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Topic: Multiplicative group of integers modulo n

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	ABSTRACT ALGEBRA ON LINE: Integers
		Any two nonzero integers a and b have a greatest common divisor, which can be expressed as the smallest positive linear combination of a and b.
		Moreover, an integer is a linear combination of a and b if and only if it is a multiple of their greatest common divisor.
		Integers a and b are said to be congruent modulo n if they have the same remainder when divided by n.
	www.math.niu.edu /~beachy/aaol/integers.html (950 words)

	Encyclopedia :: encyclopedia : Number theory (Site not responding. Last check: 2007-10-17)
		The properties of multiplicative functions such as the Möbius function and Euler's φ function are investigated; so are integer sequences such as factorials and Fibonacci numbers.
		Waring's problem (representing a given integer as a sum of squares, cubes etc.), the Twin Prime Conjecture (finding infinitely many prime pairs with difference 2) and Goldbach's conjecture (writing even integers as sums of two primes) are being attacked with analytical methods as well.
		The virtue of the machinery employed -- Galois theory, group cohomology, class field theory, group representations and L-functions -- is that it allows to recover that order partly for this new class of numbers.
	www.hallencyclopedia.com /topic/Number_theory.html (1434 words)

	Multiplicative group of integers modulo n - Wikipedia, the free encyclopedia
		In mathematics, the multiplicative group of integers modulo n is the group with multiplication as group operation and as elements the units in the ring
		Units are the numbers having multiplicative inverses in the sense of modular arithmetic, modulo n.
		A cyclic group always has a generator; a generator of the multiplicative group modulo n is called a primitive root of n.
	en.wikipedia.org /wiki/Multiplicative_group_of_integers_modulo_n (303 words)

	Springer Online Reference Works (Site not responding. Last check: 2007-10-17)
		This group is isomorphic to the multiplicative group of all roots of the equations
		A quasi-cyclic group is the union of an ascending chain of cyclic groups
		Divisible group), and each divisible Abelian group is the direct sum of a set of groups that are isomorphic to the additive group of rational numbers and to quasi-cyclic groups for certain prime numbers
	eom.springer.de /Q/q076440.htm (294 words)

	Multiplicative group Summary
		Multiplicative notation is defined as the system of symbols used in the operation of multiplication to represent numbers and actions.
		The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group, considered as a group scheme.
		That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism e that serves as the identity.
	www.bookrags.com /Multiplicative_group (1245 words)

	Multiplicative group - Wikipedia, the free encyclopedia
		See also: multiplicative group of integers modulo n, additive group.
		The group scheme of n-th roots of unity is by definition the kernel of the n-power map on the multiplicative group, considered as a group scheme.
		That is, for any integer n > 1 we can consider the morphism on the multiplicative group that takes n-th powers, and take an appropriate fiber product in the sense of scheme theory of it, with the morphism e that serves as the identity.
	en.wikipedia.org /wiki/Multiplicative_group (278 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		Applications: Primality Tests - Proving that a number n is prime generally involves finding a primitive element in the multiplicative group of integers modulo n.
		Note: Both 4 and 5 are squares (quadratic residues) modulo 11 3.
		Generally, the largest integer is either 2^31=2147483648 or 2^15=32768.
	www.math.umbc.edu /~campbell/NumbThy/CMST/NUMBTHY.txt (289 words)

	Modular arithmetic Summary
		modulo N. Suppose M is 7, e is 1000 and N is 10.
		Two integers a and b are said to be congruent modulo n, if a and b have the same remainder when divided by n, or equivalently, that their difference (a−b) is a multiple of n.
		= {..., a − 2n, a − n, a, a + n, a + 2n, a + 3n,...}.
	www.bookrags.com /Modular_arithmetic (1696 words)

	Group Theory at the Library of Math (Free Online Mathematics) (Site not responding. Last check: 2007-10-17)
		Basically a group is a set together with a single operation that satisfies certain properties: (1) there must be an identity element, (2) every element must have an inverse, and (3) the associative law must be obeyed.
		In this topic, many examples are given to explain the importance of permutation groups when the underlying set is a finite set of counting numbers; and the matrix form and cycle notation of such permutations are detailed so as to fully explore the groups of permutations of finite sets of counting numbers (called symmetric groups).
		Basically, the center of a group is the collection of elements in the group that commute with all elements in the group and the centralizer of a given element in the group is the collection of all elements in the group that commute with that given element.
	libraryofmath.com /Group_Theory.html (1788 words)

	[No title] (Site not responding. Last check: 2007-10-17)
		In Zn, the additive inverse of a is n-a, since a-(n-a) = n, which is congruent to 0 (mod n).
		The multiplicative inverse of 3 mod 10 is 7, since 7*3=21=1 (mod 10).
		a + b (mod n) = cn+d + en+f (mod n) but cn = en = 0 (mod n) (since c and e are multiples of n), so: = d + f (mod n) = a%n + b%n (mod n).
	students.cs.byu.edu /~cs465ta/lectures/primes_modular.txt (373 words)

	Number theory algorithms
		The main algorithm for the calculation of the GCD of two integers is the binary Euclidean algorithm.
		A partition of an integer n is a way of writing n as the sum of positive integers, where the order of these integers is unimportant.
		A "Gaussian integer" is a complex number of the form z=a+bI, where a and b are ordinary (rational) integers*.
	yacas.sourceforge.net /Algochapter2.html (3917 words)

	Natural to Complex Numbers
		We say that an integer (a, b) is strictly-positive iff a > b, strictly-negative iff a < b, xor zero iff a = b.
		We extend multiplication to the integers by the definition (a, b) * (c, d) = (a c + b d, a d + b c).
		Multiplication: (a, b; -b, a) * (c, d; -d, c) = (a * c - b * d, a * d + b * c; - (a * d + b * c), a * c - b * d).
	www.rism.com /Trig/natural_to_complex_numbers.htm (6128 words)

	Definition and examples
		The binary operation we shall often refer to as multiplication (even though it might be function composition) unless we are in a 'familiar' group where the operation is addition.
		The identity element is 0 and the inverse of n is -n.
		The symmetry group of the square is not abelian neither is the group of
	www.math.csusb.edu /notes/advanced/algebra/gp/node1.html (684 words)

	Fields, Finite Fields (Galois Fields) and Skew Fields - Numericana
		We understand the ternary sum of two integers as the integer whose n-th ternary digit is the sum of the two n-th ternary digits of the operands modulo 3.
		The multiplicative group of a finite field being cyclic, a finite ring is a field if and only if there's a primitive element in it (for example, the nonzero elements of the aforementioned field of order 43046721 are the distinct powers of 6561).
		Using fast exponentiation, a guess-based search for a primitive root may thus result in an efficient proof that the ring is a field or that it's not (albeit much less efficiently in the latter case, so the repeated lack of a firm conclusion is a strong indication that the ring is not a field).
	home.att.net /~numericana/answer/fields.htm (3767 words)

	very_br_txt.nb
		The values of n for which there exists a very(2,1) sequence were determined by Alles by a systematic search for n up to 50 and later extended by using the Inglis and Wiseman condition mentioned above.
		Let a be a very(3,r) sequence of length n, for r=1 or 2, and let z be a sequence of n-1 0's.
		Let a be a very(3,r) sequence of length n, for r=1,2, and let z be a sequence of n-1 0's and w a sequence of 3n-2 0's.
	www.math.vt.edu /people/layman/sequences/very_br.htm (954 words)

	APPENDIX J
		Some elements do; there is a commutative multiplicative group formed by the elements that do have inverses, and the order that group is the value of the Euler function phi(n), which equals the number of integers less than n and greater than zero that are relatively prime to n.
		F is a commutative group with respect to '+' and the additive identity is denoted by '0'.
		In this construction, the field elements (or marks in the language of [Dickson 1900]) are residue classes of the integers J modulo the prime p.
	graham.main.nc.us /~bhammel/FCCR/apdxJ.html (6145 words)

	Ring Theory
		Euler's work on the case n = 3 involved extending ordinary integer arithmetic to apply to the ring of numbers of the form a + b√-3 where a, b are integers.
		The decomposition of an integer into the product of powers of primes has an analogue in rings where prime integers are replaced by prime ideals but, rather surprisingly, powers of prime integers are not replaced by powers of prime ideals but rather by "primary ideals".
		Matrices with their laws of addition and multiplication were introduced by Cayley in 1850 while, in 1870, Pierce noted that the now familiar ring axioms held for square matrices - another early example of the axiomatic approach to rings.
	www-groups.dcs.st-and.ac.uk /~history/PrintHT/Ring_theory.html (1858 words)

	Public Key Cryptography
		If a, b, and n are integers, we say that a is congruent to b modulo n [denoted by a º b (mod n)] if n divides (a-b) or in more specific terms, a = b + kn, for some integer k, called a multiple.
		Abelian Group: a group G with the custom-defined arithmetic operation, #, commutative (i.e., for two elements a and b, a # b = b # a).
		º 1 (mod n), 1 <;= a < n.
	www.acm.org /crossroads/xrds7-1/crypto.html (4780 words)

	Modular arithmetic (Site not responding. Last check: 2007-10-17)
		For example, whilst 8 + 6 equals 14 in conventional arithmetic, in modulo 12 arithmetic the answer is two, as two is the remainder after dividing 14 by the modulus 12.
		Two integers a, b are said to be congruent modulo n if their difference is divisible by n; that is to say, if they leave the same remainder when divided by n.
		Modular arithmetic is often used as a tool for primality tests and integer factorization.
	modular-arithmetic.iqnaut.net (449 words)

	Groups (Site not responding. Last check: 2007-10-17)
		Consider the integers module N, that is, the integers 0, 1, 2, 3,..., N.
		Given an integer m, define m ⋅ n = n + ⋅ ⋅ ⋅ + n, that is, the sum of n m times.
		This is because, in general, integers do not have inverses under multiplication, there is no integer n such that 2 n = 1.
	www.ece.uwaterloo.ca /~ece250/howtos/groups.html (493 words)

	[No title]
		To accomplish this task, we construct two functions Ns and Ss, which represent the normalized, and “smoothed” sums respectively of ap ‘s defined as p+1 minus the number of solutions modulo p, where p is a prime.
		As defined in D1, if G is a finite abelian group written in multiplicative notation, a character (is a homomorphism of G into the multiplicative group of complex numbers.
		We have already mentioned that the group of rational points G(C) on the curve y2 = f(x) = x3 + ax2 +bx +c is a finitely generated abelian group.
	www.math.princeton.edu /mathlab/projects/ellcurves/op/curvecubic.doc (2824 words)

	arthritis pain relief - Schnorr group
		A Schnorr group is a large prime-order subgroup of
		Schnorr groups are useful in discrete log based cryptosystems including Schnorr signatures and DSA.
		Because the Schnorr group is of prime order, it has no non-trivial subgroups, thwarting small subgroup attacks.
	www.painreliefchat.com /arthritis-pain-relief/Schnorr_group (215 words)

	Arithmetic, Numeration, Number Theory - Numericana
		The so-called Gaussian integers are complex numbers of the form a+ib, where a and b are integers.
		The positive integers ½(z+x) and ½(z-x) are coprime (or else the sum and the difference, z and x, wouldn't be coprime).
		powers of the first n integers, note that this sum is a polynomial in n of degree p+1, which is thus fully specified by p+2 of its points.
	home.att.net /~numericana/answer/numbers.htm (7644 words)

	Fast Fourier transform (Site not responding. Last check: 2007-10-17)
		In general, such algorithms depend upon the factorization of n, but (contrary to popular misconception) there are O(''n'' log n) FFTs for all n, even prime n.
		This method (and the general idea of an FFT) was popularized by a publication of J. Cooley and J. Tukey in 1965, but it was later discovered that those two authors had independently re-invented an algorithm known to Carl Friedrich Gauss around 1805 (and subsequently rediscovered several times in limited forms).
		multiplications, proving an achievable lower bound for power-of-two sizes; unfortunately, this comes at the cost of many more additions, a tradeoff no longer favorable on modern processors with hardware multipliers.
	fast-fourier-transform.iqnaut.net (1604 words)

	Math 1301_830
		As a corollary to this theorem, in a finite group,
		Note: (1) congruence modulo n defines a relation on the set of integers; (2) congruence modulo n is reflexive, symmetric and transitive, that is, congruence modulo n is an equivalence relation on the set of integers - see theorem 10.1, page 58.
		By lemma 7.1: the identity element of a subgroup and the identity element of the group must be the same; the inverse of an element in the subgroup must be the same as the inverse element in the group.
	cms.dt.uh.edu /faculty/BecerraL/Spring2006/3306_20505_calendar.htm (3287 words)

	Math Forum - Ask Dr. Math (Site not responding. Last check: 2007-10-17)
		Date: 10/31/97 at 13:47:51 From: Doctor Rob Subject: Re: About Carmichael number Let G(N) be the multiplicative group of units modulo N. The Carmichael function lambda(N) is the exponent of this group; that is, the smallest number e such that a^e = 1 (mod N) for all a in G(N).
		A Carmichael number is defined as a composite number N for which lambda(N) divides N-1.
		Such b's are called primitive roots modulo p, and there are phi(p-1) of them.
	mathforum.org /library/drmath/view/51535.html (482 words)

	[No title]
		It is one of the oldest branches of pure mathematics, and serves as the fundamental of the underlying operations of Elliptic Curve Cryptography.
		The GCD of two positive integers a and b is the largest integer which divides both a and b.
		In general, let a and b be positive integers, with b > 0, there are unique q and r such that a = qp + r, where 0 ≤ r ≤ b-1, q is called the quotient and r the remainder.
	www.ntu.edu.sg /home/ecwchan/ecc/2_mathematic.html (1652 words)

	Residue Class Rings
		Given R, the ring of integers modulo m or an ideal of it, and an element n of R, create the ideal aZintersect Z of the ring of integers.
		Least common multiple of the elements a and b of R, that is, a generator for the R-ideal (a) intersect (b).
		Least common multiple of the sequence of elements Q, that is, a generator for the R-ideal formed by the intersection of the principal ideals generated by elements of Q. [Next][Prev] [Right] [Left] [Up] [Index] [Root]
	www.umich.edu /~gpcc/scs/magma/text538.htm (1344 words)

	MAT 534: Algebra I
		About this Course: The main goal of this course is to study in detail fundamental concepts and methods of algebra that are used in all branches of mathematics.
		Sections 0.1-0.3, 1.1, 1.4; Examples of groups: the group of integers, the group of integers modulo n, the multiplicative group of invertible integers modulo n.
		Further examples of groups: groups of permutations, dihedral groups.
	www.math.sunysb.edu /~vkiritch/MAT534.html (617 words)

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