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Modular Arithmetic and the RSA Cryptosystem

Feb 10, 2026

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This article explores how modular arithmetic forms the foundation of the RSA cryptosystem, a widely used encryption method that ensures secure communication. We delve into key concepts such as modular exponentiation, prime factorization, and the role of public and private keys in secure data transmission. By understanding these mathematical principles, readers can appreciate the security RSA provides and the underlying algorithms that protect sensitive information in the digital age.

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Modular Arithmetic and the RSA Cryptosystem

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RSA cryptosystem--preview

RSA cryptosystem--preview. Suppose n=p q and  ( n )=( p -1)( q -1), where p and q are big primes. Select (find) a and b , such that a  b= 1 mod  ( n ). K =( n ,p,q,a, b ), publicize n , b , but keep p,q,a secret. For any x,y  Z n , define

403 views • 13 slides

RSA Cryptosystem

RSA Cryptosystem. by Drs. Charles Tappert and Ron Frank The information presented here comes primarily from https ://en.wikipedia.org/wiki/RSA_(cryptosystem ) and http://www.math.uchicago.edu/~ may/VIGRE/VIGRE2007/ REUPapers /FINALAPP/Calderbank.pdf. RSA Cryptosystem.

325 views • 7 slides

RSA Asymmetric Key Cryptosystem

RSA Asymmetric Key Cryptosystem. Presented by Katherine Heller COSC 4765 University of Wyoming April 26, 2011. Asymmetric Key Cryptography. Introduced 1970’s Whitfield Diffie and Martin Hellman Known as Public Key Encryption (PKE) Eliminated need for shared private keys. RSA.

436 views • 19 slides

Modular Arithmetic and the RSA Cryptosystem

Modular Arithmetic and the RSA Cryptosystem. p-1. 1. ´ p. MAX(a,b) + MIN(a,b) = a+b. n|m means that m is a an integer multiple of n. We say that “n divides m”. True: 5|25 2|-66 7|35, False: 4|5 8|2. Greatest Common Divisor: GCD(x,y) = greatest k ¸ 1 s.t. k|x and k|y.

1.18k views • 98 slides

Section 4.4: The RSA Cryptosystem

Section 4.4: The RSA Cryptosystem. Practice HW Handwritten and Maple Exercises p. 11-13 at end of class notes.

552 views • 42 slides

The RSA Cryptosystem and Factoring Integers

The RSA Cryptosystem and Factoring Integers. OUTLINE. [1] Modular Arithmetic Algorithms [2] The RSA Cryptosystem [3] Quadratic Residues [4] Primality Testing [5] Square Roots Modulo n [6] Factoring [ 7] The Rabin Cryptosystem. [1] Modular Arithmetic Algorithms 1. The integers

644 views • 52 slides

Modular Arithmetic

Modular Arithmetic. Lecture 8. Modular Arithmetic. Def : a  b (mod n) iff n|( a - b) iff a mod n = b mod n. Modular Addition. Lemma : If a  c (mod n), and b  d (mod n) then a+b  c+d (mod n). Modular Multiplication.

991 views • 26 slides

RSA Cryptosystem

RSA Cryptosystem. What is RSA ?. RSA is a public-key cryptosystem . It was developed by Ronald Rivest, Adi Shamir, and Leonard Adleman in 1977. The security of the RSA system is based on the hard problem of factoring a large integer. Public-key cryptosystem.

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The RSA Cryptosystem and Factoring Integers (I)

The RSA Cryptosystem and Factoring Integers (I). Rong-Jaye Chen. OUTLINE. [1] Modular Arithmetic Algorithms [2] The RSA Cryptosystem [3] Quadratic Residues [4] Primality Testing [5] Square Roots Modulo n [6] Factoring Algorithms [7] Other Attacks on RSA [ 8] The Rabin Cryptosystem

739 views • 46 slides

The RSA Cryptosystem and Factoring Integers (II)

The RSA Cryptosystem and Factoring Integers (II). Rong-Jaye Chen. OUTLINE. [1] Modular Arithmetic Algorithms [2] The RSA Cryptosystem [3] Quadratic Residues [4] Primality Testing [5] Square Roots Modulo n [6] Factoring Algorithms [7] Other Attacks on RSA [8] The Rabin Cryptosystem

520 views • 29 slides

RSA Cryptosystem

RSA Cryptosystem. Outline. Euler’s theorem ( §10.1.3) RSA cryptosystem ( §10.2.3) Definition Example Security Correctness Algorithms for RSA Modular power ( §10.1.4) Modular inverse ( §10.1.5) Randomized primality testing ( §10.1.6). Euler’s Theorem.

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The RSA Cryptosystem

The RSA Cryptosystem. Dan Boneh Stanford University. The RSA cryptosystem. First published: Scientific American, Aug. 1977. (after some censorship entanglements) Currently the “work horse” of Internet security: Most Public Key Infrastructure (PKI) products.

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Modular Arithmetic and RSA Encryption

Modular Arithmetic and RSA Encryption. Stuart Reges Principal Lecturer University of Washington. Some basic terminology. Alice wants to send a secret message to Bob Eve is eavesdropping Cryptographers tell Alice and Bob how to encode their messages Cryptanalysts help Eve to break the code

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Modular Arithmetic and the RSA Cryptosystem

Modular Arithmetic and the RSA Cryptosystem. p-1. 1.  p. Starring. Adleman. Shamir. Rivest. Euler. Fermat. The RSA Cryptosystem. R ivest, S hamir, and A delman (1978)

987 views • 92 slides

The RSA Cryptosystem and Factoring Integers

The RSA Cryptosystem and Factoring Integers. Rong-Jaye Chen. OUTLINE. [1] Modular Arithmetic Algorithms [2] The RSA Cryptosystem [3] Quadratic Residues [4] Primality Testing [5] Square Roots Modulo n [6] Factoring [ 7] The Rabin Cryptosystem. [1] Modular Arithmetic Algorithms

552 views • 52 slides

Modular Arithmetic and the RSA Cryptosystem

Modular Arithmetic and the RSA Cryptosystem. p-1. 1. ´ p. MAX(a,b) + MIN(a,b) = a+b. n|m means that m is an integer multiple of n. We say that “n divides m”. True: 5|25 2|-66 7|35, False: 4|5 8|2. Greatest Common Divisor: GCD(x,y) = greatest k ¸ 1 s.t. k|x and k|y.

997 views • 97 slides

The RSA Cryptosystem and Factoring Integers (II)

317 views • 29 slides

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