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Algorithm for public-key cryptography
The RSA (Rivest–Shamir–Adleman) cryptosystem is a family of public-key cryptosystems (one of the oldest), widely used for secure data transmission. The
RSA_cryptosystem
Number of integers coprime to and less than n
is the order of the multiplicative group of integers modulo n. The RSA cryptosystem is based on this theorem: it implies that the inverse of the function
Euler's_totient_function
American computer security company
patent on the RSA cryptosystem technology granted in 1983. In 1994, RSA was against the Clipper chip during the Crypto War. In 1995, RSA sent a handful
RSA_Security
Approach to public-key cryptography
security, compared to cryptosystems based on modular exponentiation in finite fields, such as the RSA cryptosystem and ElGamal cryptosystem. Elliptic curves
Elliptic-curve_cryptography
Suite of cryptographic algorithms needed to implement a particular security service
public-key type of cryptosystem. A classical example of a cryptosystem is the Caesar cipher. A more contemporary example is the RSA cryptosystem. Another example
Cryptosystem
Form of encryption that allows computation on ciphertexts
included the following schemes: RSA cryptosystem (unbounded number of modular multiplications) ElGamal cryptosystem (unbounded number of modular multiplications)
Homomorphic_encryption
Type of cryptosystem
above and for the following schemes: Damgård–Jurik cryptosystem ElGamal Paillier cryptosystem RSA Broadcast encryption Distributed key generation Secret
Threshold_cryptosystem
Cryptographic system with public and private keys
password-authenticated key agreement techniques Paillier cryptosystem RSA encryption algorithm (PKCS#1) Cramer–Shoup cryptosystem YAK authenticated key agreement protocol
Public-key_cryptography
Method of exchanging cryptographic keys
Diffie-Hellman exchange. The method was followed shortly after by the RSA cryptosystem, an implementation of public-key cryptography using asymmetric algorithms
Diffie–Hellman_key_exchange
Topics referred to by the same term
Root System Architecture RSA (cryptosystem) (Rivest–Shamir–Adleman), for public-key encryption RSA Conference, annual gathering RSA Factoring Challenge, for
RSA
Form of public key cryptography
Several specific public-key cryptosystems were then proposed by other researchers over the next few years, such as RSA in 1977 and Merkle-Hellman in
Merkle–Hellman knapsack cryptosystem
Merkle–Hellman_knapsack_cryptosystem
Cryptographic attack on the RSA system
modulus of N. In the RSA cryptosystem, Bob might tend to use a small value of d, rather than a large random number to improve the RSA decryption performance
Wiener's_attack
Asymmetric key encryption algorithm
used in GM encryption is generated in the same manner as in the RSA cryptosystem. (See RSA, key generation for details.) Alice generates two distinct large
Goldwasser–Micali cryptosystem
Goldwasser–Micali_cryptosystem
Digital verification standard
invested effort in developing digital signature software based on the RSA cryptosystem. Nevertheless, NIST adopted DSA as a Federal standard (FIPS 186) in
Digital_Signature_Algorithm
Type of cryptographic algorithm
Merkle-Hellman Public Key Cryptosystem, one of the first public key cryptosystems, published the same year as the RSA cryptosystem. However, this system has
Knapsack_cryptosystems
Mathematics independent of applications
the problem of factoring large integers, which is the basis of the RSA cryptosystem, widely used to secure Internet communications. It follows that, currently
Pure_mathematics
Asymmetric encryption algorithm developed by Robert McEliece
improvements in information set decoding. The McEliece cryptosystem has some advantages over, for example, RSA. The encryption and decryption are faster. For
McEliece_cryptosystem
Class of cryptographic attacks
the public-key cryptosystem RSA based on the Coppersmith method. Particular applications of the Coppersmith method for attacking RSA include cases when
Coppersmith's_attack
Public-key cryptosystems use a public key for encryption and a private key for decryption. Diffie–Hellman key exchange RSA encryption Rabin cryptosystem Schnorr
List_of_cryptosystems
A prime p divides a^p–a for any integer a
used with n not prime in public-key cryptography, specifically in the RSA cryptosystem, typically in the following way: if y = x e ( mod n ) , {\displaystyle
Fermat's_little_theorem
Public-key encryption scheme
The Rabin cryptosystem is a family of public-key encryption schemes based on a trapdoor function whose security, like that of RSA, is related to the difficulty
Rabin_cryptosystem
Property of some cryptographic algorithms
{\displaystyle E(m)\oplus t=m\oplus t\oplus S(k)=E(m\oplus t)} . In the RSA cryptosystem, a plaintext m {\displaystyle m} is encrypted as E ( m ) = m e mod
Malleability_(cryptography)
However, many cryptosystems are not plaintext-aware. As an example, consider the RSA cryptosystem without padding. In the RSA cryptosystem, plaintexts and
Plaintext-aware_encryption
Unsolved problem in cryptography
eventual security of RSA-based cryptosystems—both for public-key encryption and digital signatures. More specifically, the RSA problem is to efficiently
RSA_problem
Asymmetric key encryption algorithm
hybrid cryptosystem to improve efficiency on long messages. Daniel Bleichenbacher. Chosen ciphertext attacks against protocols based on the RSA encryption
Cramer–Shoup_cryptosystem
Practice and study of secure communication techniques
numbers, such as the RSA cryptosystem, require larger keys than elliptic curve techniques. For this reason, public-key cryptosystems based on elliptic curves
Cryptography
Public-key cryptosystem
free GNU Privacy Guard software, recent versions of PGP, and other cryptosystems. The Digital Signature Algorithm (DSA) is a variant of the ElGamal signature
ElGamal_encryption
Field of knowledge
Euclid in 300 BC, had no practical application before its use in the RSA cryptosystem, now widely used for the security of computer networks. In the 19th
Mathematics
American cryptographer (born 1947)
Rivest, jointly with Adi Shamir and Leonard Adleman, introduced the RSA cryptosystem in 1978,[C1] which revolutionized modern cryptography by providing
Ron_Rivest
Public-key cryptosystem
In cryptography, a key encapsulation mechanism (KEM) is a public-key cryptosystem that allows a sender to generate a short secret key and transmit it to
Key_encapsulation_mechanism
Branch of pure mathematics
for the creation of public-key cryptography algorithms, such as the RSA cryptosystem. Number theory is the branch of mathematics that studies integers and
Number_theory
Asymmetric key encryption algorithm
terms of computation, and fares well even in comparison with cryptosystems such as RSA (depending on message length and exponent choices). However, BG
Blum–Goldwasser_cryptosystem
Process of non-randomly producing the same ciphertext for a given same plaintext and key
algorithm. Examples of deterministic encryption algorithms include RSA cryptosystem (without encryption padding), and many block ciphers when used in ECB
Deterministic_encryption
Theorem on modular exponentiation
Euler's theorem underlies the RSA cryptosystem, which is widely used in Internet communications. In this cryptosystem, Euler's theorem is used with n
Euler's_theorem
(Mathematical) decomposition into a product
much more difficult algorithmically, a fact which is exploited in the RSA cryptosystem to implement public-key cryptography. Polynomial factorization has
Factorization
Algorithm for public key cryptography
The Paillier cryptosystem, invented by and named after Pascal Paillier in 1999, is a probabilistic asymmetric algorithm for public key cryptography. The
Paillier_cryptosystem
Type of prime number
example. Some people suggest that in the key generation process in RSA cryptosystems, the modulus n should be chosen as the product of two strong primes
Strong_prime
Product of two prime numbers
calculation is an important part of the application of semiprimes in the RSA cryptosystem. For a square semiprime n = p 2 {\displaystyle n=p^{2}} , the formula
Semiprime
Measure of cryptographic strength
algorithms, which differ slightly due to different methodologies. For the RSA cryptosystem at 128-bit security level, NIST and ENISA recommend using 3072-bit
Security_level
Placeholder characters
previous articles by Rivest, Shamir, and Adleman, introducing the RSA cryptosystem, there is no mention of Alice and Bob. The choice of the first three
Alice_and_Bob
Cryptographic algorithm for digital signatures
libgcrypt LibreSSL mbed TLS Microsoft CryptoAPI OpenSSL wolfCrypt EdDSA RSA (cryptosystem) Johnson, Don; Menezes, Alfred (1999). "The Elliptic Curve Digital
Elliptic Curve Digital Signature Algorithm
Elliptic_Curve_Digital_Signature_Algorithm
American electrical engineer (1952/1953–2017)
can be regarded as a predecessor to the RSA (cryptosystem) since all that is needed to transform it into RSA is to change the arithmetic from modulo a
Stephen_Pohlig
Natural number
the RSA cryptosystem. Because it is the Fermat number Fn = 22n + 1 with n = 4, the common shorthand is "F4" or "F4". This value was used in RSA mainly
65,537
Probabilistic primality test
historical importance in showing the practical feasibility of the RSA cryptosystem. Euler proved that for any odd prime number p and any integer a, a
Solovay–Strassen primality test
Solovay–Strassen_primality_test
Type of cryptography protocol
received the message. Rabin's oblivious transfer scheme is based on the RSA cryptosystem. A more useful form of oblivious transfer called 1–2 oblivious transfer
Oblivious_transfer
Prime pair of the form (p, 2p+1)
and strong primes were useful as the factors of secret keys in the RSA cryptosystem, because they prevent the system being broken by some factorization
Safe and Sophie Germain primes
Safe_and_Sophie_Germain_primes
Calculations where numbers' precision is only limited by computer memory
307-digit key crack endangers 1024-bit RSA". "RSA Laboratories - 3.1.5 How large a key should be used in the RSA cryptosystem?". Archived from the original on
Arbitrary-precision arithmetic
Arbitrary-precision_arithmetic
Cryptographic primitives that involve lattices
widely used and known public-key schemes such as the RSA, Diffie-Hellman or elliptic-curve cryptosystems—which could, theoretically, be defeated using Shor's
Lattice-based_cryptography
non-commutative cryptography, the currently widely used public-key cryptosystems like RSA cryptosystem, Diffie–Hellman key exchange and elliptic curve cryptography
Non-commutative_cryptography
Hypothesis in computational complexity theory
becomes easy given the factorization of n {\displaystyle n} . In the RSA cryptosystem, ( n , e ) {\displaystyle (n,e)} is the public key, c {\displaystyle
Computational hardness assumption
Computational_hardness_assumption
Private research university in Cambridge, Massachusetts, United States
developed one of the first practical public-key cryptosystems, the RSA cryptosystem, and started a company, RSA Security. Digital circuits – Claude Shannon
Massachusetts Institute of Technology
Massachusetts_Institute_of_Technology
Number divisible only by 1 and itself
mathematics was shattered in the 1970s when public-key cryptography and the RSA cryptosystem were invented, using prime numbers as their basis. The increased practical
Prime_number
Degrees of separation from Paul Erdős
Cryptographers Ron Rivest, Adi Shamir, and Leonard Adleman, inventors of the RSA cryptosystem, all have Erdős number 2. The Romanian mathematician and computational
Erdős_number
Accomplishments in factoring large integers
factorisation was RSA-129, a 129-digit challenge number described in the Scientific American article of 1977 which first popularised the RSA cryptosystem. It was
Integer_factorization_records
Algebraic structure
for factorizing them in polynomial time. This is the basis of the RSA cryptosystem, widely used for secure Internet communications. In the case of K[X]
Polynomial_ring
The Benaloh Cryptosystem is an extension of the Goldwasser-Micali cryptosystem (GM) created in 1985 by Josh (Cohen) Benaloh. The main improvement of the
Benaloh_cryptosystem
American computer scientist (born 1945)
Southern California. For his contribution to the invention of the RSA cryptosystem, Adleman, along with Ron Rivest and Adi Shamir, has been a recipient
Leonard_Adleman
Cryptography framework
E(a,E(b,m)) = mab mod p = mba mod p = E(b,E(a,m)). The Massey–Omura Cryptosystem was proposed by James Massey and Jim K. Omura in 1982 as a possible improvement
Three-pass_protocol
Public university in Israel
of Interior (Yamina) Adi Shamir, cryptographer, co-inventor of the RSA cryptosystem Ariel Sharon (1928–2014), Prime Minister of Israel (Likud and Kadima)
Tel_Aviv_University
Public-key authentication standard
successfully attacked in other protocols and implementations of the RSA cryptosystem in the past. It is difficult to exploit under given conditions in the
WebAuthn
The Damgård–Jurik cryptosystem is a generalization of the Paillier cryptosystem. It uses computations modulo n s + 1 {\displaystyle n^{s+1}} where n {\displaystyle
Damgård–Jurik_cryptosystem
Cryptographer (born 1964)
and RSA public-key cryptosystems. His doctoral advisor was Ueli Maurer. Bleichenbacher is particularly notable for devising attacks against the RSA public-key
Daniel_Bleichenbacher
Annual computing prize
known as an inventor of the RSA public-key cryptosystem," wrote Dr. Dobb's editor Jonathan Erickson. "The RSA cryptosystem has formed the basis of a variety
Dr. Dobb's Excellence in Programming Award
Dr._Dobb's_Excellence_in_Programming_Award
Attack model for cryptanalysis
better approach is to use a cryptosystem which is provably secure under chosen-ciphertext attack, including (among others) RSA-OAEP secure under the random
Chosen-ciphertext_attack
Overview of and topical guide to algorithms
Encryption Standard Triple DES Blowfish (cipher) Twofish ChaCha20-Poly1305 RSA cryptosystem Diffie–Hellman key exchange Elliptic-curve cryptography Digital Signature
Outline_of_algorithms
Study of analyzing information systems in order to discover their hidden aspects
advanced computerized schemes of the present. Methods for breaking modern cryptosystems often involve solving carefully constructed problems in pure mathematics
Cryptanalysis
( 2 κ ) {\displaystyle O(2^{\kappa })} computational power. In the RSA cryptosystem, the security parameter κ {\displaystyle \kappa } denotes the length
Security_parameter
Mathematical scheme for verifying the authenticity of digital documents
invented the RSA algorithm, which could be used to produce primitive digital signatures (although only as a proof-of-concept – "plain" RSA signatures are
Digital_signature
Textbook by scientists Michael Nielsen and Isaac Chuang
Appendix 4: Number Theory Appendix 5: Public Key Cryptography and the RSA Cryptosystem Appendix 6: Proof of Lieb's Theorem Bibliography Index Peter Shor called
Quantum Computation and Quantum Information
Quantum_Computation_and_Quantum_Information
American cryptographer
Public Key Cryptosystem and A Signature Scheme Based on Discrete Logarithms" proposed the design of the ElGamal discrete log cryptosystem and of the ElGamal
Taher_Elgamal
before its common use for secure internet communications through the RSA cryptosystem. A second historical example is the theory of ellipses. They were studied
Philosophy_of_mathematics
cryptographic hash functions SHA-1 SHA-2 SHA-3 SHA-3 competition RSA (cryptosystem) X.509 Pretty Good Privacy Diffie-Hellman key exchange Blowfish (cipher)
List of cybersecurity information technologies
List_of_cybersecurity_information_technologies
Algorithms to generate prime numbers
Cryptography requires the use of very large primes: for example, with the RSA cryptosystem two primes of at least 1,024 bits (i.e. at least 21023) are recommended
Generation_of_primes
application in number theory and in cryptography: for example, in the RSA cryptosystem and Diffie–Hellman key exchange. The most common way of implementing
Kochanski_multiplication
American computer scientist
the ability of quantum computers using Shor's algorithm to break the RSA cryptosystem. She completed her Ph.D. in 2006 at Columbia University, with highest
Krysta_Svore
In cryptography, the Niederreiter cryptosystem is a variation of the McEliece cryptosystem developed in 1986 by Harald Niederreiter. It applies the same
Niederreiter_cryptosystem
Study of stealing information securely and subliminally
cryptosystem would be computationally indistinguishable from the outputs of the corresponding uninfected cryptosystem. If the infected cryptosystem is
Kleptography
General purpose C++ library
encoding/decoding engines Ciphers Elliptic curve cryptography support RSA cryptosystem support X.509 public key certificate support OpenSSL APIs DNS Service
POCO_C++_Libraries
American cryptographer, computer security expert
via a cold boot attack,[A] for her discovery that weak keys for the RSA cryptosystem are in widespread use by internet routers and other embedded devices
Nadia_Heninger
• RSA RSA • RSA-100 • RSA-1024 • RSA-110 • RSA-120 • RSA-129 • RSA-130 • RSA-140 • RSA-150 • RSA-1536 • RSA-155 • RSA-160 • RSA-170 • RSA-180 • RSA-190
Index of cryptography articles
Index_of_cryptography_articles
Security definition for digital signatures
challenger can ask for the signature of a “difficult” message. The RSA cryptosystem has the following multiplicative property: σ ( m 1 ) ⋅ σ ( m 2 ) =
Digital_signature_forgery
System that can issue, distribute and verify digital certificates
EPOC HFE IES Lamport McEliece Merkle–Hellman Naccache–Stern knapsack cryptosystem Three-pass protocol XTR SQIsign SPHINCS+ Theory Discrete logarithm cryptography
Public_key_infrastructure
Office building complex in Cambridge, Massachusetts
system, the Emacs editor, the Polaroid SX-70 camera (partly), the RSA cryptosystem (partly), the Zork computer game, the Model 204 database management
Technology Square (Cambridge, Massachusetts)
Technology_Square_(Cambridge,_Massachusetts)
Cryptographic signature scheme
this a fairly efficient digital signature scheme. The Lamport signature cryptosystem was invented in 1979 and named after its inventor, Leslie Lamport. Alice
Lamport_signature
Israeli cryptographer (born 1952)
cryptographer and inventor. He is a co-inventor of the Rivest–Shamir–Adleman (RSA) algorithm (along with Ron Rivest and Len Adleman), a co-inventor of the
Adi_Shamir
Any attack based on information gained from the implementation of a computer system
from Microsoft Research and Indiana University. Attempts to break a cryptosystem by deceiving or coercing people with legitimate access are not typically
Side-channel_attack
Process of converting plaintext to ciphertext
known as the Diffie-Hellman key exchange. RSA (Rivest–Shamir–Adleman) is another notable public-key cryptosystem. Created in 1978, it is still used today
Encryption
computationally infeasible; this assumption is required for the security of the RSA cryptosystem. The Φ-hiding assumption is a stronger assumption, namely that if p1
Phi-hiding_assumption
American computer scientist (1944–2025)
attack on the multiplicative nature of the textbook version of the RSA cryptosystem's signing was reported in the Communications of the ACM in 1984. "George
George_Davida
Security of a cryptosystem which derives purely from information theory
A cryptosystem is considered to have information-theoretic security (also called unconditional security) if the system is secure against adversaries with
Information-theoretic security
Information-theoretic_security
Key agreement protocol
EPOC HFE IES Lamport McEliece Merkle–Hellman Naccache–Stern knapsack cryptosystem Three-pass protocol XTR SQIsign SPHINCS+ Theory Discrete logarithm cryptography
Elliptic-curve_Diffie–Hellman
Mechanism for authenticating cryptographic keys
EPOC HFE IES Lamport McEliece Merkle–Hellman Naccache–Stern knapsack cryptosystem Three-pass protocol XTR SQIsign SPHINCS+ Theory Discrete logarithm cryptography
Web_of_trust
Lattice-based public key cryptosystem
NTRUEncrypt public key cryptosystem, also known as the NTRU encryption algorithm, is an NTRU lattice-based alternative to RSA and elliptic curve cryptography
NTRUEncrypt
Public-key cryptosystem that uses lattice-based cryptography
NTRU is an open-source public-key cryptosystem that uses lattice-based cryptography to encrypt and decrypt data. It consists of two algorithms: NTRUEncrypt
NTRU
Cryptographic protocol for two-party computation
oblivious transfer can be built using asymmetric cryptography like the RSA cryptosystem. Operator ∥ {\displaystyle \parallel } is string concatenation. Operator
Garbled_circuit
Asymmetric cryptographic technique based on integer factorisation
The Schmidt-Samoa cryptosystem is an asymmetric cryptographic technique, whose security, like Rabin depends on the difficulty of integer factorization
Schmidt-Samoa_cryptosystem
Archive file format for cryptography objects
of standards called Public-Key Cryptography Standards (PKCS) published by RSA Laboratories. The filename extension for PKCS #12 files is .p12 or .pfx.
PKCS_12
IEEE standardization project for public-key cryptography
of NTRU Cryptosystems, Inc., who has served since August 2001. Former chairs were Ari Singer, also of NTRU (1999–2001), and Burt Kaliski of RSA Security
IEEE_P1363
Algorithms for polynomial evaluation
integer factorization can be computed in polynomial time, breaking the RSA cryptosystem. Sometimes the computational cost of scalar multiplications (like a
Polynomial_evaluation
Swedish mathematician (1929–2014)
synthesised that course and became a standard reference for early RSA cryptosystem implementers. Outside academia he co-founded the non-profit Stockholm
Hans_Riesel
RSA CRYPTOSYSTEM
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