The last decade witnessed rapid increase in multimedia and other applications that require transmitting and protecting huge amount of data streams simultaneously.For such applications,a high-performance cryptosystem i...The last decade witnessed rapid increase in multimedia and other applications that require transmitting and protecting huge amount of data streams simultaneously.For such applications,a high-performance cryptosystem is compulsory to provide necessary security services.Elliptic curve cryptosystem(ECC)has been introduced as a considerable option.However,the usual sequential implementation of ECC and the standard elliptic curve(EC)form cannot achieve required performance level.Moreover,the widely used Hardware implementation of ECC is costly option and may be not affordable.This research aims to develop a high-performance parallel software implementation for ECC.To achieve this,many experiments were performed to examine several factors affecting ECC performance including the projective coordinates,the scalar multiplication algorithm,the elliptic curve(EC)form,and the parallel implementation.The ECC performance was analyzed using the different factors to tune-up them and select the best choices to increase the speed of the cryptosystem.Experimental results illustrated that parallel Montgomery ECC implementation using homogenous projection achieves the highest performance level,since it scored the shortest time delay for ECC computations.In addition,results showed thatNAF algorithm consumes less time to perform encryption and scalar multiplication operations in comparison withMontgomery ladder and binarymethods.Java multi-threading technique was adopted to implement ECC computations in parallel.The proposed multithreaded Montgomery ECC implementation significantly improves the performance level compared to previously presented parallel and sequential implementations.展开更多
Let E be an elliptic curve over a given number field . By Mordell’s Theorem, the torsion subgroup of E defined over Q is a finite group. Using Lutz-Nagell Theorem, we explicitly calculate the torsion subgroup E(Q)tor...Let E be an elliptic curve over a given number field . By Mordell’s Theorem, the torsion subgroup of E defined over Q is a finite group. Using Lutz-Nagell Theorem, we explicitly calculate the torsion subgroup E(Q)tors for certain elliptic curves depending on their coefficients.展开更多
For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors(Q) of rational points has been studied. When Etors(Q) are cyclic groups with even orders, the classification is...For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors(Q) of rational points has been studied. When Etors(Q) are cyclic groups with even orders, the classification is given with explicit critria, and the generators of the torsion groups are also explicitly presented in each case. These results, together with the recent re-展开更多
We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. W...We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.展开更多
Let E be an elliptic curve defined over the field of rational numbers ~. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely ...Let E be an elliptic curve defined over the field of rational numbers ~. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed（Q） is zero. Let E（k） denote the elliptic curve y2 = x3 ＋ k. Then the quadratic twist E（1）d of E（1） by d is the elliptic curve E（d3）： y2 = x3＋ k3. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d = r （rood 24） such that rankE（-d3）（Q） = 0, using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E（k3）（Q） has rank zero.展开更多
We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals.The bounds are uniform in the curve and involve the rank of the corresponding Jacobian.The method used in...We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals.The bounds are uniform in the curve and involve the rank of the corresponding Jacobian.The method used in the proof is a combination of the "determinant method" with an m-descent on the curve.展开更多
为了提高椭圆曲线标量乘法的效率,根据将求逆转换为乘法运算的思想,提出了在二进制域F2n上用仿射坐标直接计算7P的两种算法。两种算法分别通过引入公因子和除法多项式来计算7P,其运算量分别为2I+7S+14M和I+6S+20M,比Purohit等提出的算法...为了提高椭圆曲线标量乘法的效率,根据将求逆转换为乘法运算的思想,提出了在二进制域F2n上用仿射坐标直接计算7P的两种算法。两种算法分别通过引入公因子和除法多项式来计算7P,其运算量分别为2I+7S+14M和I+6S+20M,比Purohit等提出的算法(PUROHIT G N,RAWAT S A,KUMAR M.Elliptic curve point multiplication using MBNR and Point halving.International Journal of Advanced Networking and Applications,2012,3(5):1329-1337)分别节省了一次和两次求逆运算。同时还给出直接计算7kP的快速算法,该算法比重复计算k次7P更有效。最后结合半点运算和扩展多基表示形式将这些新算法应用到标量乘法中。实验结果表明,在美国国家标准技术研究所(NIST)推荐的椭圆曲线上,当预存储点的个数为2和5时,新算法比Purohit算法效率提高了30%和37%,比洪银芳等所提的算法(洪银芳,桂丰,丁勇.基于半点和多基表示的标量乘法扩展算法.计算机工程,2011,37(4):163-165)效率提高了9%和13%。新算法以增加少量的预计算存储为代价,能有效降低标量乘法的运算量。展开更多
基金Authors extend their appreciation to the Deanship of Scientific Research at Imam Mohammad Ibn Saud Islamic University for funding and supporting this work through Graduate Student Research Support Program.
文摘The last decade witnessed rapid increase in multimedia and other applications that require transmitting and protecting huge amount of data streams simultaneously.For such applications,a high-performance cryptosystem is compulsory to provide necessary security services.Elliptic curve cryptosystem(ECC)has been introduced as a considerable option.However,the usual sequential implementation of ECC and the standard elliptic curve(EC)form cannot achieve required performance level.Moreover,the widely used Hardware implementation of ECC is costly option and may be not affordable.This research aims to develop a high-performance parallel software implementation for ECC.To achieve this,many experiments were performed to examine several factors affecting ECC performance including the projective coordinates,the scalar multiplication algorithm,the elliptic curve(EC)form,and the parallel implementation.The ECC performance was analyzed using the different factors to tune-up them and select the best choices to increase the speed of the cryptosystem.Experimental results illustrated that parallel Montgomery ECC implementation using homogenous projection achieves the highest performance level,since it scored the shortest time delay for ECC computations.In addition,results showed thatNAF algorithm consumes less time to perform encryption and scalar multiplication operations in comparison withMontgomery ladder and binarymethods.Java multi-threading technique was adopted to implement ECC computations in parallel.The proposed multithreaded Montgomery ECC implementation significantly improves the performance level compared to previously presented parallel and sequential implementations.
文摘Let E be an elliptic curve over a given number field . By Mordell’s Theorem, the torsion subgroup of E defined over Q is a finite group. Using Lutz-Nagell Theorem, we explicitly calculate the torsion subgroup E(Q)tors for certain elliptic curves depending on their coefficients.
文摘For elliptic curves E over the rationals Q, the classification according to their torsion subgroups Etors(Q) of rational points has been studied. When Etors(Q) are cyclic groups with even orders, the classification is given with explicit critria, and the generators of the torsion groups are also explicitly presented in each case. These results, together with the recent re-
基金Project supported by the National Natural Science Foundation of China (Grant No. 19771052)
文摘We study the classification of elliptic curves E over the rationals Q according to the torsion subgroups E_(tors)(Q). More precisely, we classify those elliptic curves with E_(tors)(Q) being cyclic with even orders. We also give explicit formulas for generators of E_(tors)(Q). These results, together with the recent results of K. Ono for the non-cyclic E_(tors)(Q), completely solve the problem of the explicit classification and parameterization when E has a rational point of order 2.
文摘Let E be an elliptic curve defined over the field of rational numbers ~. Let d be a square-free integer and let Ed be the quadratic twist of E determined by d. Mai, Murty and Ono have proved that there are infinitely many square-free integers d such that the rank of Ed（Q） is zero. Let E（k） denote the elliptic curve y2 = x3 ＋ k. Then the quadratic twist E（1）d of E（1） by d is the elliptic curve E（d3）： y2 = x3＋ k3. Let r = 1, 2, 5, 10, 13, 14, 17, 22. Ono proved that there are infinitely many square-free integers d = r （rood 24） such that rankE（-d3）（Q） = 0, using the theory of modular forms. In this paper, we use the class number of quadratic field and Pell equation to describe these square-free integers k such that E（k3）（Q） has rank zero.
文摘We prove upper bounds for the number of rational points on non-singular cubic curves defined over the rationals.The bounds are uniform in the curve and involve the rank of the corresponding Jacobian.The method used in the proof is a combination of the "determinant method" with an m-descent on the curve.
文摘为了提高椭圆曲线标量乘法的效率,根据将求逆转换为乘法运算的思想,提出了在二进制域F2n上用仿射坐标直接计算7P的两种算法。两种算法分别通过引入公因子和除法多项式来计算7P,其运算量分别为2I+7S+14M和I+6S+20M,比Purohit等提出的算法(PUROHIT G N,RAWAT S A,KUMAR M.Elliptic curve point multiplication using MBNR and Point halving.International Journal of Advanced Networking and Applications,2012,3(5):1329-1337)分别节省了一次和两次求逆运算。同时还给出直接计算7kP的快速算法,该算法比重复计算k次7P更有效。最后结合半点运算和扩展多基表示形式将这些新算法应用到标量乘法中。实验结果表明,在美国国家标准技术研究所(NIST)推荐的椭圆曲线上,当预存储点的个数为2和5时,新算法比Purohit算法效率提高了30%和37%,比洪银芳等所提的算法(洪银芳,桂丰,丁勇.基于半点和多基表示的标量乘法扩展算法.计算机工程,2011,37(4):163-165)效率提高了9%和13%。新算法以增加少量的预计算存储为代价,能有效降低标量乘法的运算量。
