{"id":182,"date":"2018-07-12T06:25:14","date_gmt":"2018-07-12T06:25:14","guid":{"rendered":"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/?post_type=chapter&#038;p=182"},"modified":"2018-07-31T08:41:19","modified_gmt":"2018-07-31T08:41:19","slug":"groups-rings-and-fields","status":"publish","type":"chapter","link":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/chapter\/groups-rings-and-fields\/","title":{"rendered":"Groups, Rings and Fields"},"content":{"raw":"<strong>Groups, Rings and Fields<\/strong>\r\n<ul>\r\n \t<li>Groups, Rings, Fields are Fundamental elements of abstract algebra.<\/li>\r\n \t<li>Combine two elements of set, to obtain a third element of set.<\/li>\r\n<\/ul>\r\n<strong>Groups:<\/strong>\r\n<ul>\r\n \t<li>A group G, denoted by [G, \u25cf]<\/li>\r\n \t<li style=\"text-align: justify\">Set of elements with a binary operation denoted by \u25cf that associates to each ordered pair (a,b) of elements in G, an element (a \u25cf b) in G, such that following axioms are obeyed.<\/li>\r\n \t<li>(A1) Closure: If a and b belong to G, then a \u25cf b is also in G.<\/li>\r\n \t<li>(A2) Associative : a \u25cf (b \u25cf c) = (a \u25cf b) \u25cf c for all a,b,c in G.<\/li>\r\n \t<li>(A3) Identity element: element e in G such that a \u25cf e = e \u25cf a = a for all a in G<\/li>\r\n \t<li>(A4) Inverse Element : For each a in G, there is an element a\u2019 in G such that a \u25cf a\u2019= a\u2019 \u25cf a= e<\/li>\r\n<\/ul>\r\n<strong>Finite Group<\/strong>\r\n<ul>\r\n \t<li>If a group has finite number of elements, it is referred as a finite Group.<\/li>\r\n \t<li>Number of elements in the group is called the order of the group.<\/li>\r\n \t<li>A group with infinite number of elements is called infinite group. Abelian group<\/li>\r\n \t<li>A group is abelian if follows the following axiom in addition to (A1) to (A4) (A5) commutative : a \u25cf b = b \u25cf a for all a,b in G.<\/li>\r\n<\/ul>\r\n<img class=\"size-full wp-image-183 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106.png\" alt=\"\" width=\"552\" height=\"210\" \/>\r\n\r\n&nbsp;\r\n\r\nThe set of integers(positive,negative and 0) under addition follow all axioms\r\n\r\n&nbsp;\r\n\r\n(A1) Closure: adding two positive integers is positive integer, two negative integers is negative integers, positive and negative may end up in positive or negative integer.\r\n\r\n5+2 = 7-2+3=1 -3+-3=-6\r\n\r\n(A2)Associative: 3+(4+5)=(3+4)+5 -2+(-5+-6)=(-2+-5)+-6\r\n\r\n(A3)Identity element: 5+0=0+5=5 3+0=0+-3=-3\r\n\r\n(A4) Inverse element : 5+(-5)=0\r\n\r\n(A5) Commutative : 5 + -7 = -7+5\r\n<ul>\r\n \t<li>* For group operation addition, the identity element is 0, inverse element of a is \u2013a. subtraction is defined as a-b = a+(-b).<\/li>\r\n<\/ul>\r\n<img class=\"size-full wp-image-183 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106.png\" alt=\"\" width=\"552\" height=\"210\" \/>\r\n\r\n<strong>Cyclic group<\/strong>\r\n<ul>\r\n \t<li>a4= axaxaxa<\/li>\r\n \t<li>a0=e(as identity element)<\/li>\r\n \t<li>a-n=(a\u2019)n where a\u2019 is the inverse element of a within the group.<\/li>\r\n \t<li style=\"text-align: justify\">A group G is cyclic if every element of G is a power ak (k is an integer) of a fixed element a\u03b5G. The element a is said to generate the group G or to be a generator of G. A cyclic group is always abelian and may be finite or infinite.<\/li>\r\n \t<li style=\"text-align: justify\">The additive group of integers is an infinite cyclic group generated by the element 1.<\/li>\r\n \t<li>Powers are interpreted as addition so that nth power of 1.<\/li>\r\n \t<li>11+21+31+ \u2026.<\/li>\r\n<\/ul>\r\n<strong>Rings:<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">A ring R, denoted by {R, +, x} is a set of elements with two binary operations(addition and multiplication) such that all axioms are followed for all a,b,c in R .<\/li>\r\n \t<li style=\"text-align: justify\">(A1-A5) - R satisfies A1 through A5 for addition so R is an abelian group with respect to addition.<\/li>\r\n \t<li>(M1) Closure under multiplication - ab is in R if a and b belong to R.<\/li>\r\n \t<li>(M2) Associativity of multiplication \u2013 a(bc)=(ab)c for all a,b,c in R.<\/li>\r\n \t<li>(M3) distributive laws \u2013<\/li>\r\n<\/ul>\r\n<img class=\"size-full wp-image-184 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107.png\" alt=\"\" width=\"675\" height=\"547\" \/>\r\n<ol>\r\n \t<li>a(b+c) = ab+ac for all a,b,c in R.<\/li>\r\n \t<li>(a+b)c = ac+bc for all a,b,c in R.<\/li>\r\n<\/ol>\r\n<ul>\r\n \t<li style=\"text-align: justify\">* Ring can do addition, multiplication and subtraction. Subtraction is [a-b=a+(-b)].<\/li>\r\n \t<li style=\"text-align: justify\">A set of integer numbers(positive, negative and 0) is a ring, with respect to addition and multiplication.<\/li>\r\n \t<li style=\"text-align: justify\">The set of all matrices is a ring.<\/li>\r\n \t<li style=\"text-align: justify\">Ring is commutative If following axiom is satisfied.<\/li>\r\n<\/ul>\r\n(M4) commutative(multiplication): ab=ba for all a,b, in R.\r\n\r\n&nbsp;\r\n\r\n<strong>Integral domain:<\/strong>\r\n<ul>\r\n \t<li>Integral domain is a commutative ring if following axioms are satisfied.(M5) Multiplicative identity \u2013 the element 1 in R such that a1=1a=a for all a in<span style=\"text-align: initial;font-size: 1em\"> R.\u00a0<\/span>(M6)No zero divisor \u2013 if a,b in R and ab=0 then either a=0 or b=0.<\/li>\r\n \t<li>Let S be the set of integers positive, negative and 0 under operation of addition and multiplication, S is an integral domain.<\/li>\r\n<\/ul>\r\n<strong>Fields:<\/strong>\r\n<ul>\r\n \t<li>A field F denoted by [F, +,x] is set of elements with two binary operations, called addition and multiplication, such that all a,b,c in F follows following axioms.(A1-M6) : F is an integral domain if F satisfies axioms A1 through A5 and M1 through M6. (M7) Multiplicative inverse :<span style=\"text-align: initial;font-size: 1em\"> For each <\/span>a in<span style=\"text-align: initial;font-size: 1em\"> F, except <\/span>0 ,<span style=\"text-align: initial;font-size: 1em\"> there is an element a-1 in F such that aa-1 in F such that aa-1=(a-1)a=1\u00a0<\/span>In Field, addition, subtraction, multiplication and division results in the same set.<\/li>\r\n \t<li>Division is defined as a\/b=a(b-1)<\/li>\r\n \t<li>All rational, real and complex numbers are field.<\/li>\r\n<\/ul>\r\n<img class=\"size-full wp-image-185 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108.png\" alt=\"\" width=\"1458\" height=\"1067\" \/>\r\n\r\n&nbsp;\r\n\r\n<strong>Finite Fields of the Form GF(p)<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li>Finite fields are important in cryptography<\/li>\r\n \t<li>Order of a finite field that is number of elements in the field must be a power of a prime pn, where n is a positive integer.<\/li>\r\n<\/ul>\r\n<strong>GF(p<\/strong><strong>n<\/strong><strong>)<\/strong>\r\n<ul>\r\n \t<li>Finite field of order pn is GF(pn).<\/li>\r\n \t<li>GF stands for Galois field, in the honor of mathematician who studied this for the first time.<\/li>\r\n \t<li>Two special cases exist.<\/li>\r\n<\/ul>\r\n<ol>\r\n \t<li>n=1, finite field GF(p)<\/li>\r\n \t<li>n&gt;1<\/li>\r\n<\/ol>\r\n<strong>Finite fields of Order p<\/strong>\r\n<ul>\r\n \t<li style=\"text-align: justify\">For a given prime p, finite field of order p, Gf(p) as the set Zp of integers {0,1\u2026p-1} together with the arithmetic operations modulo p.<\/li>\r\n \t<li style=\"text-align: justify\">Zp is a commutative ring, with the arithmetic operations modulo p.<\/li>\r\n \t<li style=\"text-align: justify\">Any integer in Zp has multiplicative inverse if and only if that integer is relatively prime to p.<\/li>\r\n \t<li style=\"text-align: justify\">If p is prime, then all nonzero integers in Zp are relatively prime to p so for all elements multiplicative inverse exist.<\/li>\r\n<\/ul>\r\n<strong>GF(2) \u2013 addition is equivalent to XOR and multiplication is equivalent to logical AND.<\/strong>\r\n\r\n&nbsp;\r\n<table class=\"aligncenter\" style=\"height: 700px\" border=\"1\">\r\n<tbody>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><strong>+<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 69px\"><\/td>\r\n<td style=\"height: 28px;width: 55px\"><strong>1<\/strong><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 69px\"><strong>1<\/strong><\/td>\r\n<td style=\"width: 55px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 69px\"><strong>0<\/strong><\/td>\r\n<td style=\"width: 55px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><strong>x<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 69px\"><\/td>\r\n<td style=\"height: 28px;width: 55px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 69px\"><strong>0<\/strong><\/td>\r\n<td style=\"width: 55px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\r\n<td style=\"height: 28px;width: 69px\"><\/td>\r\n<td style=\"width: 55px\"><\/td>\r\n<\/tr>\r\n<tr style=\"height: 28px\">\r\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\r\n<td style=\"height: 28px;width: 68px\"><\/td>\r\n<td style=\"height: 28px;width: 69px\"><\/td>\r\n<td style=\"width: 55px\"><\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n&nbsp;\r\n\r\n<strong>Finding the multiplicative inverse in GF(p)<\/strong>\r\n\r\n&nbsp;\r\n<ul>\r\n \t<li>if a and b are relatively prime, then b has a multiplicative inverse modulo a.<\/li>\r\n \t<li>For positive integer b&lt;a there exists b-1&lt;a such that bb-1=1 mod a.<\/li>\r\n<\/ul>\r\nif by mod a=1 then y=b-1\r\n\r\n&nbsp;\r\n\r\nAddition modulo 5 GF(5)\r\n\r\n<img class=\"size-full wp-image-186 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109.png\" alt=\"\" width=\"447\" height=\"519\" \/>\r\n\r\n<img class=\"size-full wp-image-187 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110.png\" alt=\"\" width=\"507\" height=\"563\" \/>\r\n\r\n<img class=\"size-full wp-image-188 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111.png\" alt=\"\" width=\"455\" height=\"147\" \/>\r\n<ul>\r\n \t<li style=\"text-align: justify\">For a given prime p, finite field of order p, GF(p), as the set of Zp of integers [0,1,\u2026,p-1} together with the arithmetic operations modulo p.<\/li>\r\n \t<li style=\"text-align: justify\">An integer in Zp has a multiplicative inverse, if and only if that integer is relatively prime to p.<\/li>\r\n \t<li style=\"text-align: justify\">If p is prime, then all nonzero integers in Zp are relatively prime to p and therefore there exists a multiplicative inverse for all nonzero integers in Zp.Multiplicative inverse(w-1)\u00a0\u00a0\u00a0\u00a0 For each w\u03b5Zp ,<span style=\"text-align: initial;font-size: 1em\"> w\u22600, there <\/span>exist<span style=\"text-align: initial;font-size: 1em\"> a z\u03b5Zp such that wxz\u22611(mod p).\u00a0<\/span>GF(p) has following properties<\/li>\r\n \t<li style=\"text-align: justify\">GF(p) consists of p elements<\/li>\r\n \t<li style=\"text-align: justify\">The binary operations + and x are defined over set. The operations of addition, subtraction, multiplication and division can be performed without leaving the set. Each element of the set other than 0 has a multiplicative inverse.<\/li>\r\n<\/ul>\r\n<strong>Suggested Reading:<\/strong>\r\n<ol>\r\n \t<li>Cryptography and Network Security Principles and Practice by William Stallings, sixth Edition, PEARSON.<\/li>\r\n \t<li>Security in Computing by Charles Pfleeger &amp; Shari Lawrence Pfleeger, fourth Edition, PEARSON.<\/li>\r\n \t<li>Network Security by Charlie Kaufman, Radia Perlman, Mike Speciner, second Edition, PHI.<\/li>\r\n \t<li>The Complete Reference \u2013 Network Security by Roberta Bragg, Mark Rhodes-Ousley &amp; Keith Strassberg, Tata McGraw Hill<\/li>\r\n \t<li>Network Security Bible by Eric Cole, Ronald Krutz, James Conley, Wiley<\/li>\r\n \t<li>Hacking 6 Exposed by Stuart McClure, Joel Scambray &amp; George Kurtz , Tata McGraw Hill .<\/li>\r\n \t<li><a href=\"http:\/\/www.snort.org\/\">www.snort.org<\/a><\/li>\r\n \t<li><a href=\"https:\/\/nmap.org\/\">https:\/\/nmap.org<\/a><\/li>\r\n<\/ol>\r\n&nbsp;","rendered":"<p><strong>Groups, Rings and Fields<\/strong><\/p>\n<ul>\n<li>Groups, Rings, Fields are Fundamental elements of abstract algebra.<\/li>\n<li>Combine two elements of set, to obtain a third element of set.<\/li>\n<\/ul>\n<p><strong>Groups:<\/strong><\/p>\n<ul>\n<li>A group G, denoted by [G, \u25cf]<\/li>\n<li style=\"text-align: justify\">Set of elements with a binary operation denoted by \u25cf that associates to each ordered pair (a,b) of elements in G, an element (a \u25cf b) in G, such that following axioms are obeyed.<\/li>\n<li>(A1) Closure: If a and b belong to G, then a \u25cf b is also in G.<\/li>\n<li>(A2) Associative : a \u25cf (b \u25cf c) = (a \u25cf b) \u25cf c for all a,b,c in G.<\/li>\n<li>(A3) Identity element: element e in G such that a \u25cf e = e \u25cf a = a for all a in G<\/li>\n<li>(A4) Inverse Element : For each a in G, there is an element a\u2019 in G such that a \u25cf a\u2019= a\u2019 \u25cf a= e<\/li>\n<\/ul>\n<p><strong>Finite Group<\/strong><\/p>\n<ul>\n<li>If a group has finite number of elements, it is referred as a finite Group.<\/li>\n<li>Number of elements in the group is called the order of the group.<\/li>\n<li>A group with infinite number of elements is called infinite group. Abelian group<\/li>\n<li>A group is abelian if follows the following axiom in addition to (A1) to (A4) (A5) commutative : a \u25cf b = b \u25cf a for all a,b in G.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-183 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106.png\" alt=\"\" width=\"552\" height=\"210\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106.png 552w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-300x114.png 300w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-225x86.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-350x133.png 350w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p>The set of integers(positive,negative and 0) under addition follow all axioms<\/p>\n<p>&nbsp;<\/p>\n<p>(A1) Closure: adding two positive integers is positive integer, two negative integers is negative integers, positive and negative may end up in positive or negative integer.<\/p>\n<p>5+2 = 7-2+3=1 -3+-3=-6<\/p>\n<p>(A2)Associative: 3+(4+5)=(3+4)+5 -2+(-5+-6)=(-2+-5)+-6<\/p>\n<p>(A3)Identity element: 5+0=0+5=5 3+0=0+-3=-3<\/p>\n<p>(A4) Inverse element : 5+(-5)=0<\/p>\n<p>(A5) Commutative : 5 + -7 = -7+5<\/p>\n<ul>\n<li>* For group operation addition, the identity element is 0, inverse element of a is \u2013a. subtraction is defined as a-b = a+(-b).<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-183 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106.png\" alt=\"\" width=\"552\" height=\"210\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106.png 552w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-300x114.png 300w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-65x25.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-225x86.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-106-350x133.png 350w\" sizes=\"auto, (max-width: 552px) 100vw, 552px\" \/><\/p>\n<p><strong>Cyclic group<\/strong><\/p>\n<ul>\n<li>a4= axaxaxa<\/li>\n<li>a0=e(as identity element)<\/li>\n<li>a-n=(a\u2019)n where a\u2019 is the inverse element of a within the group.<\/li>\n<li style=\"text-align: justify\">A group G is cyclic if every element of G is a power ak (k is an integer) of a fixed element a\u03b5G. The element a is said to generate the group G or to be a generator of G. A cyclic group is always abelian and may be finite or infinite.<\/li>\n<li style=\"text-align: justify\">The additive group of integers is an infinite cyclic group generated by the element 1.<\/li>\n<li>Powers are interpreted as addition so that nth power of 1.<\/li>\n<li>11+21+31+ \u2026.<\/li>\n<\/ul>\n<p><strong>Rings:<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">A ring R, denoted by {R, +, x} is a set of elements with two binary operations(addition and multiplication) such that all axioms are followed for all a,b,c in R .<\/li>\n<li style=\"text-align: justify\">(A1-A5) &#8211; R satisfies A1 through A5 for addition so R is an abelian group with respect to addition.<\/li>\n<li>(M1) Closure under multiplication &#8211; ab is in R if a and b belong to R.<\/li>\n<li>(M2) Associativity of multiplication \u2013 a(bc)=(ab)c for all a,b,c in R.<\/li>\n<li>(M3) distributive laws \u2013<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-184 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107.png\" alt=\"\" width=\"675\" height=\"547\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107.png 675w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107-300x243.png 300w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107-65x53.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107-225x182.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-107-350x284.png 350w\" sizes=\"auto, (max-width: 675px) 100vw, 675px\" \/><\/p>\n<ol>\n<li>a(b+c) = ab+ac for all a,b,c in R.<\/li>\n<li>(a+b)c = ac+bc for all a,b,c in R.<\/li>\n<\/ol>\n<ul>\n<li style=\"text-align: justify\">* Ring can do addition, multiplication and subtraction. Subtraction is [a-b=a+(-b)].<\/li>\n<li style=\"text-align: justify\">A set of integer numbers(positive, negative and 0) is a ring, with respect to addition and multiplication.<\/li>\n<li style=\"text-align: justify\">The set of all matrices is a ring.<\/li>\n<li style=\"text-align: justify\">Ring is commutative If following axiom is satisfied.<\/li>\n<\/ul>\n<p>(M4) commutative(multiplication): ab=ba for all a,b, in R.<\/p>\n<p>&nbsp;<\/p>\n<p><strong>Integral domain:<\/strong><\/p>\n<ul>\n<li>Integral domain is a commutative ring if following axioms are satisfied.(M5) Multiplicative identity \u2013 the element 1 in R such that a1=1a=a for all a in<span style=\"text-align: initial;font-size: 1em\"> R.\u00a0<\/span>(M6)No zero divisor \u2013 if a,b in R and ab=0 then either a=0 or b=0.<\/li>\n<li>Let S be the set of integers positive, negative and 0 under operation of addition and multiplication, S is an integral domain.<\/li>\n<\/ul>\n<p><strong>Fields:<\/strong><\/p>\n<ul>\n<li>A field F denoted by [F, +,x] is set of elements with two binary operations, called addition and multiplication, such that all a,b,c in F follows following axioms.(A1-M6) : F is an integral domain if F satisfies axioms A1 through A5 and M1 through M6. (M7) Multiplicative inverse :<span style=\"text-align: initial;font-size: 1em\"> For each <\/span>a in<span style=\"text-align: initial;font-size: 1em\"> F, except <\/span>0 ,<span style=\"text-align: initial;font-size: 1em\"> there is an element a-1 in F such that aa-1 in F such that aa-1=(a-1)a=1\u00a0<\/span>In Field, addition, subtraction, multiplication and division results in the same set.<\/li>\n<li>Division is defined as a\/b=a(b-1)<\/li>\n<li>All rational, real and complex numbers are field.<\/li>\n<\/ul>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-185 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108.png\" alt=\"\" width=\"1458\" height=\"1067\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108.png 1458w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108-300x220.png 300w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108-768x562.png 768w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108-1024x749.png 1024w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108-65x48.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108-225x165.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-108-350x256.png 350w\" sizes=\"auto, (max-width: 1458px) 100vw, 1458px\" \/><\/p>\n<p>&nbsp;<\/p>\n<p><strong>Finite Fields of the Form GF(p)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>Finite fields are important in cryptography<\/li>\n<li>Order of a finite field that is number of elements in the field must be a power of a prime pn, where n is a positive integer.<\/li>\n<\/ul>\n<p><strong>GF(p<\/strong><strong>n<\/strong><strong>)<\/strong><\/p>\n<ul>\n<li>Finite field of order pn is GF(pn).<\/li>\n<li>GF stands for Galois field, in the honor of mathematician who studied this for the first time.<\/li>\n<li>Two special cases exist.<\/li>\n<\/ul>\n<ol>\n<li>n=1, finite field GF(p)<\/li>\n<li>n&gt;1<\/li>\n<\/ol>\n<p><strong>Finite fields of Order p<\/strong><\/p>\n<ul>\n<li style=\"text-align: justify\">For a given prime p, finite field of order p, Gf(p) as the set Zp of integers {0,1\u2026p-1} together with the arithmetic operations modulo p.<\/li>\n<li style=\"text-align: justify\">Zp is a commutative ring, with the arithmetic operations modulo p.<\/li>\n<li style=\"text-align: justify\">Any integer in Zp has multiplicative inverse if and only if that integer is relatively prime to p.<\/li>\n<li style=\"text-align: justify\">If p is prime, then all nonzero integers in Zp are relatively prime to p so for all elements multiplicative inverse exist.<\/li>\n<\/ul>\n<p><strong>GF(2) \u2013 addition is equivalent to XOR and multiplication is equivalent to logical AND.<\/strong><\/p>\n<p>&nbsp;<\/p>\n<table class=\"aligncenter\" style=\"height: 700px\">\n<tbody>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><strong>+<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 69px\"><\/td>\n<td style=\"height: 28px;width: 55px\"><strong>1<\/strong><\/td>\n<\/tr>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 69px\"><strong>1<\/strong><\/td>\n<td style=\"width: 55px\"><\/td>\n<\/tr>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 69px\"><strong>0<\/strong><\/td>\n<td style=\"width: 55px\"><\/td>\n<\/tr>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><strong>x<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 69px\"><\/td>\n<td style=\"height: 28px;width: 55px\"><\/td>\n<\/tr>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 69px\"><strong>0<\/strong><\/td>\n<td style=\"width: 55px\"><\/td>\n<\/tr>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\n<td style=\"height: 28px;width: 69px\"><\/td>\n<td style=\"width: 55px\"><\/td>\n<\/tr>\n<tr style=\"height: 28px\">\n<td style=\"height: 28px;width: 68px\"><strong>1<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><strong>0<\/strong><\/td>\n<td style=\"height: 28px;width: 68px\"><\/td>\n<td style=\"height: 28px;width: 69px\"><\/td>\n<td style=\"width: 55px\"><\/td>\n<\/tr>\n<\/tbody>\n<\/table>\n<p>&nbsp;<\/p>\n<p><strong>Finding the multiplicative inverse in GF(p)<\/strong><\/p>\n<p>&nbsp;<\/p>\n<ul>\n<li>if a and b are relatively prime, then b has a multiplicative inverse modulo a.<\/li>\n<li>For positive integer b&lt;a there exists b-1&lt;a such that bb-1=1 mod a.<\/li>\n<\/ul>\n<p>if by mod a=1 then y=b-1<\/p>\n<p>&nbsp;<\/p>\n<p>Addition modulo 5 GF(5)<\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-186 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109.png\" alt=\"\" width=\"447\" height=\"519\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109.png 447w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109-258x300.png 258w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109-65x75.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109-225x261.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-109-350x406.png 350w\" sizes=\"auto, (max-width: 447px) 100vw, 447px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-187 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110.png\" alt=\"\" width=\"507\" height=\"563\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110.png 507w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110-270x300.png 270w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110-65x72.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110-225x250.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-110-350x389.png 350w\" sizes=\"auto, (max-width: 507px) 100vw, 507px\" \/><\/p>\n<p><img loading=\"lazy\" decoding=\"async\" class=\"size-full wp-image-188 aligncenter\" src=\"http:\/\/itp4.epgpbooks.inflibnet.ac.in\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111.png\" alt=\"\" width=\"455\" height=\"147\" srcset=\"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111.png 455w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111-300x97.png 300w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111-65x21.png 65w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111-225x73.png 225w, https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-content\/uploads\/sites\/25\/2018\/07\/Untitled-111-350x113.png 350w\" sizes=\"auto, (max-width: 455px) 100vw, 455px\" \/><\/p>\n<ul>\n<li style=\"text-align: justify\">For a given prime p, finite field of order p, GF(p), as the set of Zp of integers [0,1,\u2026,p-1} together with the arithmetic operations modulo p.<\/li>\n<li style=\"text-align: justify\">An integer in Zp has a multiplicative inverse, if and only if that integer is relatively prime to p.<\/li>\n<li style=\"text-align: justify\">If p is prime, then all nonzero integers in Zp are relatively prime to p and therefore there exists a multiplicative inverse for all nonzero integers in Zp.Multiplicative inverse(w-1)\u00a0\u00a0\u00a0\u00a0 For each w\u03b5Zp ,<span style=\"text-align: initial;font-size: 1em\"> w\u22600, there <\/span>exist<span style=\"text-align: initial;font-size: 1em\"> a z\u03b5Zp such that wxz\u22611(mod p).\u00a0<\/span>GF(p) has following properties<\/li>\n<li style=\"text-align: justify\">GF(p) consists of p elements<\/li>\n<li style=\"text-align: justify\">The binary operations + and x are defined over set. The operations of addition, subtraction, multiplication and division can be performed without leaving the set. Each element of the set other than 0 has a multiplicative inverse.<\/li>\n<\/ul>\n<p><strong>Suggested Reading:<\/strong><\/p>\n<ol>\n<li>Cryptography and Network Security Principles and Practice by William Stallings, sixth Edition, PEARSON.<\/li>\n<li>Security in Computing by Charles Pfleeger &amp; Shari Lawrence Pfleeger, fourth Edition, PEARSON.<\/li>\n<li>Network Security by Charlie Kaufman, Radia Perlman, Mike Speciner, second Edition, PHI.<\/li>\n<li>The Complete Reference \u2013 Network Security by Roberta Bragg, Mark Rhodes-Ousley &amp; Keith Strassberg, Tata McGraw Hill<\/li>\n<li>Network Security Bible by Eric Cole, Ronald Krutz, James Conley, Wiley<\/li>\n<li>Hacking 6 Exposed by Stuart McClure, Joel Scambray &amp; George Kurtz , Tata McGraw Hill .<\/li>\n<li><a href=\"http:\/\/www.snort.org\/\">www.snort.org<\/a><\/li>\n<li><a href=\"https:\/\/nmap.org\/\">https:\/\/nmap.org<\/a><\/li>\n<\/ol>\n<p>&nbsp;<\/p>\n","protected":false},"author":4,"menu_order":21,"template":"","meta":{"pb_show_title":"on","pb_short_title":"","pb_subtitle":"","pb_authors":["miss-hiteishi-diwanji"],"pb_section_license":""},"chapter-type":[],"contributor":[58],"license":[],"class_list":["post-182","chapter","type-chapter","status-publish","hentry","contributor-miss-hiteishi-diwanji"],"part":3,"_links":{"self":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/chapters\/182","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/chapters"}],"about":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/wp\/v2\/types\/chapter"}],"author":[{"embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/wp\/v2\/users\/4"}],"version-history":[{"count":7,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/chapters\/182\/revisions"}],"predecessor-version":[{"id":398,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/chapters\/182\/revisions\/398"}],"part":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/parts\/3"}],"metadata":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/chapters\/182\/metadata\/"}],"wp:attachment":[{"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/wp\/v2\/media?parent=182"}],"wp:term":[{"taxonomy":"chapter-type","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/pressbooks\/v2\/chapter-type?post=182"},{"taxonomy":"contributor","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/wp\/v2\/contributor?post=182"},{"taxonomy":"license","embeddable":true,"href":"https:\/\/ebooks.inflibnet.ac.in\/itp4\/wp-json\/wp\/v2\/license?post=182"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}