INDEX

B-smooth number

-complete

-hard

-factoring method

b-sequence

kth (higher) power nonresidue

kth (higher) power residue

kth power nonresidue

kth power residue

n−1 primality test

p−1 factoring algorithm

A

additive group

additive identity

additive inverse

Advanced Encryption Standard (AES)

affine transformation

AKS algorithm

AKS primality test

algebraic computation law

algebraic equation

algebraic integer

algebraic number

almostfield

anomalous curve

arithmetic function

arithmetic progression of consecutive primes

arithmetic progression of prime numberss

associativity

asymmetric key cryptography

authentication

authorization

B

baby-step giant-step algorithm

biological (DNA) cryptography

Birch and Swinnerton-Dyer conjecture

blinding attack

block cipher

BSD conjecture

C

Caesar cipher

Carmichael’s -function

Carmichael’s theorem

CFRAC factoring algorithm

CFRAC method

character cipher

Chinese Remainder theorem (CRT)

chosen plaintext attack

chosen-ciphertext attack

chosen-plaintext attack

Church–Turing thesis

ciphers

ciphertext

ciphertext-only attack

closure

coding-based cryptography

coin-tossing states

common modulus attack

common multiple

commutative group

commutative ring

commutativity

complete system of residues

completely multiplicative function

complexity theory

composite number

composite numbers

computability theory

computation theory

computational number theory

computationally infeasible

computationally secure

conditionally unbreakable

confidentiality

congruence

congruence classes

congruent

conic

conjectured intractable problems

consecutive pairs of quadratic residues

consecutive triples of quadratic residues

Continued FRACtion (CFRAC) method

continued fraction algorithm

convergent

convergents

Converse of the Fermat little theorem

Converse of Wilson’s theorem

Cook–Karp thesis

cryptanalysis

cryptanalytic attacks

cryptographic system

cryptography

cryptology

cryptosystem

cubic Diophantine equation

cubic integer

cyclic group

D

Data Encryption Standard (DES)

decryption

degree of polynomial

deterministic cryptosystem

deterministic encryption

DHM assumption

Diffie-Hellman-Merkle key-exchange (DHM)

digital signature algorithm (DSA)

Digital Signature Standard (DSS)

digital signature system

digital signatures

Diophantine geometry

discrete logarithm

Discrete Logarithm Problem (DLP)

Disquisitiones Arithmeticae

dividend

division algorithm

division ring

divisor

DNA-based biological computation

domain

double encryption

E

ECC challenge problems

ECDLP assumption

ECM (Elliptic Curve Factoring Method)

ECPP (elliptic curve primality proving)

elementary attacks on RSA

ElGamal cryptography

ElGamal signature scheme

elite class

elliptic curve

elliptic curve cryptography (ECC)

Elliptic Curve DHM

Elliptic Curve Digital Signature Algorithm (ECDSA)

Elliptic Curve Discrete Logarithm Problem (ECDLP)

Elliptic Curve ElGamal

Elliptic Curve Massey–Omura

elliptic curve primality tests

Elliptic Curve RSA

elliptic curves

elliptic function

elliptic integral

embedding messages on elliptic curves

encryption

equivalence classes

equivalence relation

Euclid

Euclid’s algorithm

Euler’s (totient) -function

Euler’s criterion

Euler’s theorem

even number

exclusive or (XOR)

exponential-time algorithm

extended Euclid’s algorithm

F

factor

factoring by trial divisions

feasibility/infeasibility theory

Federal Information Processing Standard

Fermat’s little theorem

field

finite fields

finite group

finite order of a point on an elliptic curve

finite simple continued fraction

FIPS

FIPS

FIPS

FIPS

fixed-point

fixed-point attack

forward search attack

Function Field Sieve (FFS)

Fundamental Theorem of Arithmetic

G

Galois field

Gauss’s lemma

Gaussian integer

Gaussian prime

general purpose factoring algorithms

geometric composition law

GNFS (General Number Field Sieve)

greatest common divisor (gcd)

Gross–Zagier theorem

group

group laws on elliptic curves

guessing d attack

guessing plaintext M attack

guessing plaintext attack

H

Heegner points

height

high-order congruence

Hill n-cipher

Hill cipher

I

identity

IFP-based cryptography

incongruent

index calculus for DLP

index of a to the base g

index of an integer modulo n

infinite fields

infinite group

infinite order of a point on an elliptic curve

infinite simple continued fraction

information-theoretic security

Integer Factorization Problem (IFP)

integral domain

integrity

inverse

invertible function

irrational numbers

irreducible polynomial

J

Jacobi symbol

K

Kerckhoff principle

key bundle

known-plaintext attack

Knuth’s Factoring Challenge Problem

L

lattice-based cryptography

least common multiple (lcm)

least non-negative residue

least residue

Legendre symbol

Legendre, A. M.

Lehman’s method

Lenstra’s Elliptic Curve Method (ECM)

linear congruence

linear Diophantine equation

logarithms

M

Möbius -function

Möbius inversion formula

Massey-Omura cryptography

mathematical cryptography

McEliece’s coding-based cryptography

Menezes–Vanstone ECC

Mersenne primes

message digest

Miller-Rabin test

Miller–Rabin test

Miller–Selfridge–Rabin test

minimal polynomial

modern cryptography

modular inverse

Modular Polynomial Root Finding Problem (MPRFP)

modulus

monic

monographic cipher

MPRFP

multiple

multiple encryption

Multiple Polynomial Quadratic Sieve (MPQS)

multiplicative function

multiplicative group

multiplicative identity

multiplicative inverse

N

National Institute of Standards and Technology (NIST)

non-secret encryption

nonrepudiation

nonsingular curve

nonsingular elliptic curve

nontrivial divisor

nontrivial square root of

nonwitness

nonzero field element

norm

NTRU cryptosystem

Number Field Sieve (NFS)

number theory

number-theoretic cryptography

O

odd number

one-time pad (OTP)

one-way function

Order Finding Problem (OFP)

order of a modulo n

order of a field

order of a group

order of a point on an elliptic curve

order of an element a in group g

order of an element x modulo n

P

padding process

partial quotients

perfect secrecy

perfect square

period

periodic simple continued fraction

plaintext

Pocklington’s theorem

Pohlig–Hellman cryptosystem

point at infinity

polarization

Pollard’s factoring algorithm

Pollard’s Method

polygraphic cipher

polynomial

polynomial congruence

polynomial congruential equation

polynomial security

polynomial-time algorithm

polynomial-time computable

polynomial-time equivalent

polynomial-time reducible

polynomially secure

positive integers

post-quantum cryptography

powerful number

practical secure

practical/conjectured secure

Pratt’s primality proving

presumably intractable problems

Primality test based on order of integers

Primality test based on primitive roots

Primality test by trial divisions

Primality Test Problem (PTP)

Primality Testing Problem (PTP)

prime factor

Prime Factorization Problem (PFP)

prime field

prime number

Prime Number theorem

prime numbers

prime power

primitive root of n

privacy

private key

probabilistic encryption

probabilistic Turing machine (PTM)

proper divisor

provable intractable problems

provably secure

pseudofield

public key

public-key cryptography

public-key cryptosystem

purely periodic simple continued fraction

Q

quadratic congruence

quadratic integer

quadratic irrational

quadratic non-residue

Quadratic reciprocity law

quadratic residue

quadratic residuosity based cryptosystem

Quadratic Residuosity Problem (QRP)

Quadratic Sieve (QS)

quantum algorithm for discrete logarithms

quantum algorithm for integer factorization

quantum algorithms for elliptic curve discrete logarithms

quantum computational number theory

quantum cryptographic protocol

quantum cryptography

quantum factoring attack

Quantum Integer Factorization

quantum order finding

quantum order finding attack

quantum register

quantum resistant cryptography

qubit

quotient

R

Rabin cryptosystem

Rabin’s M2 encryption

randomized cryptosystem

randomized encryption

randomized Turing machine (RTM)

rank of an elliptic curve

rank of elliptic curve

rational integer

rational integers

rational line

rational number

rational numbers

rational point

rational prime

real base logarithm

real number

real-valued function

rectilinear polarization

reduced system of residues modulo n

reflexive

relatively prime

remainder

Repeated Doubling Method

residue

residue class

residue classes

residue of x modulo n

RFP

Riemann hypothesis

ring

ring with identity

Rivest’s Factoring Challenge Problem

Root Finding Problem (RFP)

root of polynomial

RSA assumption

RSA Cryptography

RSA cryptosystem

RSA numbers

S

salting process

secret key

secret-key cryptography

secret-key cryptosystem

security

semantic security

Shanks’ baby-step giant-step method for discrete logarithms

Shanks’ class group method

Shanks’ SQUFOF method

shift transformation

short plaintext attack

Shortest Vector Problem (SVP)

Sieve of Eratosthenes

signature generation

signature verification

Silver–Pohlig–Hellman algorithm

simple continued fraction

singular curve

size of point on elliptic curve

smooth number

SNFS (Special Number Field Sieve)

special purpose factoring algorithms

SQRT Problem

square number

square root method

Square Root Problem (SQRT)

strong probable prime

strong pseudoprimality test

strong pseudoprime

strong psudoprimality test

subexponential-time complexity

subgroup

substitution cipher

succinct primality certification

SVP

symmetric

symmetric key cryptography

T

test tube

the short d attack

theory of computations

torsion group

torsion subgroup

transitive

trapdoor

trapdoor one-way function

trial division

Triple DES (TDES)

triple prime numbers

triplet primes

trivial divisor

Turing machine

twin prime conjecture

twin prime constant

twin prime numbers

U

unbreakability

unconditionally secure

unconditionally unbreakable

US National Institute of Standards and Technology (NIST)

W

Williams’ M2 encryption

Williams’ M3 encryption

Wilson’s primality test

Wilson’s theorem

witness

X

xedni calculus for ECDLP

Z

zero of polynomial

zero-knowledge proof

zero-knowledge technique

zero-knowlege Identification