IT代写 Any integer N can be written as

Any integer N can be written as
N 􏰔 pα1 pα2 􏰓 􏰓 􏰓 pαm
N 􏰔15􏰔3􏰓5, α1 􏰔α2 􏰔1, p1 􏰔3,p2 􏰔5.
Definition 1 Greatest Common Divisor (GCD) of integers a and b is the largest integer x s.t. x|a and x|b (here | denotes ”divides without a reminder”)

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a􏰔3􏰓3􏰓2􏰔18, b􏰔3􏰓5􏰓2􏰔30 gcdpa, bq 􏰔 3 􏰓 2 􏰔 6.
Let L be the number of bits in the binary representation of N, that is N2 􏰔 n1 􏰓􏰓􏰓nL, nj 􏰔 0,1 (Ex. If N 􏰔 15, then N2 􏰔 l1o1m1o1on)
12m where αj are positive integers and pj are primes.
Let z be an integer such that
1. z2 pmod Nq 􏰔 1
2.zpmodNq􏰖1(ifzpmodNq􏰔1,thenz2 pmodNq􏰔1,butwedo not need this case)
3. zpmodNq􏰖N􏰑1(ifzpmodNq􏰔N􏰑1,thenz2 pmodNq􏰔1,and so we would like to exclude this possibility)
Theorem 1 gcdpz􏰑1, Nq or(and) gcdpz􏰐1, Nq is (are) non-trivial factor(s) of N.
Note that gcdpz 􏰑 1,Nq and gcdpz 􏰐 1,Nq can be computed using only OpL3q 􏰔 Opplog2 Nq3q operations using Euclid’s algorithm.
Theorem 2 Let x be an integer chosen uniformly randomly subject to re- quirements

1 ¤ x ¤ N 􏰑 1
x is co-prime to N, i.e. gcdpx, Nq 􏰔 1
be the order of x, i.e., xr pmod Nq􏰔1. Then
r{2 1 Prprisevenandx pmodNq􏰖N􏰑1q¥1􏰑2m.
If x is such as described in Theorem 2, then we take z 􏰔 xr{2.
Recall that m is the number of primes in factorization of N 􏰔 pα1 pα2 􏰓 􏰓 􏰓 pαm ;
Note that we do not need to check condition 3 formulated for Theo- rem 1, since since if xr{2 pmod Nq 􏰔 1 then the order of x is r{2, but we assumed that the order is r.
Algorithm for finding a factor of N
1. Randomly choose x P r1, N 􏰑 1s
2. If gcdpx,Nq ¡ 1, then RETURN gcdpx,Nq (gcdpx,Nq is a nontrivial factor of N), go to Step 7;
else: find the order r of x pmod Nq (use quantum computer here).
3. Ifrisevenandxr{2 pmodNq􏰖N􏰑1,thenassignz􏰔xr{2; elsego to Step 1.
4. Computef1 􏰔gcdpz􏰑1,Nqandf2 􏰔gcdpz􏰐1,Nq.
5. If f1|N RETURN(f1q.
6. If f2|N RETURN(f2q.
7. The end.
Example 3 N 􏰔 15. Assume we randomly took x 􏰔 7
74 pmod15q􏰔1ñr􏰔4
xr{2 􏰔72 􏰔49, 49pmod15q􏰔4ñ72 pmod15q􏰖N􏰑1􏰔14
ñz􏰔xr{2 􏰔72 􏰔49
f1 􏰔gcdpz􏰑1,15q􏰔gcdp48,15q􏰔3 f2 􏰔gcdpz􏰐1,15q􏰔gcdp50,15q􏰔5

Both f1 􏰔3 and f2 􏰔5 are factors of N 􏰔15.
Let us also find U for these N and x. Recall that
U|yy Ñ |xy pmod 15qy.
We have L 􏰔 4 and hence U is a 16 􏰒 16 permutation matrix that conducts
the mapping
y 7􏰓y 7􏰓y pmod 15q 000 177 214 14 321 6
Using the correspondence between linear algebra notation and Dirac’s no- tation, we get that the first 4 columns of U are
􏰓1000 􏰙􏰙􏰙0000 􏰙􏰙0000 􏰙􏰙0000 􏰙􏰙 0 0 0 0 􏰙􏰙0000 􏰙􏰙0001
U􏰔􏰙􏰙0100 􏰙􏰙0000
􏰙􏰙0000 􏰙􏰙0000 􏰙􏰙0000 􏰙􏰙0000 􏰙􏰘0000
􏰞􏰞 􏰓 􏰓 􏰓 􏰞􏰞
􏰞􏰞 􏰞􏰞 􏰞􏰞, 􏰞􏰞 􏰞􏰞 􏰞􏰞 􏰞􏰞 􏰞􏰞 􏰞􏰝

Indeed this matrix moves |0y to |0y:
and |1y into |7y
􏰙10􏰞 􏰙10􏰞 U 􏰓 􏰙􏰙 . 􏰞􏰞 􏰔 􏰙􏰙 . 􏰞􏰞 ,
􏰘.􏰝 􏰘.􏰝 00
U 􏰓 􏰙􏰙 . 􏰞􏰞 􏰔
􏰙􏰙0􏰞􏰞 􏰙􏰙0􏰞􏰞
􏰙􏰙􏰞􏰞 􏰙􏰙 0. 􏰞􏰞 􏰘.􏰝

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