**Abelian Group**- An abelian group is a group whose
operation is commutative, ie
. An example of an abelian group is the integers with the usual addition operation. An example of a group which is not abelian is the rotations of a cube (try it out). When dealing with an abelian group is it conventional to denote the group operation as addition (+), rather than multiplication (*) and to denote the unit element as 0 rather than 1.`a`*`b`=`b`*`a` **Carmichael Number**- A Carmichael Number is a composite number which passes the Fermat pseudoprime test for all bases. There are an infinite number of Carmichael numbers - the smallest is 561=11*17*3.
**Composite**- A composite number has non-trivial factors (ie factors other than itself and 1). Thus 13 is prime but 15=3*5 is composite. A number which is not composite is called prime. A polynomial which has non-trivial factors is called reducible.
**Euler Pseudoprime Test**- A more effective pseudoprime test
than the simpler Fermat test.
A number
`N`is called an Euler pseudoprime to base`b`if`b`^{(N-1)/2}=(`b`/`N`) (mod`N`). (Here (`b`/`N`) is the Jacobi symbol.) This test is also called the Solovay-Strassen test for its original proposers. If an integer is an Euler pseudoprime it is also a Fermat pseudoprime. **Extension**- A field
**E**is called an extension of another field**F**if**F**is contained in**E**as a subfield. Examples include the Galois fields, as they are all extensions of the integers modulo a prime`p`. **Factor**- To separate a number (or a polynomial) into a product of other numbers (also called factors). Thus 15 is factored as 15=3*5. A non-trivial factorization has no factors of 1.
**Fermat's Little Theorem**- If
`p`is prime and`b`<`p`then . Rephrased, this says that the order of`b`^{(p-1)}=1(mod`p`)`b`in the group of integers modulo`p`divides (`p`-1). **Fermat Pseudoprime Test**- The simplest (and least effective)
pseudoprime test. A number
`N`is called an Fermat pseudoprime to base`b`if . A Fermat pseudoprime is more commonly just called a pseudoprime. (The name "Fermat pseudoprime" is used as this test is equivalent to Fermat's Little Theorem.)`b`^{(N-1)}=1(mod`N`) **Field**- A field is an algebraic structure with two operators (commonly
called addition (+) and multiplication (*)) which satisfies the
following conditions:
- The elements of the field form an Abelian group under addition

**Galois Fields**- A Galois Field is a field with
finite number of elements. Galois fields take one of
two forms:
**Z**_{p}- The integers modulo some prime`p`.**F**_{p^n}- The polynomials with coefficients modulo some prime`p`with operations modulo some irreducible degree`n`polynomial`r`(`x`).

`q`=`p`^{n}elements is commonly denoted GF`q`or**F**_{q}. **Gaussian Integers**- The ring of Gaussian Integers is the
extension of the integers with a symbol
`i`which is the root of the equation . Thus this ring consists of elements of the form`x`^{2}=-1( with the additional condition that`n`+`m`*`i`) . (For example,`i`^{2}=-1(2+ , showing that 5 is not prime in the Gaussian Integers.)`i`)*(2-`i`)=5 **Group**- A field is an algebraic structure with one operator (commonly
called multiplication (*)) which satisfies the following
conditions:
- There is a distinguished unit element (commonly denoted
1, such that for any element
`a`we have .`a`*1=1*`a`=`a` - Any element
`a`has an inverse (denoted`a`^{(-1)}) such that .`a`^{(-1)}*`a`=`a`*`a`^{(-1)}=1

) is called Abelian.`a`*`b`=`b`*`a` - There is a distinguished unit element (commonly denoted
1, such that for any element
**Irreducible**- An irreducible polynomial has no
factors other than 1 (called
non-trivial factors). Thus
`x`+1 is prime but`x`^{2}-1=(`x`-1)(`x`+1) is not prime. A polynomial which is not irreducible is called composite. A number which has no factors other than 1 is called prime. **Jacobi Symbol**- The Jacobi symbol, denoted
(
`a`/`n`), is a function which is defined in terms of the Jacobi symbol. If the prime factorization of`n`is`n`=`p`1^{e1}*...*`p`k^{ek}then the Jacobi symbol is defined as (`a`/`n`) =(`a`/`p`1)^{e1}*...*(`a`/`p`k)^{ek}. **Legendre Symbol**- The Legendre symbol, denoted
(
`a`/`p`), is a function which is defined if`p`is prime. Its value is:- 0 if
`p`divides`a`. - 1 if
`a`is a quadratic residue (ie there is some integer`b`such that`p`=`b`^{2}(mod`p`)) - -1 if
`a`is not a quadratic residue.

`p`is the Jacobi symbol. - 0 if
**Lucas Pseudoprime Test**- The simplest (and least effective)
pseudoprime test. A number
`N`is called an Fermat pseudoprime to base`b`if`b`^{(N-1)}=1(mod`N`). A Fermat pseudoprime is more commonly just called a pseudoprime. **Miller-Rabin test**- Also called the
strong pseudoprime test, this test was originally
proposed by M. O. Rabin in
*Algorithms and Complexity*, J. F. Traub (Ed), Academic Press, 1976, pp 35-36., based on ideas of G. L. Miller. **Order**- Order has two related meanings:
- The order of a group is the number of elements in the group.
- The order of some element
`b`is the smallest exponent`e`such that .`b`^{e}=1

`b`in the group of integers modulo a prime`p`divides (`p`-1) **Prime**- A prime number is a number which has no factors other than 1 (called non-trivial factors). Thus 13 is prime but 15=3*5 is not prime. A number which is not prime is called composite. A polynomial which has no factors other than 1 is called irreducible.
**Primitive**- Primitivity has two meanings, a primary meaning
and a derived meaning used only in non-prime
Galois fields:
- A primitive element in a group is an element whose
powers exhaust the entire group. Thus 3 is primitive
in the group of units mod 7 as 1=3
^{6}, 2=3^{2}, 3=3^{1}, 4=3^{4}, 5=3^{5}, and 6=3^{3}, but 2 is not primitive in this group as there is no exponent`e`such that 3=2^{e}(mod 7). More commonly we say that 3 is primitive mod 7 but 2 is not. - We say that the polynomial
`p`(`x`) is primitive (mod some finite base field**F**) if the element`x`is primitive (in the previous sense) in the group of units of the polynomials mod`p`(`x`). Thus we say that the polynomial is primitive mod 2 as`x`^{2}+`x`+1 ,`x`=`x`^{1} , and`x`+1=`x`^{2}1= . However, the same polynomial is not primitive mod 5 as`x`^{3}1= `x`^{3}(mod`p`).

- A primitive element in a group is an element whose
powers exhaust the entire group. Thus 3 is primitive
in the group of units mod 7 as 1=3
**Pseudoprime**- A pseudoprime is any number which passes a pseudoprime
test for some base. Thus any prime
number is a pseudoprime for any base. Some pseudoprimes
are not prime though. (For example the
composite number 341=11*31 is
an Fermat pseudoprime for base 2
as 2
^{(341-1)}=1 (mod 341).) Pseudoprimality tests (also called compositeness tests) include:- Fermat test
- Euler test (also called Solovay-Strassen test)
- Strong Pseudoprime test (also called the Miller-Rabin test)
- Lucas test

**Ring****Solovay-Strassen test**- Also called the Euler
pseudoprime test, this test was originally
proposed by Solovay and Strassen in
*SIAM J. Computing,*.**6**(1977), 84-85 and**7**(1978), 118 **Strong Pseudoprime Test**- A pseudoprime test. Let
. If there is some`N`-1 = 2^{s}`q``r`in the range 0<`r`<`s`such that and`b`^{(N-1)/2^r}=-1(mod`N`) then`b`^{(N-1)/2^(r-1)}=1(mod`N`)`N`is called an strong pseudoprime to base`b`. This test is also called the Miller-Rabin test for its originators. If an integer is a strong pseudoprime it is also a Fermat pseudoprime and an Euler pseudoprime.

Robert Campbell Last modified: Dec 26, 1997