determining primitive roots by christoph and john c. witzgall september 15, 2015
TRANSCRIPT
DETERMININGPRIMITIVE ROOTS
byChristoph and John C. Witzgall
September 15, 2015
Divisors/Multipliers Integers will be capitalized. For A, B > 0,
gcd(A, B) = greatest common divisor:W|A and W|B => W|gcd (A, B)lcm(A, B) = least common multipleA|W and B|W => lcm(A, B)|W
(1)
A, B are relatively prime <=> gcd(A, B)=1
gcd(A, B) × lcm(A, B) = A×B
REDUCTION MODULO P>0For P > 0 , any integer X may be represented as
X = S×P + r, 0 ≤ r < P,with r the
remainder of X modulo P
and S the value of integer (“long”) division of X by P. We say that X is reduced to r modulo P.
ARITHMETIC MODULO P>0
The reduction process is used to define the arithmetic modulo P.
For remainders a, b between 0 and P-1, the operations
a + b, a - b, a×bare evaluated using their integer face value, if necessary, reducing the results modulo P.
THE GROUP GP
For P a prime, the remainders,
1, 2, 3, … , P–1
under multiplication modulo P form the group,
GP
our subject of interest.
CYCLES MODULO 71
CYCLES MODULO 7
1 2
CYCLES MODULO 7
1 2 4
CYCLES MODULO 7
1 2 4 8 -7
CYCLES MODULO 7
1 2 4 1
CYCLES MODULO 7
1 2 4 1 2
CYCLES MODULO 7
1 2 4 1 2 4
CYCLES MODULO 7
1 2 4 1 2 4 8 -7
CYCLES MODULO 7
1 2 4 1 2 4 1 . . .
CYCLES MODULO 7
(1 2 4)1
CYCLES MODULO 7
(1 2 4)1 3
CYCLES MODULO 7
(1 2 4)1 3 9 -7
CYCLES MODULO 7
(1 2 4)1 3 2
CYCLES MODULO 7
(1 2 4)1 3 2 6
CYCLES MODULO 7
(1 2 4)1 3 2 6 18 -14
CYCLES MODULO 7
(1 2 4)1 3 2 6 4
CYCLES MODULO 7
(1 2 4)1 3 2 6 4 12 -7
CYCLES MODULO 7
(1 2 4)1 3 2 6 4 5
CYCLES MODULO 7
(1 2 4)1 3 2 6 4 5 15 -14
CYCLES MODULO 7
(1 2 4)1 3 2 6 4 5 1
CYCLES MODULO 7
C(2) = (1 2 4)C(3) = (1 3 2 6 5) 3 = primitive rootC(4) = (1 4 2)C(5) = (1 5 6 2 3) 5 = primitive rootC(6) = (1 6)
GROUPS
Groups considered here are finite and abelian. The notation
|G| = “order of G”Means “number of elements.” Fundamentally,
(2) N = |G| => aN = 1 for a Є G H G => |H| divides |G|⊆
CYCLESThe cycle
C(a) = (1, a, a2, … , aN-1), aN = 1,encapsulates the period of a sequence of consecu-tive powers of an element a Є G.
(3) N = |C(a)|, aR = 1 => N|R
The entries in C(a) form a group. Such groups, generated by a single element are called cyclic.
SUBCYCLES Suppose N = ǀC(a)ǀ, KǀN, M = N/K. Then
C(aK ) = (1 aK a2K … ) is a subcycle of C(a). Its length is given by (4) Proposition: |C(aK )| = N/K = M
Proof: aKM = aN = 1. If aKJ = 1 for 0 < J ≤ Mthen N|KJ. Thus M|J, so that J = M.
SPREADS
Subgroups H1, H2 G together ⊂
generate what we call their
spread H1 ᴗ H2
Spread
H1 H2
Inter section
ORDERS OF SPREADS
For subgroups H1, H2 of abelian group G:
Theorem: |H1 ∩ H2| × |H1 H∪ 2| = |H1|×|H2|
Proof: H1×H2, |H1×H2| = |H1|×|H2|, the group of
pairs (a, b); a Є H1, b Є H₂, (a₁ b₁)×(a₂ b₂) = (a₁b₁ a₂b₂).
Φ(a₁ a₂) = a₁×a₂ is a homomorphism onto H1 H∪ 2.
Φ-1(1) isomorphic to H1 ∩ H2 : |Φ-1 (1)| = |H1 ∩ H2|
|H1 H∪ 1| = = =
COROLLARY
By (2), |H1 ∩ H2|is a common divisor of |H1|,|H2| .
Thus |H1 ∩ H2| divides gcd(|H1|,|H2|).
H1,H2 relatively prime <=> gcd(|H1|,|H2|)=1:
|H1 ∩ H2|= 1
(5) gcd(|H1|,|H2|)=1=>
|H1 H∪ 2|= |
PRIMITIVE ROOTS
(6) Primitive Root Theorem: Gp is cyclic
This means that Gp may be generated by
a single one of its elements. Each such generator is a primitive root of P. We propose:• A constructive proof based on prime
factorization of P-1• An algorithm for computing primitive roots.
APPROACHIn what follows, we aim to prove the
(7) Theorem: The spread of two cycles C(a) and C(b) in GP is cyclic
by characterizing a generator x Є GP :
__
C(a) C(b) = C(x) ∪
Successively collapsing pairs of cycles into single ones then yields a primitive root.
RELATIVELY PRIME CYCLES
We call cycles C(a), C(b) in group G relatively prime if ǀC(a)ǀ, ǀC(b)ǀ are relatively prime.
(8) Theorem: The spread of relatively prime cycles C(a) and C(b) is cyclic:
C(a) C(b) = C(a∪ ×b).
(Proof after the next slide.)
CONVENTION
For what follows in this presentation, we are using the notation
A = | C(a)| , B = | C(b)|
for the orders of cycles C(a) and C(b).
PROOF OF THEOREM (8)By (5) and (1), |C(a) C(b)| = A∪ ×B = lcm (A, B).
For M > 0: (a×b)M = 1 <=> aM = (bM ) -1 Є C(a) ∩ C(b) <=> aM = b M = 1 <=> A|M and B|M by (3)In other words, the exponent
M = lcm(A, B) = A×B is the smallest positive exponent with (a×b)M = 1. Thus C(a) C(b) = C (a×b).∪
CYCLES MODULO 11C(2) = (1 2 4 8 5 10 9 7 3 6)C(3) = (1 3 9 5 4) C(4) = (1 4 5 9 3)C(5) = (1 5 3 4 9)C(6) = (1 6 3 7 9 10 5 8 4 2)C(7) = (1 7 5 2 3 10 4 6 9 8)C(8) = (1 8 9 6 4 10 3 2 5 7)C(9) = (1 9 4 3 5)C(10) = ( 1 10)
Cycles of equal lengths have the same elements.
LAGRANGE (9) Theorem: In group G P , all cycles of equal
order contain the same set of elements.
Proof: M = ǀC(a)ǀ implies by (2) that
xM – 1 = 0 for x C(a).
By Lagrange's theorem, a polynomial of degree M has at most M distinct roots. As C(a) already has M distinct elements, there is no other group.
CONSEQUENCES Recall
A = |C(a)|, B = |C(b)|. In GP the statements:
i. A|B <=> C(a) C(b)⊆
(10) ii. |C(a) ∩ C(b)| = gcd(A, B) iii. |C(a) C(b)| = lcm∪ (A, B)
are consequences of Lagrange (9).
SEPARATION
We call U|A and V|B separators of A and B if
(11) ( i) lcm(A/U, B/V) = lcm(A, B) ( ii) gcd(A/U, B/V) = 1
(12) Theorem: If U and V separate A and B, C(a) C(b) = C(a∪ U × bV).
Proof of Theorem 12
Proof: By (4), |C(aU )|=A/U, |C(bV)|=B/V. Thus by (10)
|C(a) C(b)| = lcm(A,B)∪ |C(aU ) C(b∪ V)| = lcm(A/U, B/V)
Yielding by (11.i), C(a) C(b) = C(a∪ U ) C(b∪ V). As C(aU) C(a), C(b⊆ V) C(b), we have⊆
C(a) C(b) = C(a∪ U ) C(b∪ V )By (11.i), C(a) and C(b) are relatively prime. By (8)
C(aU ) C(b∪ V) = C(aU × bV ).
SEPARATOR PRODUCT
(13) Proposition: If U, V are separators of A, B,
U×V = gcd(A, B)
Proof: By (1) and (11),
U×V = =
=
FACTORIZATION
A = 120 B = 1260The prime factors involved in both are
2, 3, 5, 7Prime factorizations of A and B:
A = 120 = 8 × 3 × 5 × 1 B = 1260 = 4 × 9 × 5 × 7
lcm(A, B) = 8 × 9 × 5 × 7 gcd(A, B) = 4 × 3 × 5 × 1
THE GIST
A= 120 = 8 × 3 × 5 × 1 8 × 1 × 5 × 1 = 40 =A/U B=1260 = 4 × 9 × 5 × 7 1 × 9 × 1 × 7 = 63 =B/V
U = 3 × 1 = 3, V = 4 × 5 = 20
Reducing the factors of lower multiplicity to 0 leaves lcm(A, B) unchanged, while reducing A by U and B by V, effecting separation.
Stepwise Separation
We call W > 1 a partial separator of A,B if
(13) gcd(A, B/W) = gcd(A, B)/W
By (1) lcm(A, B) remains unchanged,
lcm(A, B/W) = = = lcm(A,B),
while gcd(A, B) is reduced.
CUMULATIVE MULTIPLICATION
The following is at the heart of stepwise separation:
(14)
Proposition: If W1 is a partial separator for A, B, and W2 is one for A, B/W1 , then
W1 W2
is a partial separator for A, B, too.
Proof: gcd(A, B/(W1W2 )) = gcd(A, B/W1 )/W2
= gcd(A, B)/(W1W2)).
PARTIAL SEPARATOR
(15) Theorem: With C = gcd(A, B),W = gcd(A/C, C) > 1
is a partial separator of A, B.
Proof: In view of (14), it suffices to show that any prime divisor Q|W is a partial separator of A, B. The multiplicity, mult(Q|A), of Q in A exceeds mult(Q|C). Then
gcd(Q, B/C) ≤ gcd(A/C, B/C) = 1,Implies gcd(Q, B/C) = 1 so that
mult(Q|B) = mult(Q|C) < mult (Q|A). Thus gcd(A, B/Q) = gcd(A, B)/Q.
TERMINATION(16) Theorem: If C = gcd(A, B), and W = gcd(A/C, C) = 1 then there are full separators U = C, and V = 1.
Proof: gcd(A/C) = 1 implies that the division by C removes from A all divisors common with B. Thus gcd(A/C, B) = 1 = gcd(A/U, B/V), so that
lcm(A/U, B/V) = lcm(A/C, B) = (AB)/C = lcm(A, B)
U = C and V = 1 thus meet conditions (11).
EXAMPLEWe revisit the 1. A = 120, B = 1260 C = gcd(A, B) = 60, A/C = 2 W = gcd(A/C, C) = gcd(2,60) = 22. A = 120, B = (B/2) = 630 C = gcd(A, B) = 30, A/C = 4 W = gcd(A/C, C) = gcd(4,30) = 23. A = 120, B = (B/2) = 315 C = gcd(A, B) = 15, A/C = 8 W = gcd(A/C, C) = gcd(8, 315) = 1
4. U = C = 15, V = 2 × 2 = 4
SEPARATION ALGORITHM
Given: integer A, B > 0; Wanted: separators U,V.Step 1: 1 → V, gcd(A, B) → CStep 2: If C = 1 → step 7Step 3: A/C → XStep 4: gcd (X, C) → WStep 5: If W = 1 → step 7Step 6: V×W → V, C/W → C, X×W → X, → step 4Step 7: C → U, → terminate
NUMBERS
• 1228 primes ≤ 10,000 primitive roots calculated
• 24% separation required
• 470 instances of primitive root = 2
Copies of this Power Point are available at the following link http://math.nist.gov/mcsd/Seminars/2015/2015-09-15-Witzgall.html
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