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3
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1973? C.H. Bennett ???? 1980???? R.P.
Feynman ???????????????? 1985? D. Deutsch
???????????????(???????????) 1989? D. Deutsch
R. Jozsa ???????????????????????????????
1994? P.W. Shor ?????????????????????????????
1996? L. Grover ?????????????????????
4
???????????
????????????NMR????????????
NMR(?????)?? ?????????????????
2001?12? IBM?I. Chuang? Shor?????????????????????1
5??????????
???????????7??????????????????????????????????
5
NMR??????
????????????
6
NMR?????????????
2008? Xinhua Peng, Zeyang Liao, Nanyang
Xu, Gan Qin, Xianyi Zhou, Dieter Suter, and
Jiangfeng Du ??NMR??????????????????21
(37)???? ??????3??????????????????
2011? Nanyang Xu, Jing Zhu, Dawei Lu, Xianyi
Zhou, Xinhua Peng, and Jiangfeng Du
??NMR??????????????????143 (1113)??
????????143?????????4??????????
7
???????????(??)
????????????NMR????????????
NMR(?????)?? ?????????????????
2001?12? IBM?I. Chuang? Shor?????????????????????1
5??????????
???????????7??????????????????????????????????
?????Chuang??NMR???????????????? ?????????????????
???
8
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9
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????2?????????
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??????????1???
?????????1???
0, 1
0, 1
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?????? 2??????
1??????(qubit??????)
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(?????????????)
10
? NMR????????
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(???????????????)
Chuang??? ??????????7??????????????
0
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(???)

(???)
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0?1??????????????
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11
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2????2??????????????????????????
1101
4???????????
?
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?2
1101?0100?1111 ??????????????
?
????????? (????????1?)
1101?0100?1111 ????????
3?????????1?? ??????????
????????
12
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NMR???????????
1 ?????????????????????
2 ????????????????????????????????????????????
???????????????????????? ???????????
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?????????1????
1???????????1???????
3 ???????????????2??(0,1?????)????????
13
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14
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??????????(???????) ?????p(x)?????????N???? ??N??
????p(log N)
log N N???
????????????
15
??N????????????
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???????2??????1?2??????
??
?
1 ??????? i ??????1??????
1,2???????????
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2 N?????1??????????????2??????
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???1,2??i2??????i?1????????????????????2????
????2???0
i ?N???
16
??N????????????
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(?) N77??????
2
2
1
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6
6
5
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17
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?????????????1?? ?? ???????????????????????????
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N77???
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?????
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18
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????????????????????(?)????????????
?????????????(?????)???????????????
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N???????????? f ??????????????
Shor?????????????? ?????????????????? ????????????
??????
19
Shor????????????
20
Shor???????????????????? ????????????
??N (Npq?p?q??????)
????????
??p?q
1 N ???????? a (2?a?N-1) ??????(???)???
N???????????N??????a?????? (Euclid??????)
2 ?? f(x)ax mod N ??? r ?????
???????????
????????? r ??????
3 ar/21?N?????????ar/2-1?N???????????????
???????N??????????????(Euclid??????)
21
Shor?????????????????
N15??????????????
1 N ???????? a (2?a?N-1) ??????(???)???
N???????? 2,4,7,8,11,13,14
????????a??a7???????
2 ?? f(x)ax mod N ??? r ?????
f(x)???ar 1 mod N?????? r (a?order) (N?a???????
????????????????????)
?????f(x)7x mod 15 ????
???? f(x) ??? r ????????????????????
22
????1?????2???????????
f(x)7x mod 15
23
?? r ?????????????
???????2?????????????1????2???????????????
????????
????1
????2
???????
???????
???????
???????
1 ????1??0??15??????????????????????(?4-a)
2 ????1??? x ????? f(x) ?????????????????2??????
(?4-b)
f(x)7x mod 15
f(x)ax mod N N????3???????????
??????????????????????
24
????1?????2???????????
25
?????????(??)
???????????????????????????(?4-c)
????1???
4????????
?? r 4
26
????1?????2???????????
????????
27
?????????(???)
3 ????1?????????????(?4-d)
????????????????????????
??????????
????1???
???????1??????? ?????????? r ??????
?????
?? r 4 ??????
???????
28
Shor????????????(????????)
1 N ???????? a (2?a?N-1) ??????(???)???
a7
2 ?? f(x)ax mod N ??? r ?????
?? r 4??????
?????????
3 ar/21?N?????????ar/2-1?N?????????? ?????
ar/21 721 50 ar/2-1 72-1 48
gcd(ar/21, N) gcd(50,15) 5 gcd(ar/2-1, N)
gcd(48,15) 3
Euclid??????
?? 5?3
15 53 ??????
29
Shor????????????
1 N ???????? a (2?a?N-1) ??????(???)???
(N???)2????(Euclid??????)
2 ?? f(x)ax mod N ??? r ?????
(N???)3????
C/log(N???) ????????? r ??????
3 ar/21?N?????????ar/2-1?N???????????????
(N???)2????(Euclid??????)
????? r ?????????1/2??????1?N???????????
??????C/(2 log(N???))??????????????p?q??????
1,2,3?log(N???) ?????????????1?N????????????
??????N?????????( (N???)3 log(N???)
????)? N?????????????
30
????????????
31
RSA?????
(1) ????????????????(Shamir Tromer, 2003)
TWIRL ????????????????????
(??????0.13µm1GHz?????VLSI?????????30cm?LSI)
1????????
??????
???(?)???? 512bit 1024bit
???? 10??? 1000?
????
????l ??????
RSA?????(?)??????
(2) ????????????? (R. Hughes?????, 2003)
1GHz ???????????
?1
32
???????RSA?????
1024bit???????????????????
(1) TWIRL(Shamir and Tromer, 2003)
??30cm?LSI 1000????????1?
(2) SHARK(Franke et al., 2005)
2????????1?
(3) YASD1024(Hirota et al., 2006 Geiselmann
Steinwandt, 2004)
?LSI 1000????????16???
33
??????????????
p ????g ?mod p ?????????
A 1,2,,p-1 ????Aga mod p ??? a
0,1,2,,p-2 ??????
a?A?????????
A?????????a?????????????????
????????????????????????????????
(???????????????????????????Shor????)
?????????????????????
??
ElGamal?? Diffie-Hellman???
34
???????
???? ????????????????????????
Shor???????????????????????????????
Proos Zalka???????????????????????(2003)
???????????GF(p)???? p ? ?????
???????????
GF(2m)???????( Kaye Zalka, 2004 )
35
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36
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37
10????????
NMR???????????
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NMR?????????????
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????7?????????????? 7??????
????????????1?1?????????? ????NMR?????????????????
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10????????? 1997?????????????????
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38
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??????? ?????????????????????? ??(????)
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2007??15?????????? ??????????????
?????????2012??????
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