Number Theory Used in Cryptography
Last Updated :
23 Jul, 2025
Number theory, which is the branch of mathematics relating to numbers and the rules governing them, is the mother of modern cryptography - the science of encrypting communication. Cryptography algorithms that guarantee data privacy, integrity, and authenticity derive their base from number theory, which is utilized to ensure data confidentiality and integrity during online transactions and the identification of digital users.
In this article, we show where the number theory is used in real-life applications in cryptography and how it helps to keep the digital world safe against hackers and unwelcome guests.
Applications of Number Theory in Cryptography
Fundamentals of number theory are vital for the formation of modern cryptographic algorithms that assure secure communication and data privacy in many examples of practical use. Cryptographic algorithms based on the number theory which are already in practice include RSA and elliptic curve among others and are employed in securing online transactions, digital signatures and controlled access systems.
Applications of Number Theory in Cryptography
These algorithms use the intrinsic properties of prime numbers, modular arithmetic, and the discrete logarithm to encrypt confidential information, sign digital documents, and identify the users. In addition, principles of number theory like cryptographic hash functions help in successfully verifying data integrity and preventing tampering during digital documentation or even blockchain transactions.
Organizations and individuals can shield their digital properties and also their identities and communications from cyber-attacks and unauthorized access by applying number theory in cryptography.
Real-Life Applications of Number Theory in Cryptography
Various applications of Number Theory Used in Cryptography are:
Secure Online Transactions
Number theory-based cryptographic communication protocols, like SSL/TLS (Secure Socket Layer/Transport Layer Security), serve the purpose of keeping online transactions and operations at e-commerce platforms safe and secure. These protocols use asymmetric encryption for making a two-way channel between the web browser and the server safe for protecting data like credit card information or personal details as it keeps them hidden from sniffing and manipulation.
Example: When you make an online purchase, you can recognize that your connection is encrypted using SSL/TLS protocols if you see the "https://" prefix in the URL, it will protect your financial information during the transaction so that nobody gets access to it.
Digital Identity Verification
Number theory based digital techniques like 'digital signature' and 'public key infrastructure (PKI)' are used fundamentally to verify digital identity and authentication. Digital signatures are used to sign the electronic document, the contract, and the transaction. It provides certainty in the integrity of the signed data and the authentication of the signer.
Example: When you add your digital signature to an electronically signed document, the recipients will then be able to ensure the authenticity, safety, and faithfulness by using the signer's public key.
Secure Messaging and Communication
End-to-end encryption protocols like PGP (Pretty Good Privacy) and Signal Protocol, which used number theory-related cryptographic algorithms, contribute to privacy in secure messaging platforms. These algorithms employ the public key cryptography to encrypt and decrypt the message and by doing so make sure that only intended recipients are allowed to access the encrypted data.
Example: Messaging applications like WhatsApp and Signal which use end-to-end encryption, enable the privacy of a users conversations by defying unauthorized access by third parties like service provider and hackers.
Data Integrity and Tamper Resistance
Number theory foundational principles made it possible to create hash function based on cryptographic hashing, which is used for checking the integrity and authenticity of digital data, e.g. software updates, digital certificates, and blockchain transactions. The hash values which are computed by using cryptographic hash functions are the actual unique identifiers that help in the integrity verification of the data, any unauthorized modification or tampering is detectable.
Example: Blockchain technology that crypto world uses is based on cryptographic hash functions to create the hash records of transactions which are immutable and unchangeable and transparency of distributed ledger systems used in cryptocurrencies like Bitcoin and Ethereum.
Password Security and Authentication
Cryptographic algorithms used for complex password hashing and authentication protocols have their roots in number theory concepts like prime numbers and modular arithmetic. Cryptographically secure hashing functions, like bcrypt and Argon2, are able to do this securely, while protecting against password-oriented attacks like the so-called brute force or dictionary attacks.
Example: Using password hashing algorithms, websites and online services are able to safely encrypt and store user passwords in hashed form, so that the plain text of the passwords is never available to an unauthorized view during a data breach, therefore, reducing the risk of a password being hacked.
Secure Software and Firmware Updates
Public keys on number theory principles are used to sign everything from software to firmware updates on the internet to proof them to be authentic and unchanged. Digital credentials produce by public key cryptography guarantee that software patches have not been meddled with or customized by unapproved characters, guarding against malware infusion and supply chain attacks.
Example: Operating systems like Windows and macOS use crypto signatures of downloaded software updates so as to check the authenticity of the software from the official sites to prevent it from being modified via an outsider.
Secure Access Control Systems
Public key cryptosystems based on number theory are helpful in building access control systems, encompassing physical and digital isolation, according to which only authorized people can get into certain areas, buildings and computer networks. The implementation of such mechanisms are through cryptographic protocols and algorithms that are responsible for the authentication of users, granting access authorization, and assuring confidentiality.
Example: Biometric identification systems, such as fingerprint scanners and iris recognition systems, employ cryptography based on number theory concepts, to safeguard biometric templates as well as to ascertain the authenticity of identities of people restricting entry to closed environments or confidential data.
Related Articles:
Similar Reads
Maths Mathematics, often referred to as "math" for short. It is the study of numbers, quantities, shapes, structures, patterns, and relationships. It is a fundamental subject that explores the logical reasoning and systematic approach to solving problems. Mathematics is used extensively in various fields
5 min read
Basic Arithmetic
What are Numbers?Numbers are symbols we use to count, measure, and describe things. They are everywhere in our daily lives and help us understand and organize the world.Numbers are like tools that help us:Count how many things there are (e.g., 1 apple, 3 pencils).Measure things (e.g., 5 meters, 10 kilograms).Show or
15+ min read
Arithmetic OperationsArithmetic Operations are the basic mathematical operations—Addition, Subtraction, Multiplication, and Division—used for calculations. These operations form the foundation of mathematics and are essential in daily life, such as sharing items, calculating bills, solving time and work problems, and in
9 min read
Fractions - Definition, Types and ExamplesFractions are numerical expressions used to represent parts of a whole or ratios between quantities. They consist of a numerator (the top number), indicating how many parts are considered, and a denominator (the bottom number), showing the total number of equal parts the whole is divided into. For E
7 min read
What are Decimals?Decimals are numbers that use a decimal point to separate the whole number part from the fractional part. This system helps represent values between whole numbers, making it easier to express and measure smaller quantities. Each digit after the decimal point represents a specific place value, like t
10 min read
ExponentsExponents are a way to show that a number (base) is multiplied by itself many times. It's written as a small number (called the exponent) to the top right of the base number.Think of exponents as a shortcut for repeated multiplication:23 means 2 x 2 x 2 = 8 52 means 5 x 5 = 25So instead of writing t
9 min read
PercentageIn mathematics, a percentage is a figure or ratio that signifies a fraction out of 100, i.e., A fraction whose denominator is 100 is called a Percent. In all the fractions where the denominator is 100, we can remove the denominator and put the % sign.For example, the fraction 23/100 can be written a
5 min read
Algebra
Variable in MathsA variable is like a placeholder or a box that can hold different values. In math, it's often represented by a letter, like x or y. The value of a variable can change depending on the situation. For example, if you have the equation y = 2x + 3, the value of y depends on the value of x. So, if you ch
5 min read
Polynomials| Degree | Types | Properties and ExamplesPolynomials are mathematical expressions made up of variables (often represented by letters like x, y, etc.), constants (like numbers), and exponents (which are non-negative integers). These expressions are combined using addition, subtraction, and multiplication operations.A polynomial can have one
9 min read
CoefficientA coefficient is a number that multiplies a variable in a mathematical expression. It tells you how much of that variable you have. For example, in the term 5x, the coefficient is 5 — it means 5 times the variable x.Coefficients can be positive, negative, or zero. Algebraic EquationA coefficient is
8 min read
Algebraic IdentitiesAlgebraic Identities are fundamental equations in algebra where the left-hand side of the equation is always equal to the right-hand side, regardless of the values of the variables involved. These identities play a crucial role in simplifying algebraic computations and are essential for solving vari
14 min read
Properties of Algebraic OperationsAlgebraic operations are mathematical processes that involve the manipulation of numbers, variables, and symbols to produce new results or expressions. The basic algebraic operations are:Addition ( + ): The process of combining two or more numbers to get a sum. For example, 3 + 5 = 8.Subtraction (−)
3 min read
Geometry
Lines and AnglesLines and Angles are the basic terms used in geometry. They provide a base for understanding all the concepts of geometry. We define a line as a 1-D figure that can be extended to infinity in opposite directions, whereas an angle is defined as the opening created by joining two or more lines. An ang
9 min read
Geometric Shapes in MathsGeometric shapes are mathematical figures that represent the forms of objects in the real world. These shapes have defined boundaries, angles, and surfaces, and are fundamental to understanding geometry. Geometric shapes can be categorized into two main types based on their dimensions:2D Shapes (Two
2 min read
Area and Perimeter of Shapes | Formula and ExamplesArea and Perimeter are the two fundamental properties related to 2-dimensional shapes. Defining the size of the shape and the length of its boundary. By learning about the areas of 2D shapes, we can easily determine the surface areas of 3D bodies and the perimeter helps us to calculate the length of
10 min read
Surface Areas and VolumesSurface Area and Volume are two fundamental properties of a three-dimensional (3D) shape that help us understand and measure the space they occupy and their outer surfaces.Knowing how to determine surface area and volumes can be incredibly practical and handy in cases where you want to calculate the
10 min read
Points, Lines and PlanesPoints, Lines, and Planes are basic terms used in Geometry that have a specific meaning and are used to define the basis of geometry. We define a point as a location in 3-D or 2-D space that is represented using coordinates. We define a line as a geometrical figure that is extended in both direction
14 min read
Coordinate Axes and Coordinate Planes in 3D spaceIn a plane, we know that we need two mutually perpendicular lines to locate the position of a point. These lines are called coordinate axes of the plane and the plane is usually called the Cartesian plane. But in real life, we do not have such a plane. In real life, we need some extra information su
6 min read
Trigonometry & Vector Algebra
Trigonometric RatiosThere are three sides of a triangle Hypotenuse, Adjacent, and Opposite. The ratios between these sides based on the angle between them is called Trigonometric Ratio. The six trigonometric ratios are: sine (sin), cosine (cos), tangent (tan), cotangent (cot), cosecant (cosec), and secant (sec).As give
4 min read
Trigonometric Equations | Definition, Examples & How to SolveTrigonometric equations are mathematical expressions that involve trigonometric functions (such as sine, cosine, tangent, etc.) and are set equal to a value. The goal is to find the values of the variable (usually an angle) that satisfy the equation.For example, a simple trigonometric equation might
9 min read
Trigonometric IdentitiesTrigonometric identities play an important role in simplifying expressions and solving equations involving trigonometric functions. These identities, which include relationships between angles and sides of triangles, are widely used in fields like geometry, engineering, and physics. Some important t
10 min read
Trigonometric FunctionsTrigonometric Functions, often simply called trig functions, are mathematical functions that relate the angles of a right triangle to the ratios of the lengths of its sides.Trigonometric functions are the basic functions used in trigonometry and they are used for solving various types of problems in
6 min read
Inverse Trigonometric Functions | Definition, Formula, Types and Examples Inverse trigonometric functions are the inverse functions of basic trigonometric functions. In mathematics, inverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. The inverse trigonometric functions are the inverse functions of basic trigonometric function
11 min read
Inverse Trigonometric IdentitiesInverse trigonometric functions are also known as arcus functions or anti-trigonometric functions. These functions are the inverse functions of basic trigonometric functions, i.e., sine, cosine, tangent, cosecant, secant, and cotangent. It is used to find the angles with any trigonometric ratio. Inv
9 min read
Calculus
Introduction to Differential CalculusDifferential calculus is a branch of calculus that deals with the study of rates of change of functions and the behaviour of these functions in response to infinitesimal changes in their independent variables.Some of the prerequisites for Differential Calculus include:Independent and Dependent Varia
6 min read
Limits in CalculusIn mathematics, a limit is a fundamental concept that describes the behaviour of a function or sequence as its input approaches a particular value. Limits are used in calculus to define derivatives, continuity, and integrals, and they are defined as the approaching value of the function with the inp
12 min read
Continuity of FunctionsContinuity of functions is an important unit of Calculus as it forms the base and it helps us further to prove whether a function is differentiable or not. A continuous function is a function which when drawn on a paper does not have a break. The continuity can also be proved using the concept of li
13 min read
DifferentiationDifferentiation in mathematics refers to the process of finding the derivative of a function, which involves determining the rate of change of a function with respect to its variables.In simple terms, it is a way of finding how things change. Imagine you're driving a car and looking at how your spee
2 min read
Differentiability of a Function | Class 12 MathsContinuity or continuous which means, "a function is continuous at its domain if its graph is a curve without breaks or jumps". A function is continuous at a point in its domain if its graph does not have breaks or jumps in the immediate neighborhood of the point. Continuity at a Point: A function f
11 min read
IntegrationIntegration, in simple terms, is a way to add up small pieces to find the total of something, especially when those pieces are changing or not uniform.Imagine you have a car driving along a road, and its speed changes over time. At some moments, it's going faster; at other moments, it's slower. If y
3 min read
Probability and Statistics
Basic Concepts of ProbabilityProbability is defined as the likelihood of the occurrence of any event. It is expressed as a number between 0 and 1, where 0 is the probability of an impossible event and 1 is the probability of a sure event.Concepts of Probability are used in various real life scenarios : Stock Market : Investors
7 min read
Bayes' TheoremBayes' Theorem is a mathematical formula used to determine the conditional probability of an event based on prior knowledge and new evidence. It adjusts probabilities when new information comes in and helps make better decisions in uncertain situations.Bayes' Theorem helps us update probabilities ba
13 min read
Probability Distribution - Function, Formula, TableA probability distribution is a mathematical function or rule that describes how the probabilities of different outcomes are assigned to the possible values of a random variable. It provides a way of modeling the likelihood of each outcome in a random experiment.While a Frequency Distribution shows
13 min read
Descriptive StatisticStatistics is the foundation of data science. Descriptive statistics are simple tools that help us understand and summarize data. They show the basic features of a dataset, like the average, highest and lowest values and how spread out the numbers are. It's the first step in making sense of informat
5 min read
What is Inferential Statistics?Inferential statistics is an important tool that allows us to make predictions and conclusions about a population based on sample data. Unlike descriptive statistics, which only summarize data, inferential statistics let us test hypotheses, make estimates, and measure the uncertainty about our predi
7 min read
Measures of Central Tendency in StatisticsCentral tendencies in statistics are numerical values that represent the middle or typical value of a dataset. Also known as averages, they provide a summary of the entire data, making it easier to understand the overall pattern or behavior. These values are useful because they capture the essence o
11 min read
Set TheorySet theory is a branch of mathematics that deals with collections of objects, called sets. A set is simply a collection of distinct elements, such as numbers, letters, or even everyday objects, that share a common property or rule.Example of SetsSome examples of sets include:A set of fruits: {apple,
3 min read
Practice