Android Pがコンパイラベースのセキュリティ対策を拡大
最近ベータ版がリリースされたAndroid Pは、今後のリリースでコンパイラーベースのセキュリティー対策を強化する予定であると、GoogleのエンジニアであるIvan Lozano氏は述べている。また、同氏によると、今後のリリースで、制御フローの完全性と整数オーバーフローのサニタイズも含まれるとのこと。
コンパイラベースの対策は、LLVM/Clangコンパイラによって一連の機能として提供される。対策の基本的なアイデアは、コンパイル時にランタイム操作を計測し、未定義の動作を引き起こす操作をキャッチし、安全に失敗させることである。LLVM/Clangは、ゼロ除算、NULL可能性違反、負の配列境界、制御フローの完全性を含む、いくつかのサニタイザをサポートする。
GoogleはAndroid Nougat（Android 7としてリリースされた）の時代にLLVM/Clangのサニタイズを利用し始めた。その後、Android O (Oreo、Android 8.0)で制御フローの完全性チェックを追加した。Android Pに対してGoogleは、メディアフレームワークやBluetooth、NFC、カーネルなどのセキュリティに不可欠な他のコンポーネントを含む、OSのさまざまなコンポーネントにわたって、サニタイザと制御フローの完全性の両方をサポートを拡大した、とLozano氏は述べている。
制御フローの完全性は、攻撃者が既存のコードを活用するコードリユース攻撃に対処する。たとえば、ret命令で終了する短いコードシーケンスを見つけ、それらを任意の順序で実行することによって実現される。これにより、攻撃者はコードを注入することなくスタックを制御でき、そのような攻撃を検出する複雑さが大幅に増加する。この種の攻撃に対処するために、制御フローの完全性は、コンパイル時に決定した、パスを限定したコールグラフに、プログラムの制御フローを限定することを目的としている。プログラムが予期しない呼び出し先に分岐しようとすると、安全な方法で中止される。制御フローの整合性には、コンパイル時と実行時の両方でコストがかかるが、AndroidでのGoogleのテストでは無視できることが分かったとLozano氏は述べている。
サニタイザに関して、Googleは、整数オーバーフローサニタイザの使用をlibui、libnlなどのライブラリにも広げた。整数オーバーフローは、攻撃者がプログラムの制御を奪う一般的な手段であり、以前のバージョンのAndroidに影響を与えたStagefright脆弱性の中心であった。Googleの貢献は、LLVM整数オーバーフローサニタイザを改善し、多くのケースで75％以上のオーバーヘッドを減らしたことである。
Pixel携帯電話や他のサポートされている端末を所有している場合は、Android P Betaプログラムに登録することでAndroid P betaにアクセスできる。https://www.infoq.com/jp/news/2018/08/android-p-security-mitigations
ゼロ除算の発見は日本です：
∞？？？
∞は定まった数ではない・・・・・
人工知能はゼロ除算ができるでしょうか：
とても興味深く読みました：2014年2月2日
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
ゼロ除算関係論文・本
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\begin{document}
\title{\bf Announcement 412: The 4th birthday of the division by zero $z/0=0$ \\
(2018.2.2)}
\author{{\it Institute of Reproducing Kernels}\\
Kawauchi-cho, 5-1648-16,\\
Kiryu 376-0041, Japan\\
}
\date{\today}
\maketitle
The Institute of Reproducing Kernels is dealing with the theory of division by zero calculus and declares that the division by zero was discovered as $0/0=1/0=z/0=0$ in a natural sense on 2014.2.2. The result shows a new basic idea on the universe and space since Aristotelēs (BC384 - BC322) and Euclid (BC 3 Century - ), and the division by zero is since Brahmagupta (598 - 668 ?).
In particular, Brahmagupta defined as $0/0=0$ in Brāhmasphuṭasiddhānta (628), however, our world history stated that his definition $0/0=0$ is wrong over 1300 years, but, we showed that his definition is suitable.
For the details, see the references and the site: http://okmr.yamatoblog.net/
We wrote a global book manuscript \cite{s18} with 154 pages
and stated in the preface and last section of the manuscript as follows:
\bigskip
{\bf Preface}
\medskip
The division by zero has a long and mysterious story over the world (see, for example, H. G. Romig \cite{romig} and Google site with the division by zero) with its physical viewpoints since the document of zero in India on AD 628. In particular, note that Brahmagupta (598 -668 ?) established the four arithmetic operations by introducing $0$ and at the same time he defined as $0/0=0$ in
Brhmasphuasiddhnta. Our world history, however, stated that his definition $0/0=0$ is wrong over 1300 years, but, we will see that his definition is right and suitable.
The division by zero $1/0=0/0=z/0$ itself will be quite clear and trivial with several natural extensions of the fractions against the mysterously long history, as we can see from the concepts of the Moore-Penrose generalized inverses or the Tikhonov regularization method to the fundamental equation $az=b$, whose solution leads to the definition $z =b/a$.
However, the result (definition) will show that
for the elementary mapping
\begin{equation}
W = \frac{1}{z},
\end{equation}
the image of $z=0$ is $W=0$ ({\bf should be defined from the form}). This fact seems to be a curious one in connection with our well-established popular image for the point at infinity on the Riemann sphere (\cite{ahlfors}). �As the representation of the point at infinity of the Riemann sphere by the
zero $z = 0$, we will see some delicate relations between $0$ and $\infty$ which show a strong
discontinuity at the point of infinity on the Riemann sphere. We did not consider any value of the elementary function $W =1/ z $ at the origin $z = 0$, because we did not consider the division by zero
$1/ 0$ in a good way. Many and many people consider its value by the limiting like $+\infty $ and $- \infty$ or the
point at infinity as $\infty$. However, their basic idea comes from {\bf continuity} with the common sense or
based on the basic idea of Aristotle. --
For the related Greece philosophy, see \cite{a,b,c}. However, as the division by zero we will consider its value of
the function $W =1 /z$ as zero at $z = 0$. We will see that this new definition is valid widely in
mathematics and mathematical sciences, see (\cite{mos,osm}) for example. Therefore, the division by zero will give great impacts to calculus, Euclidean geometry, analytic geometry, differential equations, complex analysis in the undergraduate level and to our basic ideas for the space and universe.
We have to arrange globally our modern mathematics in our undergraduate level. Our common sense on the division by zero will be wrong, with our basic idea on the space and the universe since Aristotle and Euclid. We would like to show clearly these facts in this book. The content is in the undergraduate level.
\bigskip
\bigskip
{\bf Conclusion}
\medskip
Apparently, the common sense on the division by zero with a long and mysterious history is wrong and our basic idea on the space around the point at infinity is also wrong since Euclid. On the gradient or on derivatives we have a great missing since $\tan (\pi/2) = 0$. Our mathematics is also wrong in elementary mathematics on the division by zero.
This book is an elementary mathematics on our division by zero as the first publication of books for the topics. The contents have wide connections to various fields beyond mathematics. The author expects the readers write some philosophy, papers and essays on the division by zero from this simple source book.
The division by zero theory may be developed and expanded greatly as in the author's conjecture whose break theory was recently given surprisingly and deeply by Professor Qi'an Guan \cite{guan} since 30 years proposed in \cite{s88} (the original is in \cite {s79}).
We have to arrange globally our modern mathematics with our division by zero in our undergraduate level.
We have to change our basic ideas for our space and world.
We have to change globally our textbooks and scientific books on the division by zero.
\bibliographystyle{plain}
\begin{thebibliography}{10}
\bibitem{ahlfors}
L. V. Ahlfors, Complex Analysis, McGraw-Hill Book Company, 1966.
\bibitem{cs}
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
\bibitem{guan}
Q. Guan, A proof of Saitoh's conjecture for conjugate Hardy H2 kernels, arXiv:1712.04207.
\bibitem{kmsy}
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$,
Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
\bibitem{ms16}
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, {\bf 6}(2016), 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.2016.62007.
\bibitem{ms18}
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Submitted for publication)
\bibitem{mms18}
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.
\bibitem{msy}
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. {\bf 6}(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
\bibitem{mos}
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, {\bf 2}8(2017); Issue 1, 2017), 1-16.
\bibitem{osm}
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), {\bf 1}(2017), 70-77.
\bibitem{os}
H. Okumura and S. Saitoh, The Descartes circles theorem and division by zero calculus. https://arxiv.org/abs/1711.04961 (2017.11.14).
\bibitem{o}
H. Okumura, Wasan geometry with the division by 0. https://arxiv.org/abs/1711.06947 International Journal of Geometry.
\bibitem{os18}
H. Okumura and S. Saitoh,
Applications of the division by zero calculus to Wasan geometry.
(Submitted for publication).
\bibitem{ps18}
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.
\bibitem{romig}
H. G. Romig, Discussions: Early History of Division by Zero,
American Mathematical Monthly, Vol. {\bf 3}1, No. 8. (Oct., 1924), pp. 387-389.
\bibitem{s79}
S. Saitoh, The Bergman norm and the Szeg$\ddot{o}$ norm, Trans. Amer. Math. Soc. {\bf 249} (1979), no. 2, 261--279.
\bibitem{s88}
S. Saitoh, Theory of reproducing kernels and its applications. Pitman Research Notes in Mathematics Series, {\bf 189}. Longman Scientific \& Technical, Harlow; copublished in the United States with John Wiley \& Sons, Inc., New York, 1988. x+157 pp. ISBN: 0-582-03564-3
\bibitem{s14}
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
\bibitem{s16}
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
\bibitem{s17}
S. Saitoh, Mysterious Properties of the Point at Infinity、
arXiv:1712.09467 [math.GM](2017.12.17).
\bibitem{s18}
S. Saitoh, Division by zero calculus (154 pages: draft): （http://okmr.yamatoblog.net/）
\bibitem{ttk}
S.-E. Takahasi, M. Tsukada and Y. Kobayashi, Classification of continuous fractional binary operations on the real and complex fields, Tokyo Journal of Mathematics, {\bf 38}(2015), no. 2, 369-380.
\bibitem{a}
https://philosophy.kent.edu/OPA2/sites/default/files/012001.pdf
\bibitem{b}
http://publish.uwo.ca/~jbell/The 20Continuous.pdf
\bibitem{c}
http://www.mathpages.com/home/kmath526/kmath526.htm
\bibitem{ann179}
Announcement 179 (2014.8.30): Division by zero is clear as z/0=0 and it is fundamental in mathematics.
\bibitem{ann185}
Announcement 185 (2014.10.22): The importance of the division by zero $z/0=0$.
\bibitem{ann237}
Announcement 237 (2015.6.18): A reality of the division by zero $z/0=0$ by geometrical optics.
\bibitem{ann246}
Announcement 246 (2015.9.17): An interpretation of the division by zero $1/0=0$ by the gradients of lines.
\bibitem{ann247}
Announcement 247 (2015.9.22): The gradient of y-axis is zero and $\tan (\pi/2) =0$ by the division by zero $1/0=0$.
\bibitem{ann250}
Announcement 250 (2015.10.20): What are numbers? - the Yamada field containing the division by zero $z/0=0$.
\bibitem{ann252}
Announcement 252 (2015.11.1): Circles and
curvature - an interpretation by Mr.
Hiroshi Michiwaki of the division by
zero $r/0 = 0$.
\bibitem{ann281}
Announcement 281 (2016.2.1): The importance of the division by zero $z/0=0$.
\bibitem{ann282}
Announcement 282 (2016.2.2): The Division by Zero $z/0=0$ on the Second Birthday.
\bibitem{ann293}
Announcement 293 (2016.3.27): Parallel lines on the Euclidean plane from the viewpoint of division by zero 1/0=0.
\bibitem{ann300}
Announcement 300 (2016.05.22): New challenges on the division by zero z/0=0.
\bibitem{ann326}
Announcement 326 (2016.10.17): The division by zero z/0=0 - its impact to human beings through education and research.
\bibitem{ann352}
Announcement 352(2017.2.2): On the third birthday of the division by zero z/0=0.
\bibitem{ann354}
Announcement 354(2017.2.8): What are $n = 2,1,0$ regular polygons inscribed in a disc? -- relations of $0$ and infinity.
\bibitem{362}
Announcement 362(2017.5.5): Discovery of the division by zero as $0/0=1/0=z/0=0$
\bibitem{380}
Announcement 380 (2017.8.21): What is the zero?
\bibitem{388}
Announcement 388(2017.10.29): Information and ideas on zero and division by zero (a project).
\bibitem{409}
Announcement 409 (2018.1.29.): Various Publication Projects on the Division by Zero.
\bibitem{410}
Announcement 410 (2018.1 30.): What is mathematics? -- beyond logic; for great challengers on the division by zero.
\end{thebibliography}
\end{document}
List of division by zero:
\bibitem{os18}
H. Okumura and S. Saitoh,
Remarks for The Twin Circles of Archimedes in a Skewed Arbelos by H. Okumura and M. Watanabe, Forum Geometricorum.
Saburou Saitoh, Mysterious Properties of the Point at Infinity、
arXiv:1712.09467 [math.GM]
arXiv:1712.09467 [math.GM]
Hiroshi Okumura and Saburou Saitoh
The Descartes circles theorem and division by zero calculus. ２０１７．１１．１４
L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063.
M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane,
New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9.
T. Matsuura and S. Saitoh,
Matrices and division by zero z/0=0,
Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58
Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt
\\ http://dx.doi.org/10.4236/alamt.2016.62007.
T. Matsuura and S. Saitoh,
Division by zero calculus and singular integrals. (Submitted for publication).
T. Matsuura, H. Michiwaki and S. Saitoh,
$\log 0= \log \infty =0$ and applications. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.)
H. Michiwaki, S. Saitoh and M.Yamada,
Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.html
H. Michiwaki, H. Okumura and S. Saitoh,
Division by Zero $z/0 = 0$ in Euclidean Spaces,
International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16.
H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$,
Journal of Technology and Social Science (JTSS), 1(2017), 70-77.
S. Pinelas and S. Saitoh,
Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics).
S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/
S. Saitoh, A reproducing kernel theory with some general applications,
Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) .
再生核研究所声明371（2017.6.27）ゼロ除算の講演― 国際会議 https://sites.google.com/site/sandrapinelas/icddea-2017 報告
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12276045402.html
1/0=0、0/0=0、z/0=0
http://ameblo.jp/syoshinoris/entry-12263708422.html
1/0=0、0/0=0、z/0=0
ソクラテス・プラトン・アリストテレス その他
Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity.
私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。
ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか
〔NHKスペシャル〕神の数式 完全版 第3回 宇宙はなぜ始まったのか
〔NHKスペシャル〕神の数式 完全版 第1回 この世は何からできているのか
NHKスペシャル 神の数式 完全版 第4回 異次元宇宙は存在するか
再生核研究所声明 411(2018.02.02): ゼロ除算発見4周年を迎えて
ゼロ除算の論文
Mysterious Properties of the Point at Infinity
Mysterious Properties of the Point at Infinity
Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197
Author: Jakub Czajko, 92(2) (2018) 171-197
WSN 92(2) (2018) 171-197
2018.3.18．午前中 最後の講演： 日本数学会 東大駒場、函数方程式論分科会 講演書画カメラ用 原稿
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より
The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18.
https://ameblo.jp/syoshinoris/entry-12361744016.html より
*057 Pinelas,S./Caraballo,T./Kloeden,P./Graef,J.(eds.):
Differential and Difference Equations with Applications:
ICDDEA, Amadora, 2017.
(Springer Proceedings in Mathematics and Statistics, Vol. 230)
May 2018 587 pp.
ゼロ除算の論文が２編、出版になりました：
ICDDEA: International Conference on Differential & Difference Equations and Applications
Differential and Difference Equations with Applications
ICDDEA, Amadora, Portugal, June 2017
• Editors
• (view affiliations)
• Sandra Pinelas
• Tomás Caraballo
• Peter Kloeden
• John R. Graef
Conference proceedingsICDDEA 2017
log0=log∞=0log0=log∞=0 and Applications
Hiroshi Michiwaki, Tsutomu Matuura, Saburou Saitoh
Pages 293-305
Division by Zero Calculus and Differential Equations
Sandra Pinelas, Saburou Saitoh
Pages 399-418
ICDDEA: International Conference on Differential & Difference Equations and Applications
Differential and Difference Equations with Applications
ICDDEA, Amadora, Portugal, June 2017
• Editors
• (view affiliations)
• Sandra Pinelas
• Tomás Caraballo
• Peter Kloeden
• John R. Graef
Conference proceedingsICDDEA 2017
log0=log∞=0log0=log∞=0 and Applications
Hiroshi Michiwaki, Tsutomu Matuura, Saburou Saitoh
Pages 293-305
Division by Zero Calculus and Differential Equations
Sandra Pinelas, Saburou Saitoh
Pages 399-418
ゼロ除算（division by zero）1/0=0、0/0=0、z/0=0
2018年05月28日(月)
テーマ：数学
テーマ：数学
これは最も簡単な 典型的なゼロ除算の結果と言えます。 ユークリッド以来の驚嘆する、誰にも分る結果では ないでしょうか？
Hiroshi O. Is It Really Impossible To Divide By Zero?. Biostat Biometrics Open Acc J. 2018; 7(1): 555703. DOI: 10.19080/BBOJ.2018.07.555703
ゼロで分裂するのは本当に不可能ですか？ - Juniper Publishers
再生核研究所 ゼロ除算の発見と重要性を指摘した：2014年2月2日
ゼロ除算の発見は日本です：
∞？？？
∞は定まった数ではない・・・・
人工知能はゼロ除算ができるでしょうか：
とても興味深く読みました：2014年2月2日
ゼロ除算の発見と重要性を指摘した：日本、再生核研究所
ゼロ除算関係論文・本
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