Title
NIST乱数検定ツールの検定項目間対角化手法
Author(s)
岩崎 淳
Citation
福岡工業大学総合研究機構研究所所報 第1巻 P39-P42
Issue Date
2018-12
URI
http://hdl.handle.net/11478/1218
Right
Type
Departmental Bulletin Paper
Textversion Publisher
福岡工業大学 機関リポジトリ
FITREPO
NIST ᩘ᳨ᐃࢶ࣮ࣝࡢ᳨ᐃ㡯┠㛫ᑐゅᡭἲ
ᒾᓮ ῟㸦ሗᕤᏛ㒊ሗᕤᏛ⛉㸧
Diagonalizing method among test items included in NIST randomness test tool
Atsushi I:$6$., 㸦Department of Computer Science and Engineering, Faculty of Information Engineering㸧 Abstract
Randomness tests included in NIST test tool are not independent each other, i.e., the distributions followed by p-value of each test item are not independent. Even if each test item has no problem, the fact makes it difficult to derive the significance level of the test tool as whole items or to introduce a rational criterion for results of the tool. In this paper, we propose a method to solve this problem. The method transforms the distribution followed by each test item’s p-value to the standard normal distribution and assumes that the joint distribution after transformation is a multidimensional normal distribution. Diagonalizing to the joint distribution, the distribution of each test item's p-vale becomes independent each other. In addition, we evaluate this method numerically.
Keywords㸸Random number, Randomness test, Diagonalization
1. ࡣࡌࡵ つ๎ᛶࡢ࡞࠸㸦ぢ࠸ࡔࡏ࡞࠸㸧ᩘิ࡛࠶ࡿᩘࡣࠊሗࢭ ࢟ࣗࣜࢸࠊᬯྕᢏ⾡ࠊࣔࣥࢸ࢝ࣝࣟィ⟬ࠊุဨ㑅ฟ࡞ ࡢᢳ㑅ࠊ᳨ရ࡞ࡉࡲࡊࡲ࡞ศ㔝࡛ᛂ⏝ࡉࢀ࡚࠸ࡿࠋࡑࡢཝ ᐦ࡞ᐃ⩏ࡸồࡵࡽࢀࡿᛶ㉁ࡣศ㔝౫Ꮡࡍࡿࠋᩘࡢ⏕ᡂ ᪉ἲࡘ࠸࡚ࡶከ✀ከᵝ࡞◊✲ࡀ࡞ࡉࢀ࡚࠸ࡿࡀࠊࢇ ࡢሙྜ࠾࠸࡚ࡣࠊཝᐦ࡞ព࡛ᐃ⩏㏻ࡾࡘせồࡉࢀ ࡿᛶ㉁ࢆࡍ࡚‶ࡓࡍᩘࢆ⏕ᡂ࡛ࡁ࡞࠸ࠊࡲࡓࡣࠊ⏕ᡂࡋ ࡚࠸ࡿಖド࡛ࡁ࡞࠸ࠋ ࡑࡢࡓࡵࠊᐇ⏝ୖࡣᵝࠎ࡞ほⅬࡽ⏕ᡂ᪉ἲࡸ⤖ᯝⓗ ⏕ᡂࡉࢀࡓᩘࢆホ౯ࡍࡿᚲせࡀ࠶ࡿࠋࡑࡢ୰ࡢ୍ࡘࡢ᪉ ἲࡀᩘ᳨ᐃࡤࢀࡿ௬ㄝ᳨ᐃ࡛࠶ࡿࠋᖐ↓௬ㄝࡣࠕ࠼ ࡽࢀࡓᩘิࡣ⌮ⓗࣛࣥࢲ࣒࡛࠶ࡿ࡛ࠖ࠶ࡾࠊᩘࡑࡢࡶ ࡢࢆホ౯ᑐ㇟ࡍࡿࠋࡇࡇ࡛ࠊࠕ⌮ⓗࣛࣥࢲ࣒࡛࠶ࡿࠖ ࡣࠊ݊-bit ࡢᩘิࢆホ౯ࡍࡿሙྜࠊࠕ࠶ࡾ࠺ࡿʹᮏࡢᩘิࡢ ୰ࡽ➼ࡋ࠸☜⋡࡛㑅ࡤࢀࡓࡳ࡞ࡏࡿ㸦࠶ࡿ࠸ࡣࠊࡑࡢࡼ ࠺⪃࠼ࡓࡋ࡚ᕪࡋᨭ࠼࡞࠸㸧ࠖ࠸࠺⌮ゎ࡛Ⰻ࠸ࠋᩘ ᳨ᐃࡣ࠶ࡃࡲ࡛ࡶᐇ㦂ⓗ࡞ホ౯᪉ἲ㐣ࡂࡎࠊࡇࢀྜ᱁ ࡋࡓࡽゝࡗ࡚ᩘࡢᛶ㉁ࡢⰋࡉࡀド᫂ࡉࢀࡿࢃࡅ࡛ࡣ ࡞࠸ࠋ୍᪉࡛ࠊ⏕ᡂ᪉ἲ౫ࡽࡎ㐺⏝࡛ࡁࡿࡓࡵỗ⏝ᛶࡀ㧗 ࠸࠸࠺Ⅼࡀ࠶ࡿࠋ≉ࠊሗࢭ࢟ࣗࣜࢸ࣭ᬯྕࡢホ౯ Ḟࡃࡇࡢ࡛ࡁ࡞࠸ࡶࡢ࡛࠶ࡾࠊᐇ㝿ࠊ⌧ᅾࡢ⡿ᅜඹ㏻㘽 ᬯྕᶆ‽AES(Advanced Encryption Standard)(1)ࡢ㑅ᐃ
ࡣࠊᩘ᳨ᐃࡢࢸࢫࢺࢭࢵࢺ࡛࠶ࡿNIST SP800-22 ᩘ ᳨ᐃࢶ࣮ࣝ(2)ࡀ⏝ࡉࢀࡓࠋ ᩘ᳨ᐃ⮬యࡣ↓ᩘసࡽࢀ࠺ࡿࡶࡢ࡛ࠊ≉ࣃ࣓࣮ࣛ ࢱࢆኚ࠼ࢀࡤ⡆༢ᩘࢆࡑࢁ࠼ࡿࡇࡀ࡛ࡁࡿࠋࡑࡢࡓࡵࠊ ⾜࠺ࡁᩘ᳨ᐃࢆࡲࡵࡓࢸࢫࢺࢭࢵࢺࡀᥦࡉࢀ࡚࠸ ࡿࠋ๓㏙ࡢ NIST ᩘ᳨ᐃࢶ࣮ࣝࡶᩘ᳨ᐃࡢࢸࢫࢺࢭࢵ ࢺࡢ୍ࡘ࡛࠶ࡾࠊ᭱ࡶᗈࡃࢃࢀ࡚࠸ࡿࠋ᭱᪂∧ࡢrevision 1a ࡛ 15 ✀㢮 188 㡯┠ࡢ᳨ᐃ࡛ᵓᡂࡉࢀ࡚࠸ࡿࡀࠊၥ㢟Ⅼ ࡀᣦࡉࢀ࡚࠸ࡿ᳨ᐃࡶྵࡲࢀ࡚࠾ࡾࠊ୍㒊ࡣಟṇࡉࢀࡓ ࡀ(3)(4)ࠊᮍࡔᮍಟṇࡢ᳨ᐃࡶṧࡗ࡚࠸ࡿࠋࡲࡓࠊಶࠎࡢ᳨ ᐃࡣูࠊࢸࢫࢺࢭࢵࢺయࡋ࡚ࡢၥ㢟ࡶᢪ࠼࡚࠸ࡿࠋ ྵࡲࢀ࡚࠸ࡿ᳨ᐃ㡯┠㛫ࡢ㛵ಀ᫂☜࡞▱ぢࡀ࡞࠸ࡇ࡛ ࠶ࡿࠋࡇࡇ᳨࡛ᐃ㡯┠㛫ࡢ㛵ಀࡣࠊp ್ࡢ⤖ྜศᕸྠ⩏ ࡛࠶ࡿࠋ࠼ࡤࠊࠕ᳨ᐃ㡯┠A ࡢ p ᳨್ᐃ㡯┠ B ࡢ p ್ ࡣṇࡢ┦㛵ࡀ࠶ࡿࠖ㸦ࡍ࡞ࢃࡕࠊࠕ᳨ᐃ㡯┠A ㏻ࡿ࡞ࡽ ᳨ᐃ㡯┠B ࡶ㏻ࡾࡸࡍ࠸ࠖ㸧࠸࠺ࡇ࡛࠶ࢀࡤࠊಶࠎࡢ ᳨ᐃࡢ᭷ពỈ‽ࢆȽタᐃࡋࡓࡋ᳨࡚ᐃ㡯┠ A B ୧᪉ ࡛ᖐ↓௬ㄝࡀ᥇ᢥࡉࢀࡿ☜⋡ࡣሺͳ െ Ƚሻଶ࡞ࡽ࡞࠸ࠋࢸࢫࢺ ࢭࢵࢺయࡋ࡚ࡢ᭷ពỈ‽ࢆồࡵࡿࠊࡲࡓࡣࠊタᐃࡋࡓ᭷ ពỈ‽࡞ࡿࡼ࠺ࢸࢫࢺࢭࢵࢺయࡋ࡚ࡢྜྰᇶ‽ࢆ ᐃࡵࡿࡇࡣᅔ㞴࡞ࡿࠋࡲࡓࠊᙉ࠸┦㛵ࡀࡳࡽࢀࡿ࡞ࡽ」 ᩘࡢ᳨ᐃࢆᐇࡍࡿࡇࡣィ⟬㈨※ࡢ↓㥏㐵࠸ゝ࠼ࡼ ࠺ࠋNIST ᩘ᳨ᐃࢶ࣮ࣝ࠾࠸࡚ࡣࠊඛ⾜◊✲᳨࡛ᐃ㡯┠ 㛫ᴫࡡṇࡢ┦㛵ࡀ࠶ࡿࡇࡀሗ࿌ࡉࢀ࡚ࡁࡓ(5)(6)ࠋࡍ࡞ࢃ ࡕࠊ⌮ⓗ࡞ᩘࡀྜ᱁ࡍࡿ᳨ᐃ㡯┠ᩘࡣࠊ㡯┠ࡀ࠸⊂ ❧࡛࠶ࡿሙྜẚࡿከࡃ࡞ࡿഴྥࡀ࠶ࡿࡇࢆពࡋ ࡚࠸ࡿࠋ ᮏ✏࡛ࡣ NIST ᩘ᳨ᐃࢶ࣮ࣝࡢ᳨ᐃ㡯┠㛫ࡢ㠀⊂❧ᛶ ࡢၥ㢟ᑐࡋ࡚ࠊࡑࢀࢆྲྀࡾ㝖ࡃࣉ࣮ࣟࢳ࡛ྲྀࡾ⤌ࡴࠋල యⓗࡣࠊ」ᩘࡢ᳨ᐃ㡯┠ࢆ㏻ࡌ࡚ᚓࡽࢀࡓ p ್ࡢࢭࢵࢺ
ᒾᓮ ῟ ࢆኚࡋࠊ⊂❧࡞ศᕸᚑ࠺ࡼ࠺ࡍࡿࡇࢆ┠ᣦࡍࠋ 2. NIST SP800-22 ᩘ᳨ᐃࢶ࣮ࣝ NIST SP800-22 ᩘ᳨ᐃࢶ࣮ࣝࡢ᭱᪂∧ revision 1a ࡣ ௨ୗࡢ15 ✀㢮ࡢ᳨ᐃ࡛ᵓᡂࡉࢀ࡚࠸ࡿ㸸 1. Frequency Test
2. Frequency Test within a Block 3. Runs Test
4. Tests for the Longest-Run-of-Ones in a Block 5. Binary Matrix Rank Test
6. Discrete Fourier Transform Test
7. Non-overlapping Template Matching Test 8. Overlapping Template Matching Test 9. Maurer’s “Universal Statistical” Test 10. Linear Complexity Test
11. Serial Test
12. Approximate Entropy Test 13. Cumulative Sums Test 14. Random Excursions Test 15. Random Excursion Variant Test
ࡇࢀࡽ 15 ✀㢮ࡢ᳨ᐃࡀᐇࡉࢀࡓࢧࣥࣉࣝࣉࣟࢢ࣒ࣛࡀ NIST ࡽᥦ౪ࡉࢀ࡚࠸ࡿࠋࢧࣥࣉࣝࣉࣟࢢ࣒࡛ࣛࡣࠊࣃࣛ ࣓࣮ࢱࢆኚ࠼ࡿ࡞ࡋ࡚ 1 ✀㢮ᑐࡋ࡚」ᩘࡢ᳨ᐃࢆᐇ ࡍࡿࡶࡢࡶྵࡲࢀ࡚࠾ࡾࠊ⥲ィ188 㡯┠ࡢ᳨ᐃࡀ⾜ࢃࢀࡿࠋ NIST ᩘ᳨ᐃࢶ࣮ࣝࡣ」ᩘᮏࡢᩘิᑐࡋ᳨࡚ᐃࢆ⾜ ࠺ࠋࡍ࡞ࢃࡕࠊ㛗ࡉ݊-bitൈ ݉ᮏ࠸࠺ᙧ᳨࡛ᐃᑐ㇟࡞ࡿᩘ ิࡀ࠼ࡽࢀࠊྛ᳨ᐃ୍࡛ᮏ୍ᮏࡢᩘิᑐࡋ࡚ p ್ࡀィ ⟬ࡉࢀࡿࠋࡓࡔࡋࠊእⓗRandom Excursions Test Random Excursion Variant Test ࡢ 2 ✀ 16 㡯┠㛵ࡋ࡚ࡣ ᩘิࡼࡗ࡚ࡣ᳨ᐃࢆ⾜ࢃ࡞࠸࠸࠺࢜ࣉࢩࣙࣥࡀ࠶ࡾࠊ݉ ᮏࡢᩘิࡍ࡚ᑐࡋ࡚p ್ࡀィ⟬ࡉࢀࡿࢃࡅ࡛ࡣ࡞࠸ࠋ ྛ᳨ᐃ㡯┠ࡢྜྰࡣࠊࡑࡢ᳨ᐃ࡛ᚓࡽࢀࡓࡍ࡚ࡢ p ್ ࢆ⏝࠸࡚ࠊࡉࡽ௨ୗࡢ2 ㏻ࡾࡢ᳨ᐃࢆ⾜࠸ุᐃࡉࢀࡿ㸸 z [Proportion Test] ͲǤͲͳࢆୗᅇࡗࡓ p ್ࡢಶᩘࡀᖹᆒ 㸦ͲǤͲͳ݉㸧േ ͵ ൈᶆ‽೫ᕪ㸦ඥሺͲǤͲͳሻሺͲǤͻͻሻ݉㸧ࡢ⠊ᅖෆ ࡞ࡽྜ᱁ࠊࡑ࠺࡛࡞ࡅࢀࡤᖐ↓௬ㄝࢆᲠ༷ࡍࡿࠋ z [Uniformity Test] Ͳࡽͳࡢ༊㛫ࢆͳͲಶࡢࣅࣥ➼ศ ࡋࠊྛࣅࣥධࡗ࡚࠸ࡿ S ್ࡢಶᩘࢆ᳨࢝ᐃ ࡅࠊࡑࡢp ್ࡀͲǤͲͲͲͳࡼࡾࡁࡅࢀࡤྜ᱁ࠊࡑ࠺࡛ ࡞ࡅࢀࡤᖐ↓௬ㄝࢆᲠ༷ࡍࡿࠋ ⌮ⓗ࡞ሙྜ࠾࠸࡚ࠊProportion Test ࡢ᭷ពỈ‽ࡣ⣙ ͲǤͲͲʹ࡛࠶ࡿࠋUniformity Test ࡣ᭷ពỈ‽ࡀͲǤͲͲͲͳᴟࡵ ࡚ᑠࡉ࠸ࡢ࡛ࠊ⌮ⓗ࡞ᩘࢆᲠ༷ࡍࡿࡇࡣࡵࡗࡓ㉳ ࡇࡽ࡞࠸ࠋ 3. ᥦᡭἲ ᳨ᐃ㡯┠㛫ࡢ㠀⊂❧ᛶࡽ⏕ࡌࡿၥ㢟ᑐࡋ࡚ゎỴἲࢆ ⪃࠼࡚࠸ࡃࠋࡲࡎࠊࠕ᳨ᐃ㛫ࡢ┦㛵ࢆྲྀࡾ㝖ࡅࡤࠊ㸦ᐇ⏝ୖ㸧 ⊂❧ࡳ࡞ࡏࡿࠖ࠸࠺ࡊࡗࡃࡾࡋࡓ௬ᐃࢆ࠾ࡇ࠺ࠋ୍⯡ ㄽࡋ࡚㠀⊂❧ᛶࢆྲྀࡾ㝖ࡃࡇࡣ㞴ࡋ࠸ࠋࡋࡋࠊ ┦㛵ࢆྲྀࡾ㝖ࡃࡇࡣඹศᩓ⾜ิࡢᑐゅࡼࡾ⾜࠼ࡿࡢ ࡛ࠊࡇࡢ௬ᐃࡼࡾၥ㢟ࡣࡎ࠸ࡪࢇ⡆༢࡞ࡿࠋ
࡞࠾ࠊRandom Excursions Test Random Excursion Variant Test ࡣእⓗᢅ࠸ࡃ࠸ࡓࡵࠊࡇࢀࡽࢆ㝖࠸ࡓ 13✀ 162᳨ᐃࡢࡳࢆ⪃࠼ࡿࡇࡍࡿࠋࡲࡓࠊྛ᳨ᐃࡣ ࡃၥ㢟ࡣ࡞ࡃࠊᖐ↓௬ㄝࡢୗ࡛ࡢྛ᳨ᐃࡢ p ್ࡢศᕸࡣ ሾͲǡͳሿ̿ᵝศᕸ࡛࠶ࡿࡍࡿࠋ ࠑ3㺃1ࠒඹศᩓ⾜ิࡢᑐゅ ྛ᳨ᐃ㡯┠ 1ࡽ 162ࡲ࡛ࡢ␒ྕࢆࡅࡿࡇࡋࡼ ࠺ࠋ㸦ලయⓗࡣࢧࣥࣉࣝࣉࣟࢢ࣒ࣛࡢ⾲♧㡰ࡍࡿࠋ㸧ࡲ ࡓࠊ᳨ᐃᑐ㇟ࡢ݉ᮏࡢᩘิࡶͳࡽ݉ࡲ࡛ࡢ␒ྕࡀࡽࢀ ࡚࠸ࡿࡍࡿࠋ᳨ᐃ㡯┠j࡛ᩘิ݅ᑐࡋ࡚ồࡲࡗࡓ p ್ࢆ ǡ᭩ࡃࡇࡋࠊ ܲǣ ൌ ൭ ଵǡଵ ڮ ǡଵ ڭ ڰ ڭ ଵǡଵଶ ڮ ǡଵଶ ൱ ᐃ⩏ࡍࡿࠋ ᳨ᐃ㡯┠݆݅ࡢ p ್ࡢඹศᩓࢆܿǡࡋ࡚ࠊሺ݅ǡ ݆ሻ-ᡂศࡀܿǡ ࡛࠶ࡿͳʹ ൈ ͳʹࡢ⾜ิܥࢆඹศᩓ⾜ิࡍࡿࠋ⾜ิܥࡣᙜ↛ ᐇᑐ⛠⾜ิ࡛࠶ࡿࡽࠊ࠶ࡿ┤⾜ิܮࡀᏑᅾࡋ࡚ܮܥܮᑐ ゅ࡛ࡁࡿࠋ ḟࠊඛᐃ⩏ࡋࡓܲܮࢆ⏝࠸࡚ ܲᇱؔ ܮܲ ᐃ⩏ࡋࠊܲᇱࡢྛ⾜ࢆ᪂ࡓྛ᳨ᐃ㡯┠࡛ᚓࡽࢀࡓ p ್ࡢ ࢭࢵࢺࡔᛮ࠸࡞࠾ࡍࠋ㸦Ỵࡋ࡚ྛ⾜ᑐᛂࡍࡿ᳨ᐃࡀ࠶ࡿ ࢃࡅ࡛ࡣ࡞࠸ࠋ࠶ࡃࡲ࡛ࡶࠊ༢౽ᐅⓗࡑࡢࡼ࠺⪃࠼ࡿ ࡔࡅ࡛࠶ࡿࠋ㸧ࡑ࠺ࡍࡿࠊྛ᳨ᐃ㡯┠ࡢp ್ࡢศᕸࡣ↓┦ 㛵࡞ࡿࠋ ࠑ3㺃2ࠒၥ㢟Ⅼ ඛ♧ࡋࡓඹศᩓ⾜ิࡢᑐゅἲࡣ௨ୗࡢࡼ࠺࡞ၥ㢟 ࡀ࠶ࡿࠋ z ᖐ↓௬ㄝࡢୗ࡛ࡢp ್ࡢ⤖ྜศᕸࡀᮍ▱࡞ࡢ࡛ࠊඹ ศᩓ⾜ิܥࢆ⌮ㄽⓗồࡵࡿࡇࡣฟ᮶࡞࠸ࠋ z ኚᚋࡢྛ᳨ᐃ㸦ܲᇱࡢྛ⾜㸧࡛ࡣp ್ࡀ୍ᵝศᕸ ࡞ࡽ࡞࠸ࠋ ࡇࡢ࠺ࡕ୍Ⅼ┠㛵ࡋ࡚ࡣࠊᐇ㦂ⓗᚓࡽࢀࡿศᕸࡽィ ⟬ࡉࢀࡿඹศᩓ⾜ิ࡛௦⏝ࡍࡿࡇ࡛ࡦࡲࡎゎỴ࡛ࡁࡼ ࠺ࠋⅬ┠㛵ࡋ࡚ࡣ௨ୗࡢࡼ࠺࡞᪉㔪࡛ᑐฎࡍࡿ㸸 1. p ್ࢆ㐃⥆࡞㛵ᩘࢆ⏝࠸࡚ኚࡋࠊྛ᳨ᐃࡢ p ್ࡢศ ᕸࢆᶆ‽ṇつศᕸࡍࡿࠋࡉࡽࠕ」ᩘࡢ᳨ᐃ࡛ࡢ p ್ࡢ⤖ྜศᕸࡣከḟඖṇつศᕸᚑࡗ࡚࠸ࡿࠖࡢ௬ ᐃࢆ࠾ࡃࠋ 2. ࡇࡢ≧ែ࡛ඹศᩓ⾜ิࢆồࡵࠊࡑࢀࢆᑐゅࡍࡿ┤ ⾜ิࢆ⏝࠸࡚p ್ࢆኚࡍࡿࠋከḟඖṇつศᕸࡢᛶ㉁ ࡽኚᚋࡢศᕸࡣࠊྛ࿘㎶ศᕸࡀ⊂❧࡞ṇつศᕸ ࡞ࡗ࡚࠸ࡿࡣࡎ࡛࠶ࡿࠋ 3. ᭱ᚋྛ࿘㎶ศᕸࢆⓑⰍࡋࠊศᕸࢆࢫࢸࢵࣉ1 ࡢ㏫ ኚ࡛ሾͲǡͳሿ୍ᵝศᕸ࡞࠾ࡍࠋ ࢫࢸࢵࣉ 2 ࡛ྛ࿘㎶ศᕸࡀ⊂❧࡞ࡿࡢ࡛ࠊ௨ୖࡢᡭ㡰ࡢ ኚ࡛ྛ᳨ᐃࡢ⤖ᯝࡣ⊂❧࡞ࡿࠋ
ࠑ3㺃3ࠒኚἲ ௨ୖ㏙࡚ࡁࡓࡇࢆࡲࡵࠊኚἲࢆලయⓗ㏙࡚ ࠾ࡇ࠺ࠋ௨ୗࡢࡼ࠺࡞ᡭ㡰࡛࠶ࡿ㸸 1. ⾜ิܲࡢྛᡂศࢆ հ ିଵሺʹ െ ͳሻ ኚࡋࡓ⾜ิܳࢆồࡵࡿࠋࡇࡇ࡛ erf ࡣㄗᕪ㛵ᩘ࡛࠶ ࡿࠋ 2. ⾜ิܳࡽඹศᩓ⾜ิܥመࢆィ⟬ࡍࡿࠋ 3. ඹศᩓ⾜ิܥመࢆᑐゅࡍࡿ┤⾜ิܮࢆồࡵࠊ ܳᇱؔ ܮܳ ࡍࡿࠋ 4. ⾜ิܳᇱࡢ➨݅⾜ࡢᡂศࡽࡑࡢᖹᆒ್ߤ ᶆ‽೫ᕪߪ ࢆィ⟬ࡋࠊྛሺ݅ǡ ݆ሻ-ᡂศݍǡࢆ ݍǡհ ݍǡെ ߤ ߪ ኚࡋࡓ⾜ิܳᇱᇱࢆồࡵࡿࠋ 5. ⾜ิܳᇱᇱࡢྛᡂศࢆ ݍᇱᇱհሺݍ ᇱᇱሻ ͳ ʹ ኚࡋࡓ⾜ิܲᇱࢆồࡵࡿࠋ㸦⾜ิܲᇱࡢྛ⾜ࡀኚᚋࡢ ྛ᳨ᐃ㡯┠ࡢp ್࡞ࡿࠋ㸧 ௨ୖࡢᡭ㡰࡛ྛ᳨ᐃࡀ⊂❧࡛ࠊࡘࠊS ್ࡀ୍ᵝศᕸᚑ࠺ ࡼ࠺࡞ࡿࠋ⨨ࢀ࡚࠸ࡿ௬ᐃࢆ☜ㄆࡋ࡚࠾ࡃࠊ z 㸦ኚ๓ࡢ㸧ྛ᳨ᐃ㡯┠࡛ࡢp ್ࡢศᕸࡣሾͲǡͳሿ୍ᵝศ ᕸ࡛࠶ࡿ z ྛ᳨ᐃ㡯┠ࡢp ್ࡢศᕸࢆᶆ‽ṇつศᕸኚࡋࡓ ࡁ」ᩘࡢ᳨ᐃ࡛ࡢp ್ࡢ⤖ྜศᕸࡣከḟඖṇつศᕸ ࡞ࡗ࡚࠸ࡿ ࡢ Ⅼ࡛࠶ࡿࠋ୍Ⅼ┠㛵ࡋ࡚ࡣࠊྛ✀ၥ㢟ࡣ࠶ࡿࡅࢀ ࡶࠊࡑࡶࡑࡶྛ᳨ᐃࡣ p ್ࡀ୍ᵝศᕸ࡞ࡿࡇࢆᮇᚅࡋ ࡚㸦࠶ࡿ࠸ࡣࠊ┠ᣦࡋ࡚㸧タィࡉࢀ࡚࠸ࡿࡢ࡛⮬↛࡞௬ᐃ ࡣ࠸࠼࡞࠸࡛࠶ࢁ࠺ࠋⅬ┠ࡢ௬ᐃࡣ௨ୗࡢࡼ࠺⪃࠼ ࢀࡤࡼ࠸ࠋࡍ࡞ࢃࡕࠊ᳨ᐃ㡯┠㛫ࡢ㠀⊂❧ᛶࡣᐇ⏝ୖࡢࢿࢵ ࢡ࡞ࡗ࡚࠸ࡿࡅࢀࡶࠊࡶࡶྛ᳨ᐃ㡯┠ࡣ⊂❧㏆ ࡃࠊ㠀⊂❧ᛶࡢࢇࡣ⥺ᙧ࡞┦㛵ᵓ㐀ࡋ࡚ࡽ࠼ࡽ ࢀࡿࠋࡑ࠺ࡍࡿࠊⅬ┠ࡢ௬ᐃࡶ㏆ఝⓗጇᙜ࠸࠼ࡿࡔ ࢁ࠺ࠋ 4. ᐇ㦂 ᥦᡭἲࡢᩘ್ⓗ࡞ホ౯ࢆ⾜࠺ࠋ ࠑ4㺃1ࠒᐇ㦂㸯 ࣓ࣝࢭࣥࢾࢶࢫࢱ ࢆ⏝࠸࡚ᩘิࢆసᡂࡋࠊᐇ㦂ࢆ⾜ࡗ ࡓࠋ1,67 ࡣ᳨ᐃᑐ㇟࡞ࡿᩘิࡢࣅࢵࢺ㛗 ݊ ൌ ͳͲͲͲͲͲͲࠊ ᳨ᐃᮏᩘ݉ ൌ ͳͲͲͲࢆᩘ᳨ᐃࢶ࣮ࣝ⏝ࡢ᥎ዡ್ࡋ࡚ ࠸ࡿࠋࡇࢀࢆ㋃ࡲ࠼࡚ࠊࡲࡎ ݊ ൌ ͳͲͲͲͲͲͲࠊ݉ ൌ ͳͲͲͲͲͲͲ ࡋ᳨࡚ᐃࡢᐇࠊᥦᡭἲࡼࡿኚࢆ⾜࠸ࠊኚᚋࡢ⤖ᯝ ࢆྛࢭࢵࢺͳͲͲͲᮏࡎࡘࡢͳͲͲͲࢭࢵࢺศࡋ࡚ホ౯ࡋࡓࠋ ࡲࡓࠊඹศᩓ⾜ิࢆᑐゅࡍࡿ┤⾜ิࡢ㑅ࡧ᪉ࡣ௵ព ᛶࡀṧࡿࡀࠊኚᚋࡢᅛ᭷್ࡀࡁࡉࡢ㡰࡛ᑐゅ⾜ิ୪ ࡪࡼ࠺㑅ࡪࡇࡍࡿࠋ ᅗ ࡣኚᚋࡢྛ᳨ᐃ㡯┠࡛ 3URSRUWLRQ 7HVW ࡼࡾᲠ ༷ࡉࢀࡓࢭࢵࢺᩘࢆ♧ࡋ࡚࠸ࡿࠋࡁ࡞ᅛ᭷್ᑐᛂࡍࡿ ᳨ᐃ㡯┠ᑠࡉ࡞ᅛ᭷್ᑐᛂࡍࡿ᳨ᐃ㡯┠㸦୧➃ࡢ᳨ᐃ 㡯┠㸧࡛ㄗᕪ㸦➨୍✀㐣ㄗ㸧ࡢ⠊ᅖࢆࡁࡃ㉸࠼࡚Რ༷ࡉࢀ ࡚࠸ࡿࡢࡀぢ࡚ྲྀࢀࡿࠋᥦᡭἲ࡛࠾ࢀ࡚࠸ࡿ ࡘࡢ௬ ᐃࡀ㸦ᑡ࡞ࡃࡶཝᐦࡣ㸧‶ࡓࡉࢀ࡚࠸࡞࠸ࡋࢃᐤࡏࡀࡁ ࡚࠸ࡿࡶࡢゎ㔘࡛ࡁࡿࠋ㏫ゝ࠺ࠊኚ๓ࡣከࡃࡢ᳨ᐃ 㡯┠ரࡗ࡚Ꮡᅾࡋ࡚࠸ࡓၥ㢟Ⅼࢆࠊኚࡼࡾ୧➃ࡢ᳨ ᐃ㡯┠㞟ࡵࡿࡇࡀ࡛ࡁ࡚࠸ࡿࡶ࠸࠼ࡿࠋ ࠑ4㺃2ࠒᐇ㦂 2 ᐇ㦂 ࢆ㋃ࡲ࠼ࡿࠊᥦᡭἲࡼࡿኚࢆ⾜ࡗࡓᚋ࡛ࠊ ୧➃ࡢ᳨ᐃ㡯┠ࢆྲྀࡾ㝖ࡅࡤ᭷ព⩏࡞᳨ᐃ⤖ᯝࢆྲྀࡾฟࡏ ࡿࡶࡢᮇᚅ࡛ࡁࡿࠋࣃ࣓࣮ࣛࢱࡣᐇ㦂 ྠᵝࡋࠊ࣓ࣝ ࢭࣥࢾࢶࢫࢱ ࢆ⏝࠸ࡓ᳨ᐃኚࢆ ᅇࠊ$(6 ࢆ &75 ࣮ࣔࢻ࡛⏝ࡋ࡚ ᅇ⾜ࡗࡓࠋࡑࡢᚋࠊ୧➃ࡽ᳨ᐃࢆ ࡘ ࡎࡘ㸦ࡍ࡞ࢃࡕࠊ࣡ࣥࢫࢸࢵࣉ࡛ ࡘ㸧๐㝖ࡋ࡚࠸ࡁᣲືࢆ ᅗ1 ኚᚋࡢྛ᳨ᐃ㡯┠࠾ࡅࡿᲠ༷ࢭࢵࢺᩘ Fig. 1. Number of sets rejected by each test
item after transformation.
ᅗ 2 ྲྀࡾ㝖࠸ࡓ᳨ᐃ㡯┠ᩘ㡯┠ྜ᱁ࡋࡓࢭࢵࢺᩘ
ࡢ㛵ಀ
Fig. 2. Relationship between number of removed test items and number of sets passing all test items.
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