The Chrysalis project is a proposed method for post-quantum cryptography using the Riemann sphere. To this end, Riemann primitives are introduced in addition to a novel implementation of this new method. Chrysalis itself is the first cryptographic scheme to rely on Holomorphic Learning with Errors, which is a complex form of Learning with Errors relying on the Gauss Circle Problem within the Riemann sphere. The principle security reduction proposed by this novel cryptographic scheme applies complex analysis of a Riemannian manifold along with tangent bundles relative to a disjoint union of subsets based upon a maximal element. A surjective function allows the mapping of multivariate integrals onto subspaces. The proposed NP-Hard problem for security reduction is the non-commutative Grothendieck problem. The reduction of this problem is achieved by applying bilinear matrices in terms of the holomorphic vector bundle such that coordinate systems are intersected via surjective functions between each holomorphic expression. The result is an arbitrarily selected set of points within constraints of bilinear matrix inequalities approximate to the non-commutative problem. This is achieved by applying the quadratic form of bilinear matrices to a linear matrix inequality.

翻译：Chrysalis项目是一种利用黎曼球面实现后量子密码学的新方法。为此，在引入黎曼原语的同时，还提出了一种该新方法的新颖实现方案。Chrysalis本身是首个依赖全纯学习误差的密码方案，它是学习误差的一种复杂形式，基于黎曼球面中的高斯圆问题。该新型密码方案提出的主要安全归约方法涉及对黎曼流形进行复分析，并结合基于最大元素的子集不交并上的切丛。一个满射函数允许将多元积分映射到子空间上。用于安全归约的NP难问题是非交换格罗滕迪克问题。该问题的归约通过在全纯向量丛上应用双线性矩阵实现，使得每个全纯表达式之间的坐标系通过满射函数相交。其结果是在双线性矩阵不等式约束下近似于非交换问题的一组任意选择点集。这通过将双线性矩阵的二次型应用于线性矩阵不等式来实现。