椭圆曲线与伪随机序列的构造
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摘要
伪随机序列在扩频通信、多址通信(CDMA)、测量测距、软件测试、雷达导航、序列密码与公钥密码等领域都有广泛的应用．特别在序列密码中，密钥流序列的随机性与不可预测性完全决定了序列密码系统的安全性．构造适合各种用途的伪随机序列一直是研究的热点．本文研究利用椭圆曲线构造的伪随机序列(伪随机数)，主要利用有限域上椭圆曲线有理点群的指数和估计讨论椭圆曲线序列的密码性质——分布、相关性、(非)线性复杂度等，得到如下主要结果：
(1)系统讨论椭圆曲线-线性同余序列的一致分布性质，即该类序列是渐近一致分布的，并给出了它的非线性复杂度下界；
(2)讨论两类由椭圆曲线构造的二元序列的“良性”分布(well distribution)与高阶(κ阶)相关性(correlation of order κ)，这两类序列具有“优”的密码性质，也正面回答了Goubin等提出的公开问题；
(3)利用椭圆曲线及其挠曲线构造一类二元序列，其周期为4p(其中椭圆曲线定义在有限域F_p上)，0-1分布基本平衡，线性复杂度至少为周期的四分之一；
(4)讨论了剩余类环Z_(pq)上的椭圆曲线的有理点的分布估计，并用于分析一类由剩余类环Z_(pq)上椭圆曲线构造的二元序列的伪随机性；
(5)讨论椭圆曲线-幂生成器序列的相关性及椭圆曲线-二次生成器序列的一致分布；
(6)讨论椭圆曲线-子集和序列的一致分布；
(7)讨论椭圆曲线上的线性反馈移位寄存器序列的分布，线性复杂度等性质．
Pseudorandom sequences have wide applications in spread-spectrum communication systems, code division multiple-access systems, ranging systems, software testing, radar systems, stream ciphers and public cryptosystems. Especially the security of stream ciphers heavily depends on the randomness and the unpredictability of the key streams. It is a hot-spot to construct pseudo-random sequences for use. This dissertation investigates the construction of pseudo-random sequences (pseudo-random numbers) from elliptic curves and mainly analyzes their cryptographic properties by using exponential sums over rational points along elliptic curves. The main results are as follows:
(1) The uniform distribution of the elliptic curve linear congruential generator is discussed and the lower bound of its nonlinear complexity is given.
(2) Two large families of binary sequences are constructed from elliptic curves. The well distribution measure and the correlation measure of order k of the resulting sequences are studied. The results indicate that they are "good" binary sequences which give a positive answer to a conjecture proposed by Goubin et al.
(3) A kind of binary sequences from an elliptic curve and its twisted curves over a prime field F_p. The length of the sequences is 4p. The '1' and '0' occur almost the same times. The linear complexity is at least one-fourth the period (length).
(4) The exponential sums over rational points along elliptic curves over ring Z_(pq) are estimated and are used to estimate the well distribution measure and the correlation measure of order k of a family of binary sequences from elliptic curves over ring Z_(pq).
(5) The correlation of the elliptic curve power number generator is given. It is proved that the sequences produced by the elliptic curve quadratic generator are asymptotically uniformly distributed.
(6) The uniform distribution of the elliptic curve subset sum generator is considered.
(7) We apply the linear feedback shift register over elliptic curves to produce sequences with long periods. The distribution and the linear complexity of the resulting sequences are also considered.
(1)系统讨论椭圆曲线-线性同余序列的一致分布性质，即该类序列是渐近一致分布的，并给出了它的非线性复杂度下界；
(2)讨论两类由椭圆曲线构造的二元序列的“良性”分布(well distribution)与高阶(κ阶)相关性(correlation of order κ)，这两类序列具有“优”的密码性质，也正面回答了Goubin等提出的公开问题；
(3)利用椭圆曲线及其挠曲线构造一类二元序列，其周期为4p(其中椭圆曲线定义在有限域F_p上)，0-1分布基本平衡，线性复杂度至少为周期的四分之一；
(4)讨论了剩余类环Z_(pq)上的椭圆曲线的有理点的分布估计，并用于分析一类由剩余类环Z_(pq)上椭圆曲线构造的二元序列的伪随机性；
(5)讨论椭圆曲线-幂生成器序列的相关性及椭圆曲线-二次生成器序列的一致分布；
(6)讨论椭圆曲线-子集和序列的一致分布；
(7)讨论椭圆曲线上的线性反馈移位寄存器序列的分布，线性复杂度等性质．
Pseudorandom sequences have wide applications in spread-spectrum communication systems, code division multiple-access systems, ranging systems, software testing, radar systems, stream ciphers and public cryptosystems. Especially the security of stream ciphers heavily depends on the randomness and the unpredictability of the key streams. It is a hot-spot to construct pseudo-random sequences for use. This dissertation investigates the construction of pseudo-random sequences (pseudo-random numbers) from elliptic curves and mainly analyzes their cryptographic properties by using exponential sums over rational points along elliptic curves. The main results are as follows:
(1) The uniform distribution of the elliptic curve linear congruential generator is discussed and the lower bound of its nonlinear complexity is given.
(2) Two large families of binary sequences are constructed from elliptic curves. The well distribution measure and the correlation measure of order k of the resulting sequences are studied. The results indicate that they are "good" binary sequences which give a positive answer to a conjecture proposed by Goubin et al.
(3) A kind of binary sequences from an elliptic curve and its twisted curves over a prime field F_p. The length of the sequences is 4p. The '1' and '0' occur almost the same times. The linear complexity is at least one-fourth the period (length).
(4) The exponential sums over rational points along elliptic curves over ring Z_(pq) are estimated and are used to estimate the well distribution measure and the correlation measure of order k of a family of binary sequences from elliptic curves over ring Z_(pq).
(5) The correlation of the elliptic curve power number generator is given. It is proved that the sequences produced by the elliptic curve quadratic generator are asymptotically uniformly distributed.
(6) The uniform distribution of the elliptic curve subset sum generator is considered.
(7) We apply the linear feedback shift register over elliptic curves to produce sequences with long periods. The distribution and the linear complexity of the resulting sequences are also considered.
引文
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