Mathematical backgroundnumber theorem

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Mathematical backgroundnumber theorem

• Integers Z ,-3,-2,-1,0,1,2,3,
• Division and properties of divisibility
• Modular a mod b a b ?a/b?
• Greatest common divisor dgcd(a,b)
• Whenever ca and cb, then cd.
• Least common multiple dlcm(a,b)
• Whenever ac and bc, then dc.
• Fact if agt0, bgt0, then lcm(a,b)ab /gcd(a,b)

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Prime

• If pgt2, the only divisors of p are 1 and p, then
  p is called prime, otherwise, composite
• Prime number theorem ?(x) denotes ?primes ? x
• ?(x) /(x/ln x) ? 1
• x?17, then ?(x) ? x/ln x
• x ? 1, then ?(x) ? 1.25506 x/ln x

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Relatively prime (co-prime)

• If gcd(a,b) 1, then a and b are co-prime.
• Zn0,1,,n-1
• Zn 0? b ? n gcd(b,n)1
• Fact (Zn,?) forms a group.

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Fundamental theorem of arithmetic

• Any number gt2 has a unique factorization as a
  product of prime powers, regardless of order of
  primes.
• gcd and lcm facts
• Example
• let a48642819, b3458271319 then
• gcd(a,b) 219 38
• lcm(a,b) 28 71319 442624.
• Facts
• if p is prime and pab, the pa or pb.
• If ma and na, and gcd(m,n)1, then mna.

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Algorithms in Z and complexity

• Number of bits k lg n
• Bit operations supposue kmax(lg a, lg b)
• a b O(k)
• a - b O(k)
• a ? b O(k2) (precisely, O(lg a lg b))
• a / b O(k2)
• Example estimate the complexity of n!.

(n-2) (nk k) O(n2 (lg n)2)
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Euclidean algorithm

• Fact if agt0, bgt0 and agtb, then gcd(a,b)gcd(b,a
  mod b)
• Algorithm
• While b ? 0 do the following
• Set r ? a mod b, a?b, b?r.
• Return a.
• Complexity

• O((lg a)3), more precisely, O((lg a)2)

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Modulo, congruence and Zn

• a ? b mod n iff n divides a-b. called a is
  congruent to b modulo n.
• Facts
• a ? b mod n iff a and b have the same remainder
  when divided by n.
• Reflexivity
• Symmetry
• Transitivity, therefore, equivalent class
• If a1 ? b1 mod n and a2 ? b2 mod n
• then a1a2 ? b1b2 mod n and
• a1a2 ? b1b2 mod n
• Define Zn 0,1,,n-1

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Multiplicative inverse

• Given a and n, the multiplicative inverse of a is
  an integer x such that ax ? 1 mod n.
• If exist, then unique, denoted as xa-1
• Given a,b ? Zn , define a/b mod n ab-1 mod n,
  only if b has multiplicative inverse.
• Fact a ? Zn , then a is invertible iff
  gcd(a,n)1.

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Extended Euclidean Algorithm

• Algorithm computer b-1 mod a
• t0 ? 0, t1 ? 1.
• While b ? 0 do the following
• Set q ? ?a/b?, t ? t0 ? q?t1, t0 ?t1, t1 ?t.
• Set r ? a mod b, a?b, b?r. ,
• If a is 1
• then t0 is b-1 mod a (or a ? t0)
• else no inverse
• Complexity O((lg a)3), more precisely, O((lg
  a)2)

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Chinese remainder theorem

• Suppose m1,,mr are pairwise co-primes
• Then the following congruence system
• x ? a1 (mod m1)
• x ? a2 (mod m2)
• .
• x ? ar (mod mr)
• has a unique solution modulo Mm1??mr, which is
  given by
• x ? ?i1raiMiyi mod M. Where MiM/mi and yi ?
  Mi-1 mod mi.

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Specific CRT fact

• If gcd(n1,n2) 1, then
• x a mod n1
• x a mod n2
• Has a unique solution
• x a mod n1n2

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Multiplicative group Zn

• Zn a ? Zn gcd(a, n)1
• In particular, if n is prime, Zn 1,,n-1
• Order of Zn is defined as Zn .
• If a ? Zn , then the order of a, denoted as
  ord(a), is the least positive integer t such that
  at ? 1 mod n.
• Fact for any multiplicative group G of order N
  and g ? G, ord(g)N. (Lagrange theorem)
• Fact if ord(a)t and as ? 1 mod n, then ts.

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Euler phi function ?

• ?(n) denotes the integers co-prime to n.
• ?(n) Zn
• Fact
• If p is prime, then ?(p) p-1
• If p is prime and a?1, then ?(pa) pa pa-1
• If gcd(m,n)1, then ?(mn) ?(m) ?(n)
• If np1e1 p2e2 pkek , then ?(n) n
  (1-1/p1)(1-1/p2) (1-1/pk)
• Fact
• For n?5, ?(n) gtn / (6 ln ln n)

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Euler theorem and Fermat theorem

• Fact let n?2,
• If a ? Zn , then a?(n) ?1 mod n. (Euler theorem)
• If n is a product of distinct primes and r ?s mod
  ?(n) then ar ? as mod n for all integers a.
• Fact let p be a prime
• If gcd(a, p)1, then ap-1 ? 1 mod p. (Fermat
  theorem)
• r ?s mod p-1 then ar ? as mod p for all integers
  a
• In particular, ap ? a mod p

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Generator or primitive element

• let ?? Zn , if ord(?) ?(n), called generator.
• In this case, Zn ?i mod n 0? i ??(n) -1
• If Zn has a generator, then Zn is said to be
  cyclic.

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Properties of generators

• Zn has a generator iff n2,4,pk or 2pk where p
  is a prime and kgt1.
• In particular, if p is a prime, Zp has a
  generator.
• If ? is a generator of Zn , then b ? ?i mod n is
  also a generator iff gcd(i, ?(n))1.
• If Zn is cyclic, the number of generators is ?
  (?(n)).
• ? is a generator of Zn iff ??(n)/p ?1 mod n for
  each prime divisor p of ?(n).

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Quadratic residue modulo n

• let a? Zn, a is said to be a quadratic residue
  modulo n or a square modulo n if there exists an
  x such that x2 ? a mod n. otherwise a is called
  quadratic non-residue modulo n.
• The set of all quadratic residue modulo n is
  denoted by Qn, The set of all quadratic
  non-residue modulo n is denoted by Qn

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Quadratic residue (cont.)

• Fact let p be an odd prime and ? is a generator
  of Zp , then
• a is a quadratic residue iff a(p-1)/2 ? 1 mod p
• a ? Zp is a quadratic residue iff a ? ?i mod p
  and i is even.
• Therefore Qp Qp (p-1)/2.
• Fact let npq and p, q are odd primes, then
• a ? Zn is a quadratic residue iff a ? Qp and a
  ? Qq .
• Therefore Qn Qp Qq (p-1)(q-1)/4.

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Square root

• let a? Zn, if x ? Zn satisfies x2 ? a mod n,
  then x is called the square root of a modulo n.
• Fact
• if p is an odd prime, and a? Qp, then a has
  exactly two square roots.
• More generally, let np1e1 p2e2 pkek where
  the pi are distinct primes and eigt0. If a? Qp,
  then a has exactly 2k square roots.

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Algorithms in Zn and complexity

• If a, b ? Zn , then
• (ab) mod nab or ab-n , O(lg a)
• ab mod n, O((lg a)2)
• Multiplicative inverse O((lg a)3) or O((lg a)2)
• Modular exponentiation ak mod n
• square-and-multiply algorithm
• O((lg a)3)

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Legendre symbol

• Let p be an odd prime and a an integer, Legendre
  symbol is defined as

0, if pa,
1, if a ? Qp

-1, if a ? Qp

• Fact properties of Legendre symbol (page 72 of
  textbook HAC)

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Jacobi symbol

• Let n ? 3 be odd with prime factorization
• np1e1 p2e2 pkek , then Jacobi sysbol is
  defined as

In particular, if p is a prime, then Jacobi
symbol is just Legendre symbol.

• Facts Properties of Jacobi symbol (See page 73
  of textbook HAC)

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Abstract Algebra--group

• Definition (G, )
• Closed
• Associative
• Identity element 1,
• Inverse element
• If commutative, called abelian group.
• Example
• (Z, ) infinite group,
• (Zn , ) group of order n (modulo n).
• (Zn, ?) is not a group
• (Zn, ?) is a group of order ?(n).

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Abstract algebra--ring

• Definition (R, , ?)
• (R, ) abelian group with identity denoted 0.
• ? is associative
• Exist a multiplicative identity 1 ? 0
• ? is distributive over .
• If a ? b b ? a, then commutative ring.
• Example
• (Z, , ?) infinite commutative ring
• (Zn , , ?) commutative ring (modulo n)
• (Zn, , ?) is not a ring.
• An element a of a ring R is called a unit of an
  invertible element if a has an inverse.
• The set of units in a ring R forms a group under
  ?, called the group of unit of R.
• Example the group of units of Zn is Zn.

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Abstract algebra--field

• Definition (F, , ?)
• A commutative ring in which all non-zero elements
  have multiplicative inverse.
• Example
• (Z, , ?) is not a field.
• (R, , ?) is a field.
• Fact
• (Zn, , ?) is a field iff n is a prime.

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Polynomial rings

• Definition
• Degree, Division, remainder
• Irreducible
• Modulo
• Extended Euclidean algorithm