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Tennis players with most titles in the Open Era
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This page lists all tennis players who have won at least 30 top-level professional tournament titles since the Open Era began in 1968.
Titles can be any combination of singles and doubles, so the combined total is the default sorting of the lists. The current top-level events are on the ATP World Tour for men and the WTA Tour for women.
Contents
1 Key
2 Men
3 Women
4 See also
5 References
Key[edit]
Player Name and nationality
Span Years (any gaps omitted)
Sing. Singles titles
Doub. Same-gender doubles titles
Mix. Mixed doubles titles
Total Combined number of titles
† Tennis Hall of Fame member
* Player is currently active
Men[edit]
John McEnroe won the most titles overall.
Jimmy Connors won the most singles titles.
Bob and Mike Bryan won 116 doubles titles together.
Player
Span
Sing.
Doub.
Mix.
Total
Ref.
John McEnroe† 1978–06 77 78 1 156 [1]
Jimmy Connors† 1972–96 109 15 0 124
Mike Bryan* 1998– 0 119 4 123 [2]
Bob Bryan* 1998– 0 116 7 123 [3]
Rod Laver† 1968–76 74 37 0 111 [4][1]
Ilie Năstase† 1968–89 58 51 2 111 [5][1]
Tom Okker 1968–82 31 78 0 109 [1]
Roger Federer* 1998– 98 8 0 106 [6]
/ Ivan Lendl† 1978–94 94 6 0 100 [7]
Daniel Nestor* 1991– 0 91 4 95 [8]
Stan Smith† 1968–86 38 54 0 92 [9][1]
Todd Woodbridge† 1988–05 2 83 6 91 [10]
Rafael Nadal* 2001– 80 11 0 91 [11]
Raúl Ramírez 1973–83 19 62 0 81 [12][1]
Frew McMillan† 1968–83 2 74 4 80 [13][1]
Brian Gottfried 1972–85 25 54 0 79 [14][1]
Guillermo Vilas† 1968–92 62 16 0 78 [15]
Mark Woodforde† 1984–00 4 67 5 76 [16]
Bob Hewitt 1968–83 4 65 5 74 [1]
Marty Riessen 1968–81 6 60 7 73 [1]
Novak Djokovic* 2003– 69 1 0 70 [17]
Peter Fleming 1973–88 3 66 0 69 [1]
Björn Borg† 1973–83 64 4 0 68 [18]
Wojtek Fibak 1973–89 15 52 0 67 [19]
Anders Järryd 1980–96 8 59 0 67
Pete Sampras† 1988–02 64 2 0 66 [20]
Leander Paes* 1991– 1 55 10 66
John Newcombe† 1968–83 32 33 0 65
Emilio Sánchez 1984–97 15 50 0 65 [21]
Tomáš Šmíd 1976–92 9 55 0 64
Boris Becker† 1984–99 49 15 0 64 [22]
Andre Agassi† 1986–06 60 1 0 61
Jonas Björkman 1991–08 6 54 0 60
Mahesh Bhupathi 1995–16 0 52 8 60
Stefan Edberg† 1983–96 41 18 0 59 [23]
/ Nenad Zimonjić* 1995– 0 54 5 59
Max Mirnyi* 1996– 1 52 5 58
Ken Rosewall† 1968–80 43 14 0 57 [24]
Mark Knowles 1992–12 0 55 1 56
Manuel Orantes† 1968–83 33 22 0 55
Paul Haarhuis 1989–03 1 54 0 55
Sherwood Stewart 1968–85 1 51 2 54
Andrés Gómez 1979–95 21 33 0 54
Yevgeny Kafelnikov 1992–03 26 27 0 53
Robert Lutz 1968–85 9 43 0 52
Rick Leach 1987–06 0 46 4 50
Sergio Casal 1981–95 1 47 1 49
Jacco Eltingh 1988–98 4 44 0 48
Arthur Ashe† 1968–79 33 14 0 47
Andy Murray* 2005– 45 2 0 47
Thomas Muster 1985–99 44 1 0 45
Mark Edmondson 1975–88 6 34 0 40
Mats Wilander† 1981–96 33 7 0 40
Martin Damm 1990–11 0 40 0 40
Balázs Taróczy 1972–90 13 26 0 39
Yannick Noah† 1977–96 23 16 0 39
Guy Forget 1982–00 11 28 0 39
John Fitzgerald 1980–95 6 30 2 38
Ken Flach 1983–96 0 34 2 36
Patrick Galbraith 1987–00 0 36 0 36
Cyril Suk 1988–06 0 32 4 36
Andy Roddick† 2000–12 32 4 0 36
Vitas Gerulaitis 1972–86 26 9 0 35
Pavel Složil 1973–92 2 32 1 35
Kevin Ullyett 1990–10 0 34 1 35
Horia Tecău* 1998– 0 34 1 35
Sandy Mayer 1970–85 10 24 0 34
Kevin Curren 1979–93 5 26 3 34
Michael Chang† 1988–03 34 0 0 34
Lleyton Hewitt 1998–16 30 3 0 33
František Čermák* 1998– 0 31 1 32
/ Goran Ivanišević 1988–04 22 9 0 31
Fabrice Santoro 1989–10 6 24 1 31
Michaël Llodra 1999–14 5 26 0 31
Bruno Soares* 2000– 0 28 3 31 [25]
David Rikl 1989–05 0 30 0 30
Marcelo Melo* 1998– 0 30 0 30 [26]
Women[edit]
Martina Navratilova won the most singles and doubles titles.
Player
Span
Sing.
Doub.
Mix.
Total
Ref.
/ Martina Navratilova† 1975–06 167 177 10 354 [27]
Billie Jean King† 1968–83 67 101 7 175 [28]
Chris Evert† 1972–89 157 18 0 175 [29]
Margaret Court† 1968–77 92 48 7 147 [30]
Pam Shriver† 1979–97 21 106 1 128 [31]
Rosemary Casals† 1968–91 11 112 3 126 [32][33]
Steffi Graf† 1982–99 107 11 0 118 [34]
Martina Hingis† 1994–17 43 64 7 114 [35]
Jana Novotná† 1987–99 24 76 4 104 [36]
A. Sánchez Vicario† 1985–02 29 69 4 102 [37]
Serena Williams* 1995– 72 23 2 97 [38]
Lindsay Davenport† 1993–08 55 38 0 93 [39]
Lisa Raymond 1993–15 4 79 5 88 [40]
/ Natasha Zvereva† 1988–02 4 80 2 86 [41]
Helena Suková† 1981–98 10 69 5 84 [42]
E. Goolagong Cawley† 1968–83 68 11 1 80 [43]
Venus Williams* 1994– 49 22 2 73 [44]
Gigi Fernández† 1983–97 2 69 0 71 [45]
/ Larisa Neiland 1982–99 2 65 4 71 [46]
Wendy Turnbull 1970–89 9 55 5 69 [47]
Cara Black* 1998– 1 60 5 66 [48]
Rennae Stubbs 1986–11 0 60 2 62 [49]
/ / Monica Seles† 1989–08 53 6 0 59 [50]
Virginia Wade† 1968–86 55 0 0 55 [51]
/ Liezel Huber 1993–17 0 53 2 55 [52]
Kim Clijsters† 1999–12 41 11 0 52 [53]
Paola Suárez 1991–12 4 44 0 48 [54]
Katarina Srebotnik* 1995– 4 39 5 48 [55]
V. Ruano Pascual 1992–10 3 43 1 47 [56]
/ Hana Mandlíková† 1978–90 27 19 0 46 [57]
Conchita Martínez 1988–06 33 13 0 46 [58]
Ai Sugiyama 1992–09 6 38 1 45 [59]
Justine Henin† 1999–11 43 2 0 45 [60]
Sania Mirza* 2003– 1 41 3 45 [61]
Lori McNeil 1983–02 10 33 1 44 [62]
Gabriela Sabatini† 1985–96 27 14 0 41 [63]
A. Medina Garrigues* 1997– 11 28 0 39 [64]
Maria Sharapova* 2001– 36 3 0 39 [65]
Elizabeth Smylie 1982–97 2 32 3 37 [66]
Nadia Petrova 1999–17 13 24 0 37 [67]
Samantha Stosur* 1999– 9 24 3 36 [68]
Tracy Austin† 1978–94 30 4 1 35 [69]
Roberta Vinci 1999–18 10 25 0 35 [70]
Sara Errani* 2002– 9 26 0 35 [71]
Svetlana Kuznetsova* 2000– 18 16 0 34 [72]
Claudia Kohde-Kilsch 1980–94 8 25 0 33 [73]
Zina Garrison 1982–97 11 19 3 33 [74]
Nathalie Tauziat 1984–03 8 25 0 33 [75]
Caroline Wozniacki* 2005– 29 2 0 31 [76]
Květa Peschke* 1993– 1 30 0 31 [77]
Latisha Chan* 2004– 0 29 1 30 [78]
See also[edit]
Tennis portal
Open Era tennis records – men's singles
Open Era tennis records – women's singles
List of Open Era tennis records
References[edit]
^ Jump up to: a b c d e f g h i j k "2016 ATP Media Guide - Statistical Information (Page 11)" (PDF). ATPWorldTour.com.
Jump up ^ "Mike Bryan – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Bob Bryan – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ listed at Rod Laver career statistics
Jump up ^ "Ilie Nastase – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Roger Federer – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Ivan Lendl – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Daniel Nestor – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Stan Smith – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Todd Woodbridge – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Rafael Nadal – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Raúl Ramírez – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 17, 2015.
Jump up ^ "Frew McMillan – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Brian Gottfried – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Guillermo Vilas – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Mark Woodforde – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Novak Djokovic – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 18, 2012.
Jump up ^ "Bjorn Borg – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Wojtek Fibak – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Pete Sampras – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Emilio Sanchez – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Boris Becker – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ "Stefan Edberg – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved May 19, 2012.
Jump up ^ listed at Ken Rosewall career statistics
Jump up ^ "Bruno Soares – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved March 4, 2018.
Jump up ^ "Marcelo Melo – Titles/Finals". ATPWorldTour.com – Tennis Players. ATP World Tour. Retrieved June 25, 2018.
Jump up ^ Martina Navratilova Archived 2012-11-03 at the Wayback Machine. WTA profile.
Jump up ^ Billie Jean King WTA profile.
Jump up ^ Chris Evert WTA profile.
Jump up ^ Margaret Court WTA profile
Jump up ^ Pam Shriver WTA profile.
Jump up ^ Rosemary Casals Encyclopedia.com
Jump up ^ Joel Drucker (March 3, 2009). "Casals far more than King's sidekick". sports.espn.go.com. ESPN.
Jump up ^ Steffi Graf WTA profile
Jump up ^ Martina Hingis WTA profile.
Jump up ^ Jana Novotná WTA profile
Jump up ^ Arantxa Sánchez Vicario WTA profile
Jump up ^ Serena Williams WTA profile.
Jump up ^ Lindsay Davenport WTA profile
Jump up ^ Lisa Raymond WTA profile.
Jump up ^ Natasha Zvereva WTA profile
Jump up ^ Helena Suková WTA profile.
Jump up ^ Evonne Goolagong Cawley WTA profile.
Jump up ^ Venus Williams WTA profile.
Jump up ^ Gigi Fernandez WTA profile.
Jump up ^ Larisa Savchenko Neiland WTA profile.
Jump up ^ Wendy Turnbull WTA profile.
Jump up ^ Cara Black WTA profile.
Jump up ^ Rannae Stubbs WTA profile.
Jump up ^ Monica Seles WTA profile.
Jump up ^ Virginia Wade WTA profile.
Jump up ^ Liezel Huber WTA profile.
Jump up ^ Kim Clijsters WTA profile.
Jump up ^ Paola Suárez WTA profile.
Jump up ^ Katarina Srebotnik WTA profile.
Jump up ^ Virginia Ruano Pascual WTA profile.
Jump up ^ Hana Mandlíková WTA profile.
Jump up ^ Conchita Martínez WTA profile.
Jump up ^ Ai Sugiyama WTA profile.
Jump up ^ Justine Henin WTA profile.
Jump up ^ Sania Mirza WTA profile.
Jump up ^ Lori McNeil WTA profile.
Jump up ^ Gabriela Sabatini WTA profile.
Jump up ^ Anabel Medina Garrigues WTA profile.
Jump up ^ Maria Sharapova WTA profile.
Jump up ^ Elizabeth Smylie WTA profile.
Jump up ^ Nadia Petrova WTA profile.
Jump up ^ Samantha Stosur WTA profile.
Jump up ^ Tracy Austin WTA profile.
Jump up ^ Roberta Vinci WTA profile.
Jump up ^ Sara Errani WTA profile.
Jump up ^ Svetlana Kuznetsova WTA profile.
Jump up ^ Claudia Kohde-Kilsch WTA profile.
Jump up ^ Zina Garrison WTA profile.
Jump up ^ Nathalie Tauziat WTA profile.
Jump up ^ Caroline Wozniacki WTA profile.
Jump up ^ Kveta Peschke WTA profile.
Jump up ^ Latisha Chan WTA profile.
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text:
Second moment of area
From Wikipedia, the free encyclopedia
Jump to: navigation, search
This article is about the geometrical property of an area, termed the second moment of area. For the moment of inertia dealing with the rotation of an object with mass, see Mass moment of inertia.
For a list of equations for second moments of area of standard shapes, see List of second moments of area.
The 2nd moment of area, also known as moment of inertia of plane area, area moment of inertia, or second area moment, is a geometrical property of an area which reflects how its points are distributed with regard to an arbitrary axis. The second moment of area is typically denoted with either an I {\displaystyle I} for an axis that lies in the plane or with a J {\displaystyle J} for an axis perpendicular to the plane. In both cases, it is calculated with a multiple integral over the object in question. Its dimension is L (length) to the forth power. Its unit of dimension when working with the International System of Units is meters to the fourth power, m4, or inches to the fourth power, in4, when working in the Imperial System of Units.
In the field of structural engineering, the second moment of area of a beam is an important property used in the calculation of the beam's deflection and the calculation of stress caused by a moment applied to the beam. The planar second moment of area provides insight into a beam's resistance to bending due to an applied moment, force, or distributed load perpendicular to its neutral axis, as a function of its shape. The polar second moment of area provides insight into a beam's resistance to torsional deflection, due to an applied moment parallel to its cross-section, as a function of its shape. This resistance to deflection or torsion is only a function of shape, and not due to a beam's material stiffness, or Elastic Modulus.
Note: Different disciplines use the term moment of inertia (MOI) to refer to either the x, y, or both of the planar second moments of area, I = ∬ R x 2 d A {\displaystyle I=\textstyle \iint _{R}x^{2}\,\mathrm {d} A} , where x is the distance to some reference plane, or the polar second moment of area, I = ∬ R r 2 d A {\displaystyle I=\textstyle \iint _{R}r^{2}\,\mathrm {d} A} , where r is the distance to some reference axis. In each case the integral is over all the infinitesimal elements of area, dA, in some two-dimensional cross-section. In math and physics, moment of inertia is strictly the second moment of mass with respect to distance from an axis: I = ∫ Q r 2 d m {\displaystyle I=\textstyle \int _{Q}r^{2}\mathrm {d} m} , where r is the distance to some potential rotation axis, and the integral is over all the infinitesimal elements of mass, dm, in a three-dimensional space occupied by an object Q. The MOI, in this sense, is the analog of mass for rotational problems. In engineering (especially mechanical and civil), moment of inertia commonly refers to the second moment of the area.[1]
Contents
[hide]
1 Definition
1.1 Product moment of area
2 Parallel axis theorem
3 Perpendicular axis theorem
4 Composite shapes
5 Examples
5.1 Rectangle with centroid at the origin
5.2 Annulus centered at origin
5.3 Any polygon
6 See also
7 References
Definition[edit]
An arbitrary shape. ρ is the radial distance to the element dA, with projections x and y on the axes.
The second moment of area for an arbitrary shape R with respect to an arbitrary axis B B ′ {\displaystyle BB'} is defined as
J B B ′ = ∬ R ρ 2 d A {\displaystyle J_{BB'}=\iint \limits _{R}{\rho }^{2}\,\mathrm {d} A}
where
d A {\displaystyle \mathrm {d} A} is the differential area of the arbitrary shape, and
ρ {\displaystyle \rho } is the distance from the axis B B ′ {\displaystyle BB'} to d A {\displaystyle \mathrm {d} A} .[2]
For example, when the desired reference axis is the x-axis, the second moment of area I x x {\displaystyle I_{xx}} (often denoted as I x {\displaystyle I_{x}} ) can be computed in Cartesian coordinates as
I x = ∬ R y 2 d x d y {\displaystyle I_{x}=\iint \limits _{R}y^{2}\,\mathrm {d} x\,\mathrm {d} y}
The second moment of the area is crucial in Euler–Bernoulli theory of slender beams.
Product moment of area[edit]
More generally, the product moment of area is defined as[3]
I x y = ∬ R y x d x d y {\displaystyle I_{xy}=\iint \limits _{R}yx\,\mathrm {d} x\,\mathrm {d} y}
Parallel axis theorem[edit]
A shape with centroidal axis x. The parallel axis theorem can be used to obtain the second moment of area with respect to the x' axis.
Main article: Parallel axis theorem
It is sometimes necessary to calculate the second moment of area of a shape with respect to an x ′ {\displaystyle x'} axis different to the centroidal axis of the shape. However, it is often easier to derive the second moment of area with respect to its centroidal axis, x {\displaystyle x} , and use the parallel axis theorem to derive the second moment of area with respect to the x ′ {\displaystyle x'} axis. The parallel axis theorem states
I x ′ = I x + A d 2 {\displaystyle I_{x'}=I_{x}+Ad^{2}}
where
A {\displaystyle A} is the area of the shape, and
d {\displaystyle d} is the perpendicular distance between the x {\displaystyle x} and x ′ {\displaystyle x'} axes.[4][5]
A similar statement can be made about a y ′ {\displaystyle y'} axis and the parallel centroidal y {\displaystyle y} axis. Or, in general, any centroidal B {\displaystyle B} axis and a parallel B ′ {\displaystyle B'} axis.
Perpendicular axis theorem[edit]
Main article: Perpendicular axis theorem
For the simplicity of calculation, it is often desired to define the polar moment of area (with respect to a perpendicular axis) in terms of two area moments of inertia (both with respect to in-plane axes). The simplest case relates J z {\displaystyle J_{z}} to I x {\displaystyle I_{x}} and I y {\displaystyle I_{y}} .
J z = ∬ R ρ 2 d A = ∬ R ( x 2 + y 2 ) d A = ∬ R x 2 d A + ∬ R y 2 d A = I x + I y {\displaystyle J_{z}=\iint \limits _{R}\rho ^{2}\,\mathrm {d} A=\iint \limits _{R}\left(x^{2}+y^{2}\right)\,\mathrm {d} A=\iint \limits _{R}x^{2}\,\mathrm {d} A+\iint \limits _{R}y^{2}\,\mathrm {d} A=I_{x}+I_{y}}
This relationship relies on the Pythagorean theorem which relates x {\displaystyle x} and y {\displaystyle y} to ρ {\displaystyle \rho } and on the linearity of integration.
Composite shapes[edit]
For more complex areas, it is often easier to divide the area into a series of "simpler" shapes. The second moment of area for the entire shape is the sum of the second moment of areas of all of its parts about a common axis. This can include shapes that are "missing" (i.e. holes, hollow shapes, etc.), in which case the second moment of area of the "missing" areas are subtracted, rather than added. In other words, the second moment of area of "missing" parts are considered negative for the method of composite shapes.
Examples[edit]
See list of second moments of area for other shapes.
Rectangle with centroid at the origin[edit]
Rectangle with base b and height h
Consider a rectangle with base b {\displaystyle b} and height h {\displaystyle h} whose centroid is located at the origin. I x {\displaystyle I_{x}} represents the second moment of area with respect to the x-axis; I y {\displaystyle I_{y}} represents the second moment of area with respect to the y-axis; J z {\displaystyle J_{z}} represents the polar moment of inertia with respect to the z-axis.
I x = ∬ R y 2 d A = ∫ − b 2 b 2 ∫ − h 2 h 2 y 2 d y d x = ∫ − b 2 b 2 1 3 h 3 4 d x = b h 3 12 I y = ∬ R x 2 d A = ∫ − b 2 b 2 ∫ − h 2 h 2 x 2 d y d x = ∫ − b 2 b 2 h x 2 d x = b 3 h 12 {\displaystyle {\begin{aligned}I_{x}&=\iint \limits _{R}y^{2}\,\mathrm {d} A=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}\int _{-{\frac {h}{2}}}^{\frac {h}{2}}y^{2}\,\mathrm {d} y\,\mathrm {d} x=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}{\frac {1}{3}}{\frac {h^{3}}{4}}\,\mathrm {d} x={\frac {bh^{3}}{12}}\\I_{y}&=\iint \limits _{R}x^{2}\,\mathrm {d} A=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}\int _{-{\frac {h}{2}}}^{\frac {h}{2}}x^{2}\,\mathrm {d} y\,\mathrm {d} x=\int _{-{\frac {b}{2}}}^{\frac {b}{2}}hx^{2}\,\mathrm {d} x={\frac {b^{3}h}{12}}\end{aligned}}}
Using the perpendicular axis theorem we get the value of J z {\displaystyle J_{z}} .
J z = I x + I y = b h 3 12 + h b 3 12 = b h 12 ( b 2 + h 2 ) {\displaystyle J_{z}=I_{x}+I_{y}={\frac {bh^{3}}{12}}+{\frac {hb^{3}}{12}}={\frac {bh}{12}}\left(b^{2}+h^{2}\right)}
Annulus centered at origin[edit]
Annulus with inner radius r1 and outer radius r2
Consider an annulus whose center is at the origin, outside radius is r 2 {\displaystyle r_{2}} , and inside radius is r 1 {\displaystyle r_{1}} . Because of the symmetry of the annulus, the centroid also lies at the origin. We can determine the polar moment of inertia, J z {\displaystyle J_{z}} , about the z {\displaystyle z} axis by the method of composite shapes. This polar moment of inertia is equivalent to the polar moment of inertia of a circle with radius r 2 {\displaystyle r_{2}} minus the polar moment of inertia of a circle with radius r 1 {\displaystyle r_{1}} , both centered at the origin. First, let us derive the polar moment of inertia of a circle with radius r {\displaystyle r} with respect to the origin. In this case, it is easier to directly calculate J z {\displaystyle J_{z}} as we already have r 2 {\displaystyle r^{2}} , which has both an x {\displaystyle x} and y {\displaystyle y} component. Instead of obtaining the second moment of area from Cartesian coordinates as done in the previous section, we shall calculate I x {\displaystyle I_{x}} and J z {\displaystyle J_{z}} directly using polar coordinates.
I x , c i r c l e = ∬ R y 2 d A = ∬ R ( r sin  θ ) 2 d A = ∫ 0 2 π ∫ 0 r ( r sin  θ ) 2 ( r d r d θ ) = ∫ 0 2 π ∫ 0 r r 3 sin 2  θ d r d θ = ∫ 0 2 π r 4 sin 2  θ 4 d θ = π 4 r 4 J z , c i r c l e = ∬ R r 2 d A = ∫ 0 2 π ∫ 0 r r 2 ( r d r d θ ) = ∫ 0 2 π ∫ 0 r r 3 d r d θ = ∫ 0 2 π r 4 4 d θ = π 2 r 4 {\displaystyle {\begin{aligned}I_{x,circle}&=\iint \limits _{R}y^{2}\,dA=\iint \limits _{R}(r\sin {\theta )}^{2}\,\mathrm {d} A=\int _{0}^{2\pi }\int _{0}^{r}(r\sin {\theta })^{2}\left(r\,\mathrm {d} r\,\mathrm {d} \theta \right)\\&=\int _{0}^{2\pi }\int _{0}^{r}r^{3}\sin ^{2}{\theta }\,\mathrm {d} r\,\mathrm {d} \theta =\int _{0}^{2\pi }{\frac {r^{4}\sin ^{2}{\theta }}{4}}\,\mathrm {d} \theta ={\frac {\pi }{4}}r^{4}\\J_{z,circle}&=\iint \limits _{R}r^{2}\,\mathrm {d} A=\int _{0}^{2\pi }\int _{0}^{r}r^{2}\left(r\,\mathrm {d} r\,\mathrm {d} \theta \right)=\int _{0}^{2\pi }\int _{0}^{r}r^{3}\,\mathrm {d} r\,\mathrm {d} \theta \\&=\int _{0}^{2\pi }{\frac {r^{4}}{4}}\,\mathrm {d} \theta ={\frac {\pi }{2}}r^{4}\end{aligned}}}
Now, the polar moment of inertia about the z {\displaystyle z} axis for an annulus is simply, as stated above, the difference of the second moments of area of a circle with radius r 2 {\displaystyle r_{2}} and a circle with radius r 1 {\displaystyle r_{1}} .
J z = J z , r 2 − J z , r 1 = π 2 r 2 4 − π 2 r 1 4 = π 2 ( r 2 4 − r 1 4 ) {\displaystyle J_{z}=J_{z,r_{2}}-J_{z,r_{1}}={\frac {\pi }{2}}r_{2}^{4}-{\frac {\pi }{2}}r_{1}^{4}={\frac {\pi }{2}}\left(r_{2}^{4}-r_{1}^{4}\right)}
Alternatively, we could change the limits on the d r {\displaystyle \mathrm {d} r} integral the first time around to reflect the fact that there is a hole. This would be done like this.
J z = ∬ R r 2 d A = ∫ 0 2 π ∫ r 1 r 2 r 2 ( r d r d θ ) = ∫ 0 2 π ∫ r 1 r 2 r 3 d r d θ = ∫ 0 2 π [ r 2 4 4 − r 1 4 4 ] d θ = π 2 ( r 2 4 − r 1 4 ) {\displaystyle {\begin{aligned}J_{z}&=\iint \limits _{R}r^{2}\,\mathrm {d} A=\int _{0}^{2\pi }\int _{r_{1}}^{r_{2}}r^{2}\left(r\,\mathrm {d} r\,\mathrm {d} \theta \right)=\int _{0}^{2\pi }\int _{r_{1}}^{r_{2}}r^{3}\,\mathrm {d} r\,\mathrm {d} \theta \\&=\int _{0}^{2\pi }\left[{\frac {r_{2}^{4}}{4}}-{\frac {r_{1}^{4}}{4}}\right]\,\mathrm {d} \theta ={\frac {\pi }{2}}\left(r_{2}^{4}-r_{1}^{4}\right)\end{aligned}}}
Any polygon[edit]
A simple polygon
The second moment of area for any simple polygon on the XY-plane can be computed in general by summing contributions from each segment of the polygon. A polygon is assumed to have n {\displaystyle n} vertices, numbered in counter-clockwise fashion. If polygon vertices are numbered clockwise, returned values will be negative, but absolute values will be correct.
I x = 1 12 ∑ i = 1 n ( y i 2 + y i y i + 1 + y i + 1 2 ) ( x i y i + 1 − x i + 1 y i ) I y = 1 12 ∑ i = 1 n ( x i 2 + x i x i + 1 + x i + 1 2 ) ( x i y i + 1 − x i + 1 y i ) I x y = 1 24 ∑ i = 1 n ( x i y i + 1 + 2 x i y i + 2 x i + 1 y i + 1 + x i + 1 y i ) ( x i y i + 1 − x i + 1 y i ) {\displaystyle {\begin{aligned}I_{x}&={\frac {1}{12}}\sum _{i=1}^{n}\left(y_{i}^{2}+y_{i}y_{i+1}+y_{i+1}^{2}\right)\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\\I_{y}&={\frac {1}{12}}\sum _{i=1}^{n}\left(x_{i}^{2}+x_{i}x_{i+1}+x_{i+1}^{2}\right)\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\\I_{xy}&={\frac {1}{24}}\sum _{i=1}^{n}\left(x_{i}y_{i+1}+2x_{i}y_{i}+2x_{i+1}y_{i+1}+x_{i+1}y_{i}\right)\left(x_{i}y_{i+1}-x_{i+1}y_{i}\right)\end{aligned}}}
where x i , y i {\displaystyle x_{i},y_{i}} are the coordinates of the i {\displaystyle i} -th polygon vertex, for 1 ≤ i ≤ n {\displaystyle 1\leq i\leq n} . Also, x n + 1 , y n + 1 {\displaystyle x_{n+1},y_{n+1}} are assumed to be equal to the coordinates of the first vertex, i.e., x n + 1 = x 1 {\displaystyle x_{n+1}=x_{1}} and y n + 1 = y 1 {\displaystyle y_{n+1}=y_{1}} .[6][7]
See also[edit]
List of second moments of area
List of moments of inertia
Moment of inertia
Parallel axis theorem
Perpendicular axis theorem
Radius of gyration
References[edit]
Jump up ^ Beer, Ferdinand P. (2013). Vector Mechanics for Engineers (10th ed.). New York: McGraw-Hill. p. 471. ISBN 978-0-07-339813-6. The term second moment is more proper than the term moment of inertia, since, logically, the latter should be used only to denote integrals of mass (see Sec. 9.11). In engineering practice, however, moment of inertia is used in connection with areas as well as masses.
Jump up ^ Pilkey, Walter D. (2002). Analysis and Design of Elastic Beams. John Wiley & Sons, Inc. p. 15. ISBN 978-0-471-38152-5.
Jump up ^ Beer, Ferdinand P. (2013). "Chapter 9.8: Product of inertia". Vector Mechanics for Engineers (10th ed.). New York: McGraw-Hill. p. 495. ISBN 978-0-07-339813-6.
Jump up ^ Hibbeler, R. C. (2004). Statics and Mechanics of Materials (Second ed.). Pearson Prentice Hall. ISBN 0-13-028127-1.
Jump up ^ Beer, Ferdinand P. (2013). "Chapter 9.6: Parallel-axis theorem". Vector Mechanics for Engineers (10th ed.). New York: McGraw-Hill. p. 481. ISBN 978-0-07-339813-6.
Jump up ^ Joaquin Obregon (2012). "Mechanical Symmetry".
Jump up ^ Bourke, Paul (1997). "Polygons & Meshes".
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Side-channel attack
From Wikipedia, the free encyclopedia
Jump to: navigation, search
An attempt to decode RSA ... Key bits using power analysis. The left peak represents the CPU power variations during the step of the algorithm without multiplication, the right (broader) peak – step with multiplication, allowing an attacker to read bits 0, 1.
In computer security, a side-channel attack is any attack based on information gained from the implementation of a computer system, rather than weaknesses in the implemented algorithm itself (e.g. cryptanalysis and software bugs). Timing information, power consumption, electromagnetic leaks or even sound can provide an extra source of information, which can be exploited.
Some side-channel attacks require technical knowledge of the internal operation of the system, although others such as differential power analysis are effective as black-box attacks. The rise of Web 2.0 applications and software-as-a-service has also significantly raised the possibility of side-channel attacks on the web, even when transmissions between a web browser and server are encrypted (e.g., through HTTPS or WiFi encryption), according to researchers from Microsoft Research and Indiana University.[1] Many powerful side-channel attacks are based on statistical methods pioneered by Paul Kocher.[2]
Attempts to break a cryptosystem by deceiving or coercing people with legitimate access are not typically considered side-channel attacks: see social engineering and rubber-hose cryptanalysis. For attacks on computer systems themselves (which are often used to perform cryptography and thus contain cryptographic keys or plaintexts), see computer security.
Contents
[hide]
1 General
2 Examples
3 Countermeasures
4 See also
5 References
6 Further reading
7 External links
General[edit]
General classes of side channel attack include:
Cache attack — attacks based on attacker's ability to monitor cache accesses made by the victim in a shared physical system as in virtualized environment or a type of cloud service.
Timing attack — attacks based on measuring how much time various computations (such as, say, comparing an attacker's given password with the victim's unknown one) take to perform.
Power-monitoring attack — attacks that make use of varying power consumption by the hardware during computation.
Electromagnetic attack — attacks based on leaked electromagnetic radiation, which can directly provide plaintexts and other information. Such measurements can be used to infer cryptographic keys using techniques equivalent to those in power analysis or can be used in non-cryptographic attacks, e.g. TEMPEST (aka van Eck phreaking or radiation monitoring) attacks.
Acoustic cryptanalysis — attacks that exploit sound produced during a computation (rather like power analysis).
Differential fault analysis — in which secrets are discovered by introducing faults in a computation.
Data remanence — in which sensitive data are read after supposedly having been deleted. (i.e. Cold boot attack)
Software-initiated fault attacks — Currently a rare class of side-channels, Row hammer is an example in which off-limits memory can be changed by accessing adjacent memory too often (causing state retention loss).
Optical - in which secrets and sensitive data can be read by visual recording using a high resolution camera, or other devices that have such capabilities (see examples below).
In all cases, the underlying principle is that physical effects caused by the operation of a cryptosystem (on the side) can provide useful extra information about secrets in the system, for example, the cryptographic key, partial state information, full or partial plaintexts and so forth. The term cryptophthora (secret degradation) is sometimes used to express the degradation of secret key material resulting from side-channel leakage.
Examples[edit]
A cache side-channel attack works by monitoring security critical operations such as AES T-table entry[3][4][5] or modular exponentiation multiplicand accesses.[6] Attacker then is able to recover the secret key depending on the accesses made (or not made) by the victim, deducing the encryption key. Also, unlike some of the other side-channel attacks, this method does not create a fault in the ongoing cryptographic operation and is invisible to the victim.
In 2017, cache-based vulnerabilities have been discovered in CPUs (dubbed Meltdown and Spectre), which allow an attacker to leak memory contents of other processes and the operating system itself.
A timing attack watches data movement into and out of the CPU or memory on the hardware running the cryptosystem or algorithm. Simply by observing variations in how long it takes to perform cryptographic operations, it might be possible to determine the entire secret key. Such attacks involve statistical analysis of timing measurements and have been demonstrated across networks.[7]
A power-analysis attack can provide even more detailed information by observing the power consumption of a hardware device such as CPU or cryptographic circuit. These attacks are roughly categorized into simple power analysis (SPA) and differential power analysis (DPA).
Fluctuations in current also generate radio waves, enabling attacks that analyze measurements of electromagnetic emanations. These attacks typically involve similar statistical techniques as power-analysis attacks.
Historical analogues to modern side-channel attacks are known. A recently declassified NSA document reveals that as far back as 1943, an engineer with Bell telephone observed decipherable spikes on an oscilloscope associated with the decrypted output of a certain encrypting teletype.[8] According to former MI5 officer Peter Wright, the British Security Service analyzed emissions from French cipher equipment in the 1960s.[9] In the 1980s, Soviet eavesdroppers were suspected of having planted bugs inside IBM Selectric typewriters to monitor the electrical noise generated as the type ball rotated and pitched to strike the paper; the characteristics of those signals could determine which key was pressed.[10]
Power consumption of devices causes heating, which is offset by cooling effects. Temperature changes create thermally induced mechanical stress. This stress can create low level acoustic (i.e. noise) emissions from operating CPUs (about 10 kHz in some cases). Recent research by Shamir et al. has suggested that information about the operation of cryptosystems and algorithms can be obtained in this way as well. This is an acoustic attack; if the surface of the CPU chip, or in some cases the CPU package, can be observed, infrared images can also provide information about the code being executed on the CPU, known as a thermal-imaging attack.[citation needed]
Optical side-channel attack examples include gleaning information from the hard disk activity indicator[11] to reading a small number of photons emitted by transistors as they change state.[12]
Countermeasures[edit]
Because side-channel attacks rely on the relationship between information emitted (leaked) through a side channel and the secret data, countermeasures fall into two main categories: (1) eliminate or reduce the release of such information and (2) eliminate the relationship between the leaked information and the secret data, that is, make the leaked information unrelated, or rather uncorrelated, to the secret data, typically through some form of randomization of the ciphertext that transforms the data in a way that can be undone after the cryptographic operation (e.g., decryption) is completed.
Under the first category, displays with special shielding to lessen electromagnetic emissions, reducing susceptibility to TEMPEST attacks, are now commercially available. Power line conditioning and filtering can help deter power-monitoring attacks, although such measures must be used cautiously, since even very small correlations can remain and compromise security. Physical enclosures can reduce the risk of surreptitious installation of microphones (to counter acoustic attacks) and other micro-monitoring devices (against CPU power-draw or thermal-imaging attacks).
Another countermeasure (still in the first category) is to jam the emitted channel with noise. For instance, a random delay can be added to deter timing attacks, although adversaries can compensate for these delays by averaging multiple measurements (or, more generally, using more measurements in the analysis). As the amount of noise in the side channel increases, the adversary needs to collect more measurements.
Another countermeasure under the first category is to use security analysis software to identify certain classes of side channel attacks that can be found during the design stages of the underlying hardware itself. Timing attacks and cache attacks are both identifiable through certain commercially available security analysis software platforms, which allow for testing to identify the attack vulnerability itself, as well as the effectiveness of the architectural change to circumvent the vulnerability. The most comprehensive method to employ this countermeasure is to create a Secure Development Lifecycle for hardware, which includes utilizing all available security analysis platforms at their respective stages of the hardware development lifecycle.[13]
In the case of timing attacks against targets whose computation times are quantized into discrete clock cycle counts, an effective countermeasure against is to design the software to be isochronous, that is to run in an exactly constant amount of time, independently of secret values. This makes timing attacks impossible.[14] Such countermeasures can be difficult to implement in practice, since even individual instructions can have variable timing on some CPUs.
One partial countermeasure against simple power attacks, but not differential power-analysis attacks, is to design the software so that it is "PC-secure" in the "program counter security model". In a PC-secure program, the execution path does not depend on secret values. In other words, all conditional branches depend only on public information. (This is a more restrictive condition than isochronous code, but a less restrictive condition than branch-free code.) Even though multiply operations draw more power than NOP on practically all CPUs, using a constant execution path prevents such operation-dependent power differences (differences in power from choosing one branch over another) from leaking any secret information.[14] On architectures where the instruction execution time is not data-dependent, a PC-secure program is also immune to timing attacks. [15][16]
Another way in which code can be non-isochronous is that modern CPUs have a memory cache: accessing infrequently used information incurs a large timing penalty, revealing some information about the frequency of use of memory blocks. Cryptographic code designed to resist cache attacks attempts to use memory in only a predictable fashion (such as accessing only the input, outputs and program data, and doing so according to a fixed pattern). For example, data-dependent look-up tables must be avoided because the cache could reveal which part of the look-up table was accessed.
Other partial countermeasures attempt to reduce the amount of information leaked from data-dependent power differences. Some operations use power that is correlated to the number of 1 bits in a secret value. Using a constant-weight code (such as using Fredkin gates or dual-rail encoding) can reduce the leakage of information about the Hamming weight of the secret value, although exploitable correlations are likely to remain unless the balancing is perfect. This "balanced design" can be approximated in software by manipulating both the data and its complement together.[14]
Several "secure CPUs" have been built as asynchronous CPUs; they have no global timing reference. While these CPUs were intended to make timing and power attacks more difficult,[14] subsequent research found that timing variations in asynchronous circuits are harder to remove[citation needed].
A typical example of the second category (decorrelation) is a technique known as blinding. In the case of RSA decryption with secret exponent d {\displaystyle d} and corresponding encryption exponent e {\displaystyle e} and modulus m {\displaystyle m} , the technique applies as follows (for simplicity, the modular reduction by m is omitted in the formulas): before decrypting, that is, before computing the result of y d {\displaystyle y^{d}} for a given ciphertext y {\displaystyle y} , the system picks a random number r {\displaystyle r} and encrypts it with public exponent e {\displaystyle e} to obtain r e {\displaystyle r^{e}} . Then, the decryption is done on y ⋅ r e {\displaystyle y\cdot r^{e}} to obtain ( y ⋅ r e ) d = y d ⋅ r e ⋅ d = y d ⋅ r {\displaystyle {(y\cdot r^{e})}^{d}=y^{d}\cdot r^{e\cdot d}=y^{d}\cdot r} . Since the decrypting system chose r {\displaystyle r} , it can compute its inverse modulo m {\displaystyle m} to cancel out the factor r {\displaystyle r} in the result and obtain y d {\displaystyle y^{d}} , the actual result of the decryption. For attacks that require collecting side-channel information from operations with data controlled by the attacker, blinding is an effective countermeasure, since the actual operation is executed on a randomized version of the data, over which the attacker has no control or even knowledge.
Another more general countermeasure (in that it is effective against all side-channel attacks) is the masking countermeasure. The principle of masking is to avoid manipulating any sensitive value y {\displaystyle y} directly, but rather manipulate a sharing of it: a set of variables (called "shares") y 1 , . . . , y d {\displaystyle y_{1},...,y_{d}} such that y = y 1 ⊕ . . . ⊕ y d {\displaystyle y=y_{1}\oplus ...\oplus y_{d}} (where ⊕ {\displaystyle \oplus } is the XOR operation). An attacker must recover all the values of the shares to get any meaningful information.[17]
See also[edit]
Brute-force attack
Computer surveillance
Covert channel
Side effect
References[edit]
Jump up ^ Shuo Chen; Rui Wang; XiaoFeng Wang & Kehuan Zhang (May 2010). "Side-Channel Leaks in Web Applications: a Reality Today, a Challenge Tomorrow" (PDF). IEEE Symposium on Security & Privacy 2010.
Jump up ^ Kocher, Paul (1996). "Timing attacks on implementations of Diffie-Hellman, RSA, DSS, and other systems". Advances in Cryptology—CRYPTO’96. Lecture Notes in Computer Science. 1109: 104–113. doi:10.1007/3-540-68697-5_9. Retrieved 14 April 2014.
Jump up ^ Ashokkumar C.; Ravi Prakash Giri; Bernard Menezes (2016). "Highly Efficient Algorithms for AES Key Retrieval in Cache Access Attacks".
Jump up ^ Gorka Irazoqui; Mehmet Sinan Inci; Thomas Eisenbarth; Berk Sunar, Wait a minute! A fast, Cross-VM attack on AES (PDF), retrieved 2018-01-07
Jump up ^ Yuval Yarom; Katrina Falkner, Flush+Reload: a High Resolution, Low Noise, L3 Cache Side-Channel Attack (PDF), retrieved 2018-01-07
Jump up ^ Mehmet S. Inci; Berk Gulmezoglu; Gorka Irazoqui; Thomas Eisenbarth; Berk Sunar, Cache Attacks Enable Bulk Key Recovery on the Cloud (PDF), retrieved 2018-01-07
Jump up ^ David Brumley; Dan Boneh (2003). "Remote timing attacks are practical" (PDF).
Jump up ^ "Declassified NSA document reveals the secret history of TEMPEST". Wired.com. April 29, 2008.
Jump up ^ [1]
Jump up ^ Church, George (April 20, 1987). "The Art of High-Tech Snooping". Time. Retrieved January 21, 2010.
Jump up ^ Eduard Kovacs (February 23, 2017), "Hard Drive LED Allows Data Theft From Air-Gapped PCs", Security Week, retrieved 2018-03-18
Jump up ^ J. Ferrigno; M. Hlaváč (September 2008), "When AES blinks: introducing optical side channel", IET Information Security, 2 (3), p. 94–98, doi:10.1049/iet-ifs:20080038
Jump up ^ Tortuga Logic (2018). "Identifying Isolation Issues in Modern Microprocessor Architectures".
^ Jump up to: a b c d "A Network-based Asynchronous Architecture for Cryptographic Devices" by Ljiljana Spadavecchia 2005 in sections "3.2.3 Countermeasures", "3.4.2 Countermeasures", "3.5.6 Countermeasures", "3.5.7 Software countermeasures", "3.5.8 Hardware countermeasures", and "4.10 Side-channel analysis of asynchronous architectures".
Jump up ^ "The Program Counter Security Model: Automatic Detection and Removal of Control-Flow Side Channel Attacks" by David Molnar, Matt Piotrowski, David Schultz, David Wagner (2005).
Jump up ^ "The Program Counter Security Model: Automatic Detection and Removal of Control-Flow Side Channel Attacks" USENIX Work-in-Progress presentation of paper
Jump up ^ "Masking against Side-Channel Attacks: A Formal Security Proof" by Emmanuel Prouff, Matthieu Rivain in Advances in Cryptology – EUROCRYPT 2013.
Further reading[edit]
Books
Ambrose, Jude et al. (2010). Power Analysis Side Channel Attacks: The Processor Design-level Context. VDM Verlag. ISBN 9783836485081. CS1 maint: Uses authors parameter (link)
Articles
[2], Differential Power Analysis, P. Kocher, J. Jaffe, B. Jun, appeared in CRYPTO'99.
[3], Timing Attacks on Implementations of Diffie-Hellman, RSA, DSS, and Other Systems, P. Kocher.
[4], Introduction to Differential Power Analysis and Related attacks, 1998, P Kocher, J Jaffe, B Jun.
Nist.gov, a cautionary Note Regarding Evaluation of AES Candidates on Smart Cards, 1999, S Chari, C Jutla, J R Rao, P Rohatgi
DES and Differential Power Analysis, L Goubin and J Patarin, in Proceedings of CHES'99, Lecture Notes in Computer Science Nr 1717, Springer-Verlag
Grabher, Philipp et al. (2007). "Cryptographic Side-Channels from Low-power Cache Memory". In Galbraith, Steven D. Cryptography and coding: 11th IMA International Conference, Cirencester, UK, December 18-20, 2007 : proceedings, Volume 11. Springer. ISBN 9783540772712. CS1 maint: Uses authors parameter (link)
Kamal, Abdel Alim; Youssef, Amr M. (2012). "Fault analysis of the NTRUSign digital signature scheme". Cryptography and Communications. 4 (2): 131–144. doi:10.1007/s12095-011-0061-3.
Daniel Genkin; Adi Shamir; Eran Tromer (December 18, 2013). "RSA Key Extraction via Low-Bandwidth Acoustic Cryptanalysis". Tel Aviv University. Retrieved October 15, 2014.
External links[edit]
Sima, Mihai; Brisson, André (2015), Whitenoise Encryption Implementation with Increased Robustness against Side-Channel Attacks
Brisson, André (2015),University of Victoria, British Columbia Side Channel Attack Resistance study of Whitenoise
New side channel attack techniques
COSADE Workshop International Workshop on Constructive Side-Channel Analysis and Secure Design
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