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                           <rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers An elliptic curve is defined by an equation in two variables x &amp; y, y2" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3 + ax + b Where x, y, a, b are all real numbers 4a3+27b20 The set of points" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E(a, b) forms an abelian group with respect to “addition” operation defined as follows: P+Q is reflection of the intersection" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="R O (Infinity) acts as additive identity To double a point P, find intersection of tangent and curve Closure: P+Q" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ε E Associativity: P+(Q+R) = (P+Q)+R Identity: P+O=P Inverse: -P ε E Commutative: P+Q = Q+P R" xmpDM:startTime="0" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont) Slope of line PQ is: D = (yQ-yP)/(xQ-xP) The sum R=P+Q is: xR=D2-xP-xQ yR=-yp+D(xP-xR)" xmpDM:startTime="621033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="P+P=2P=R" xmpDM:startTime="621033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables &amp; coefficients are defined over GF Prime curves: Ep(a,b) defined" xmpDM:startTime="810967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="over Zp Use integers modulo a prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="810967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = 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prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="820133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = Exponentiation Easy to" xmpDM:startTime="820133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="compute Q=P+P+…+P=kP, where Q, P ε E Hard to find k given Q, P (Similar to discrete log)" xmpDM:startTime="820133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont) Slope of line PQ is: D = (yQ-yP)/(xQ-xP) The sum R=P+Q is: xR=D2-xP-xQ yR=-yp+D(xP-xR)" xmpDM:startTime="820633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="P+P=2P=R" xmpDM:startTime="820633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers An elliptic curve is defined by an equation in two variables x &amp; y, y2" xmpDM:startTime="821333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3 + ax + b Where x, y, a, b are all real numbers 4a3+27b20 The set of points" xmpDM:startTime="821333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E(a, b) forms an abelian group with respect to “addition” operation defined as follows: P+Q is reflection of the intersection" xmpDM:startTime="821333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="R O (Infinity) acts as additive identity To double a point P, find intersection of tangent and curve Closure: P+Q" xmpDM:startTime="821333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ε E Associativity: P+(Q+R) = (P+Q)+R Identity: P+O=P Inverse: -P ε E Commutative: P+Q = Q+P R" xmpDM:startTime="821333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont) Slope of line PQ is: D = (yQ-yP)/(xQ-xP) The sum R=P+Q is: xR=D2-xP-xQ yR=-yp+D(xP-xR)" xmpDM:startTime="822133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="P+P=2P=R" xmpDM:startTime="822133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables &amp; coefficients are defined over GF Prime curves: Ep(a,b) defined" xmpDM:startTime="825333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="over Zp Use integers modulo a prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="825333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = Exponentiation Easy to" xmpDM:startTime="825333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="compute Q=P+P+…+P=kP, where Q, P ε E Hard to find k given Q, P (Similar to discrete log)" xmpDM:startTime="825333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont) Slope of line PQ is: D = (yQ-yP)/(xQ-xP) The sum R=P+Q is: xR=D2-xP-xQ yR=-yp+D(xP-xR)" xmpDM:startTime="957367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="P+P=2P=R" xmpDM:startTime="957367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables &amp; coefficients are defined over GF Prime curves: Ep(a,b) defined" xmpDM:startTime="962167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="over Zp Use integers modulo a prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="962167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = Exponentiation Easy to" xmpDM:startTime="962167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="compute Q=P+P+…+P=kP, where Q, P ε E Hard to find k given Q, P (Similar to discrete log)" xmpDM:startTime="962167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example Ep(a,b): y2=x3+ax+b mod p E23(1,1): y2=x3+x+1 mod 23 Consider only +ve x and y R=P+Q xR=(l2-xP-xQ)" xmpDM:startTime="1061433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod p yR=(l(xP-xR)-yP) mod p Where Example: (3,10)+(3,10) xR=(62-3-3) mod 23 =7 yR=(6(3-7)-10) mod 23 =12" xmpDM:startTime="1061433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables &amp; coefficients are defined over GF Prime curves: Ep(a,b) defined" xmpDM:startTime="1134000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="over Zp Use integers modulo a prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="1134000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = Exponentiation Easy to" xmpDM:startTime="1134000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="compute Q=P+P+…+P=kP, where Q, P ε E Hard to find k given Q, P (Similar to discrete log)" xmpDM:startTime="1134000" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont) Slope of line PQ is: D = (yQ-yP)/(xQ-xP) The sum R=P+Q is: xR=D2-xP-xQ yR=-yp+D(xP-xR)" xmpDM:startTime="1134400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="P+P=2P=R" xmpDM:startTime="1134400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers An elliptic curve is defined by an equation in two variables x &amp; y, y2" xmpDM:startTime="1136200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3 + ax + b Where x, y, a, b are all real numbers 4a3+27b20 The set of points" xmpDM:startTime="1136200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E(a, b) forms an abelian group with respect to “addition” operation defined as follows: P+Q is reflection of the intersection" xmpDM:startTime="1136200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="R O (Infinity) acts as additive identity To double a point P, find intersection of tangent and curve Closure: P+Q" xmpDM:startTime="1136200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ε E Associativity: P+(Q+R) = (P+Q)+R Identity: P+O=P Inverse: -P ε E Commutative: P+Q = Q+P R" xmpDM:startTime="1136200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve Cryptography Majority of public-key crypto (RSA, D-H) use either integer or polynomial arithmetic with very large numbers/polynomials Imposes" xmpDM:startTime="1137200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="a significant load in storing and processing keys and messages An alternative is to use elliptic curves Offers same security" xmpDM:startTime="1137200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="with smaller bit sizes Newer, but not as well analyzed" xmpDM:startTime="1137200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers An elliptic curve is defined by an equation in two variables x &amp; y, y2" xmpDM:startTime="1138033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="= x3 + ax + b Where x, y, a, b are all real numbers 4a3+27b20 The set of points" xmpDM:startTime="1138033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="E(a, b) forms an abelian group with respect to “addition” operation defined as follows: P+Q is reflection of the intersection" xmpDM:startTime="1138033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="R O (Infinity) acts as additive identity To double a point P, find intersection of tangent and curve Closure: P+Q" xmpDM:startTime="1138033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ε E Associativity: P+(Q+R) = (P+Q)+R Identity: P+O=P Inverse: -P ε E Commutative: P+Q = Q+P R" xmpDM:startTime="1138033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont) Slope of line PQ is: D = (yQ-yP)/(xQ-xP) The sum R=P+Q is: xR=D2-xP-xQ yR=-yp+D(xP-xR)" xmpDM:startTime="1140167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="P+P=2P=R" xmpDM:startTime="1140167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables &amp; coefficients are defined over GF Prime curves: Ep(a,b) defined" xmpDM:startTime="1141067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="over Zp Use integers modulo a prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="1141067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = Exponentiation Easy to" xmpDM:startTime="1141067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="compute Q=P+P+…+P=kP, where Q, P ε E Hard to find k given Q, P (Similar to discrete log)" xmpDM:startTime="1141067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example Ep(a,b): y2=x3+ax+b mod p E23(1,1): y2=x3+x+1 mod 23 Consider only +ve x and y R=P+Q xR=(l2-xP-xQ)" xmpDM:startTime="1141700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod p yR=(l(xP-xR)-yP) mod p Where Example: (3,10)+(3,10) xR=(62-3-3) mod 23 =7 yR=(6(3-7)-10) mod 23 =12" xmpDM:startTime="1141700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="1323267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="1323267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="1323267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="1323267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="1323267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="1323267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1324533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1325167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="1325633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="1325633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="1325633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1326367" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="1326700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="1327300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="1327300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="1327300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="1327300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="1327300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="1327300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example Ep(a,b): y2=x3+ax+b mod p E23(1,1): y2=x3+x+1 mod 23 Consider only +ve x and y R=P+Q xR=(l2-xP-xQ)" xmpDM:startTime="1327467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod p yR=(l(xP-xR)-yP) mod p Where Example: (3,10)+(3,10) xR=(62-3-3) mod 23 =7 yR=(6(3-7)-10) mod 23 =12" xmpDM:startTime="1327467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="1441433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="1441433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="1441433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="1441433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="1441433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="1441433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example Ep(a,b): y2=x3+ax+b mod p E23(1,1): y2=x3+x+1 mod 23 Consider only +ve x and y R=P+Q xR=(l2-xP-xQ)" xmpDM:startTime="1915800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod p yR=(l(xP-xR)-yP) mod p Where Example: (3,10)+(3,10) xR=(62-3-3) mod 23 =7 yR=(6(3-7)-10) mod 23 =12" xmpDM:startTime="1915800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves Elliptic curve cryptography uses curves whose variables &amp; coefficients are defined over GF Prime curves: Ep(a,b) defined" xmpDM:startTime="1932500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="over Zp Use integers modulo a prime Easily implemented in software Binary curves: E2n(a,b) defined over GF(2n) Use polynomials with" xmpDM:startTime="1932500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="binary coefficients Easily implemented in hardware Cryptography: Addition in elliptic = multiplication in Integer Repeated addition = Exponentiation Easy to" xmpDM:startTime="1932500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="compute Q=P+P+…+P=kP, where Q, P ε E Hard to find k given Q, P (Similar to discrete log)" xmpDM:startTime="1932500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example Ep(a,b): y2=x3+ax+b mod p E23(1,1): y2=x3+x+1 mod 23 Consider only +ve x and y R=P+Q xR=(l2-xP-xQ)" xmpDM:startTime="1933933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="mod p yR=(l(xP-xR)-yP) mod p Where Example: (3,10)+(3,10) xR=(62-3-3) mod 23 =7 yR=(6(3-7)-10) mod 23 =12" xmpDM:startTime="1933933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="1934233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="1934233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="1934233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="1934233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="1934233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="1934233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2020067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2265933" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="2267300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="2267300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="2267300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2268233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="2298300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="2298300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="2298300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Security Relies on elliptic curve logarithm problem Can use much smaller key sizes than with RSA etc For equivalent" xmpDM:startTime="2298633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="key lengths computations are roughly equivalent Hence for similar security ECC offers significant computational advantages" xmpDM:startTime="2298633" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="2500233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="2500233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="2500233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2500833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2502533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2558333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2639533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="2642733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="2642733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="2642733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="2642733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="2642733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="2642733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2644067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2648467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2821433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="2822333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="2822333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="2822333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="2822333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="2822333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="2822333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2836300" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2836733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="2837200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="2837200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="2837200" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="2838800" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="2886533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="2886533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="2886533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3468033" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3468567" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3472267" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="3472867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="3472867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="3472867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3550467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3550767" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example E211(0, -4), y2=x3+ax+b=x3-4 G=(2,2), Calculate 121G 121=1111001  64G+32G+16G+8G+G 1 2 3 Numerator of ¸ ¸ 1 2" xmpDM:startTime="3551533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="3 xR yR Point G 2 2 2G=G+G 12 4 53 3 5 200 4G=2G+2G 75 189 163 198 159" xmpDM:startTime="3551533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="114 8G=4G+4G 94 17 149 80 174 163 16G=8G+8G 98 115 200 188 181 209 32G=16G+16G 168 207 158 169" xmpDM:startTime="3551533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="136 11 64G=32G+32G 206 22 48 182 147 97 9G=G+8G 161 172 119 169 111 145 25G=9G+16G 64 70 208" xmpDM:startTime="3551533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="19 69 20 57G=25G+32G 202 67 63 66 142 15 121G=57G+64G 82 5 169 143 115 48 Denominator of ¸" xmpDM:startTime="3551533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="(Denominator of ¸)-1" xmpDM:startTime="3551533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3592867" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="3593133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="3593500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="3593500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="3593500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Security Relies on elliptic curve logarithm problem Can use much smaller key sizes than with RSA etc For equivalent" xmpDM:startTime="3594433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="key lengths computations are roughly equivalent Hence for similar security ECC offers significant computational advantages" xmpDM:startTime="3594433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="PRNG based on Asymmetric Ciphers Asymmetric encryption algorithms produce apparently random output Hence can be used to build a pseudorandom" xmpDM:startTime="3595167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="number generator (PRNG) Much slower than symmetric algorithms Hence only use to generate a short pseudorandom bit sequence (e.g.," xmpDM:startTime="3595167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="key)" xmpDM:startTime="3595167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Security Relies on elliptic curve logarithm problem Can use much smaller key sizes than with RSA etc For equivalent" xmpDM:startTime="3596133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="key lengths computations are roughly equivalent Hence for similar security ECC offers significant computational advantages" xmpDM:startTime="3596133" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example E257(0, -4), Pm=(112,26), nB=101 G=(2, 2) PB=nBG = 101(2, 2) =(197, 167) k=41, C1=kG=41(2,2)=(136, 128) C2=Pm+kPB= (112," xmpDM:startTime="3596467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="26) + 41(197, 167) =(112, 26)+(68, 84) = (246, 174) Cm={C1, C2} = {(136,128),(246, 174)} Pm=C2-nBC1 = (246, 174)-101(136, 128)" xmpDM:startTime="3596467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="=(246, 174)-(68, 84) = (112, 26)" xmpDM:startTime="3596467" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Security Relies on elliptic curve logarithm problem Can use much smaller key sizes than with RSA etc For equivalent" xmpDM:startTime="3600433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="key lengths computations are roughly equivalent Hence for similar security ECC offers significant computational advantages" xmpDM:startTime="3600433" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="PRNG based on Asymmetric Ciphers Asymmetric encryption algorithms produce apparently random output Hence can be used to build a pseudorandom" xmpDM:startTime="3607533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="number generator (PRNG) Much slower than symmetric algorithms Hence only use to generate a short pseudorandom bit sequence (e.g.," xmpDM:startTime="3607533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="key)" xmpDM:startTime="3607533" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="PRNG based on RSA Micali-Schnorr PRNG using RSA in ANSI X9.82" xmpDM:startTime="3684500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and ISO 18031" xmpDM:startTime="3684500" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="PRNG based on ECC Dual elliptic curve PRNG NIST SP 800-9, ANSI X9.82" xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and ISO 18031 Some controversy on security /inefficiency Notation: x(P) = x coordinate of P." xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="lsbi(x)=i least sig bits of x Algorithm for i = 1 to k do set si = x(si-1 P )" xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="set ri = lsb240 (x(si Q)) end for return r1 , ." xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="." xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="." xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", rk Only use if just have ECC P Q × × si-1 si=x(si-1P) ri=lsb240[x(siQ)] × P Q ×" xmpDM:startTime="3766167" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="PRNG based on RSA Micali-Schnorr PRNG using RSA in ANSI X9.82" xmpDM:startTime="3868967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and ISO 18031" xmpDM:startTime="3868967" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="PRNG based on ECC Dual elliptic curve PRNG NIST SP 800-9, ANSI X9.82" xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and ISO 18031 Some controversy on security /inefficiency Notation: x(P) = x coordinate of P." xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="lsbi(x)=i least sig bits of x Algorithm for i = 1 to k do set si = x(si-1 P )" xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="set ri = lsb240 (x(si Q)) end for return r1 , ." xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="." xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="." xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name=", rk Only use if just have ECC P Q × × si-1 si=x(si-1P) ri=lsb240[x(siQ)] × P Q ×" xmpDM:startTime="3875233" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Summary Diffie-Hellman key exchange allows creating a secret in public based on exponentiation ElGamal cryptography uses D-H Elliptic Curve cryptography" xmpDM:startTime="4041700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="is based on defining addition of points on an elliptic curve in GF(p) or GF(2n) Public key cryptography (both RSA" xmpDM:startTime="4041700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="and ECC) can also be used to generate cryptographically secure pseudorandom numbers." xmpDM:startTime="4041700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4041700" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Homework 10 1." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Consider an Elgamal scheme with a common prime q=71 and a primitive root ®=7." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="A." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If B has public key YB=3 and A choose the random integer k=2, what is the ciphertext of M=30? B." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="If A now chooses a different value of k so that the encoding of M=30 is C=(59,C2)." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="What is the integer C2? 2." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="For an elliptic curve cryptography using E11(1,6) and G=(2,7)." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="B’s private key nB=7." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="A." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Find B’s Public key PB B." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="A wishes to encrypt the message Pm=(10, 9) and chooses the random value k=3." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Determine the ciphertext Cm C." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Show the calculation by which B recovers Pm from Cm." xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4134733" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Lab 10: Kali Linux Prepare a bootable USB drive with Kali Linux See instructions at: http://docs.kali.org/downloading/kali-linux-live-usb-install" xmpDM:startTime="4175833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="You will need a 4GB or larger USB 3 flash drive Also, you will need to change the boot sequence" xmpDM:startTime="4175833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="in your computer to allow booting from the USB drive No other changes are required to your disk or computer." xmpDM:startTime="4175833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Explore Kali and submit the list of penetration tools available in Kali Note: Kali is a goddess that destroys evil" xmpDM:startTime="4175833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Ref: https://en.wikipedia.org/wiki/Kali_Linux" xmpDM:startTime="4175833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4175833" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Acronyms ANSI American National Standards Institute DEC Dual Elliptic Curve DSS Digital Signature Standard ECC Elliptic curve cryptography GF Galvois" xmpDM:startTime="4382067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Field IEEE Institute of Electrical and Electronic Engineers ISO International Standards Organization MIME Multipurpose Internet Multimedia Email NIST National Institute" xmpDM:startTime="4382067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="of Science and Technology OFB Output feedback mode PRF Pseudo-random function PRNG Pseudo-Random Number Generator RSA Rivest, Shamir, and Adleman" xmpDM:startTime="4382067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="SP Standard Practice VPN Virtual Private Network" xmpDM:startTime="4382067" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Scan This to Download These Slides Raj Jain http://rajjain.com" xmpDM:startTime="4383333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4383333" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Related Modules Audio/Video Recordings and Podcasts of Professor Raj Jain's Lectures, https://www.youtube.com/channel/UCN4-5wzNP9-ruOzQMs-8NUw" xmpDM:startTime="4384400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE473S: Introduction to Computer Networks (Fall 2016), http://www.cse.wustl.edu/~jain/cse473-16/index.html" xmpDM:startTime="4384400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Wireless and Mobile Networking (Spring 2016), http://www.cse.wustl.edu/~jain/cse574-16/index.html" xmpDM:startTime="4384400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE571S: Network Security (Fall 2014), http://www.cse.wustl.edu/~jain/cse571-14/index.html" xmpDM:startTime="4384400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="CSE571S: Network Security (Spring 2017), http://www.cse.wustl.edu/~jain/cse571-17/index.html" xmpDM:startTime="4384400" xmpDM:duration="1000"/></rdf:li><rdf:li><rdf:Description xmpDM:name="" xmpDM:startTime="4384400" xmpDM:duration="1000"/></rdf:li></rdf:Seq>
                     </xmpDM:markers>
                  </rdf:Description>
               </rdf:li>
               <rdf:li>
                  <rdf:Description xmpDM:trackType="TableOfContents" xmpDM:frameRate="f1000" xmpDM:trackName="Table of Contents">
                     <xmpDM:markers>
                        <rdf:Seq>
                           <rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers" xmpDM:startTime="0" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="0" tscDM:imageoffset="0" tscDM:imagerect="0, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont)" xmpDM:startTime="621033" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="1" tscDM:imageoffset="0" tscDM:imagerect="75, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="810967" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="2" tscDM:imageoffset="0" tscDM:imagerect="150, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example" xmpDM:startTime="816667" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="3" tscDM:imageoffset="0" tscDM:imagerect="225, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="820133" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="4" tscDM:imageoffset="0" tscDM:imagerect="300, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont)" xmpDM:startTime="820633" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="5" tscDM:imageoffset="0" tscDM:imagerect="375, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers" xmpDM:startTime="821333" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="6" tscDM:imageoffset="0" tscDM:imagerect="450, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont)" xmpDM:startTime="822133" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="7" tscDM:imageoffset="0" tscDM:imagerect="525, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="825333" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="8" tscDM:imageoffset="0" tscDM:imagerect="600, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont)" xmpDM:startTime="957367" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="9" tscDM:imageoffset="0" tscDM:imagerect="675, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="962167" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="10" tscDM:imageoffset="0" tscDM:imagerect="750, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example" xmpDM:startTime="1061433" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="11" tscDM:imageoffset="0" tscDM:imagerect="825, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="1134000" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="12" tscDM:imageoffset="0" tscDM:imagerect="900, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont)" xmpDM:startTime="1134400" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="13" tscDM:imageoffset="0" tscDM:imagerect="975, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers" xmpDM:startTime="1136200" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="14" tscDM:imageoffset="0" tscDM:imagerect="1050, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve Cryptography" xmpDM:startTime="1137200" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="15" tscDM:imageoffset="0" tscDM:imagerect="1125, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curves over Real Numbers" xmpDM:startTime="1138033" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="16" tscDM:imageoffset="0" tscDM:imagerect="1200, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Elliptic Curve over Real Numbers (Cont)" xmpDM:startTime="1140167" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="17" tscDM:imageoffset="0" tscDM:imagerect="1275, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="1141067" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="18" tscDM:imageoffset="0" tscDM:imagerect="1350, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example" xmpDM:startTime="1141700" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="19" tscDM:imageoffset="0" tscDM:imagerect="1425, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example" xmpDM:startTime="1323267" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="20" tscDM:imageoffset="0" tscDM:imagerect="1500, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="1324533" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="21" tscDM:imageoffset="0" tscDM:imagerect="1575, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="1325167" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="22" tscDM:imageoffset="0" tscDM:imagerect="1650, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example" xmpDM:startTime="1325633" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="23" tscDM:imageoffset="0" tscDM:imagerect="1725, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="1326367" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="24" tscDM:imageoffset="0" tscDM:imagerect="1800, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="1326700" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="25" tscDM:imageoffset="0" tscDM:imagerect="1875, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example" xmpDM:startTime="1327300" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="26" tscDM:imageoffset="0" tscDM:imagerect="1950, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example" xmpDM:startTime="1327467" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="27" tscDM:imageoffset="0" tscDM:imagerect="2025, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example" xmpDM:startTime="1441433" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="28" tscDM:imageoffset="0" tscDM:imagerect="2100, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example" xmpDM:startTime="1915800" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="29" tscDM:imageoffset="0" tscDM:imagerect="2175, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curves" xmpDM:startTime="1932500" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="30" tscDM:imageoffset="0" tscDM:imagerect="2250, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="Finite Elliptic Curve Example" xmpDM:startTime="1933933" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="31" tscDM:imageoffset="0" tscDM:imagerect="2325, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example" xmpDM:startTime="1934233" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="32" tscDM:imageoffset="0" tscDM:imagerect="2400, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="2020067" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="33" tscDM:imageoffset="0" tscDM:imagerect="2475, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="2265933" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="34" tscDM:imageoffset="0" tscDM:imagerect="2550, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example" xmpDM:startTime="2267300" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="35" tscDM:imageoffset="0" tscDM:imagerect="2625, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="2268233" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="36" tscDM:imageoffset="0" tscDM:imagerect="2700, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example" xmpDM:startTime="2298300" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="37" tscDM:imageoffset="0" tscDM:imagerect="2775, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Security" xmpDM:startTime="2298633" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="38" tscDM:imageoffset="0" tscDM:imagerect="2850, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption Example" xmpDM:startTime="2500233" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="39" tscDM:imageoffset="0" tscDM:imagerect="2925, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="2500833" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="40" tscDM:imageoffset="0" tscDM:imagerect="3000, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="2502533" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="41" tscDM:imageoffset="0" tscDM:imagerect="3075, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="2558333" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="42" tscDM:imageoffset="0" tscDM:imagerect="3150, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="2639533" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="43" tscDM:imageoffset="0" tscDM:imagerect="3225, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example" xmpDM:startTime="2642733" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="44" tscDM:imageoffset="0" tscDM:imagerect="3300, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="2644067" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="45" tscDM:imageoffset="0" tscDM:imagerect="3375, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="2648467" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="46" tscDM:imageoffset="0" tscDM:imagerect="3450, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="2821433" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="47" tscDM:imageoffset="0" tscDM:imagerect="3525, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Example" xmpDM:startTime="2822333" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="48" tscDM:imageoffset="0" tscDM:imagerect="3600, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Diffie-Hellman" xmpDM:startTime="2836300" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="49" tscDM:imageoffset="0" tscDM:imagerect="3675, 0, 75, 56"/></rdf:li><rdf:li><rdf:Description xmpDM:name="ECC Encryption/Decryption" xmpDM:startTime="2836733" tscDM:image="l_10opk2_Thumbnails.png" tscDM:imageindex="50" 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