Public defence in the field of Mathematics and Statistics, M.Sc.(Tech.) Okko Makkonen

Title of the thesis: Algebraic methods for secure coded computing

Thesis defender: Okko Makkonen 
Opponent: Professor Gretchen Matthews, Virginia Tech, USA
Custos: Professor Camilla Hollanti, Aalto University School of Science

Coding theory is the study of error-correcting codes, which are used to transmit information over a noisy communication channel. Over the last decade, these codes have found their way into distributed cloud computing systems to provide robustness against node failures and large latencies. Recently, so-called coded computing has been developed to utilize the vast resources of these systems to distribute large computational tasks over a number of computing nodes, while providing robustness against slow and unresponsive nodes by using error-correcting codes. These methods allow for more efficient use of available resources to speed up large computational tasks across many industries.

Secret sharing is a cryptographic method that is used to divide a message into a number of pieces such that any sufficiently small number of the pieces does not reveal anything about the message. As such, secret sharing is complementary to data encryption as the security is not based on access to a decryption key, but rather on access to parts of the data itself. In fact, combining secret sharing with coded computing can enable cloud computing with data that is proprietary or confidential, as no data is leaked to the cloud computing provider.

In this thesis, we focus on methods for computing bilinear functions, such as matrix multiplication, which is called secure distributed matrix multiplication (SDMM). We study these methods through the componentwise product, or star product, of linear codes. We are able to design constructions that use fewer computing nodes compared to the literature by using linear codes coming from the function fields of algebraic curves. The improvements arise from the algebraic structure in the star products of these codes. The constructions presented in this thesis enhance understanding of the fundamental limits of SDMM.

In addition to coding-theoretic tools using finite field arithmetic, we also focus on arithmetic over the real numbers, which presents issues with numerical stability. In order to tackle these problems, we study the numerical stability of polynomial interpolation and design constructions that are more numerically stable than previous constructions. These methods allow secure coded computing to be used in real-world applications where arithmetic is over the real numbers without suffering from precision loss due to finite field quantization or bad numerical stability.

Thesis available for public display 7 days prior to the defence at Aaltodoc. 

Contact information: okko.makkonen@aalto.fi