental Genesis adds a dual perspective, treating GeoGebra both as a tool students learn to operate and as an artifact that actively reshapes their mathematical thinking.

One unconventional element is the use of ChatGPT to provide an accessible, natural-language definition of a lattice at the start of the session. The researchers are careful to frame this as a didactical support tool rather than a replacement for instruction: prompts are carefully designed, and outputs are systematically checked by teachers against established academic sources. The activity also connects the abstract material to the physical world through an aerial view of Barcelona, whose famously regular urban grid is overlaid with lattice points at street intersections. Students are challenged to navigate from point A to point B using only lattice points, discovering that there is no single straight-line solution but multiple valid routes, a vivid demonstration of the taxi distance and the non-unique structure of lattice problems.

To evaluate the activity, the team ran experiments with 102 secondary school students in Spain, spanning the final year of compulsory education and both years of the Baccalaureate program. The cohort was 43.14 percent female and 53.92 percent male, with specializations split across technology, natural sciences, social sciences, and arts. A baseline survey revealed that 94.12 percent of participants had no prior experience with cryptography, making the group an ideal test bed for whether digital tools can open access to advanced content. The study followed a pre-post design: a short pre-test, a 30-minute introductory lecture on post-quantum cryptography, roughly one hour with the GeoGebra activity, and then a post-test. Ethical approval was obtained from the university ethics committee, with informed consent from students and their legal guardians, and the dataset was made publicly available on Zenodo.

The statistical results were striking. Using paired two-sample Student’s t-tests at a significance level of 0.05, the researchers rejected both null hypotheses. Students’ self-assessed knowledge of post-quantum cryptography rose from a median of 1 to a median of 3 on a five-point scale, and reported interest increased significantly. Correct answers on the question of what post-quantum cryptography is based on jumped from 3.92 percent to 82.35 percent, while understanding of its mathematical foundations climbed from 23.53 percent to 82.35 percent. The gains extended to prerequisite mathematics: correct responses on the conceptual definition of vectors rose from 40.20 percent to 81.37 percent, vector addition from 32.35 percent to 81.37 percent, and scalar multiplication from 33.33 percent to 84.31 percent. Notably, the largest improvements appeared in questions addressed through dynamic visualization and direct manipulation, suggesting that hands-on exploration with immediate visual feedback consolidates learning more effectively than definitions and examples alone.

The timing of this work is significant beyond the classroom. Spain’s latest PISA results showed its lowest mathematics score since the assessment began, with students dropping eight points compared to the previous edition, creating what the researchers call an urgent need for innovative and useful mathematics materials. The activity aligns with Spain’s current education law, LOMLOE, and its evaluation criteria on selecting digital tools and forms of representation, as well as basic knowledge areas covering dynamic geometry software and computational thinking. It also echoes the Informatics for All Coalition’s vision of citizens as critical users and active creators of technology rather than mere consumers. Existing resources for learning lattice cryptography, such as CrypTool, SageMath materials, and IBM Quantum’s online course, target advanced learners and professionals, leaving a gap at earlier educational stages that this study directly addresses.

The researchers see the activity as a promising model for teacher training initiatives seeking to bring advanced topics into classrooms through technology, and they plan to extend it with more advanced content, including the determinant of a lattice’s fundamental parallelepiped and the Learning With Errors problem. Further workshops with students and teachers are planned to gather additional data and refine the approach, with an eye toward tailoring activities to individual differences in prior knowledge for more equitable outcomes. As the 2035 deadline for the post-quantum transition approaches, the study makes a compelling case that the mathematics protecting tomorrow’s secrets can, with the right interactive tools, be taught today in an ordinary high school classroom.

Subject of Research: Teaching lattice-based post-quantum cryptography to secondary school students through an interactive GeoGebra activity

Article Title: An interactive tool for teaching lattice-based post-quantum cryptography

Article References: An interactive tool for teaching lattice-based post-quantum cryptography. (n.d.). https://doi.org/10.1007/s44322-026-00068-x

Image Credits: AI Generated

DOI: 10.1007/s44322-026-00068-x

Keywords: post-quantum cryptography, lattice-based cryptography, GeoGebra, mathematics education, quantum computing, NIST standards, secondary education, CRYSTALS-Kyber, interactive learning, vectors, taxi distance, digital security