Hardware Implementation of Extended Stein’s Algorithm
Location
Ada, Ohio
Start Date
3-12-2024 12:00 AM
End Date
3-12-2024 12:00 AM
Description
Bėzout's Constant’s are an important mathematical concept for the field of cryptography. The development of cryptographic systems utilizing asymmetric algorithms requires the calculation of Bėzout’s constants. Bėzout’s Identity states that if a and b are integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form az + bt are exactly the multiples of d. Bėzout’s constants are traditionally computed using Extended Euclid’s Algorithm (EEA) due to its simplicity and minimal number of iterations. However, Euclid's heavy use of multiplication and division operations presents problems for computer scientists/engineers. Stein’s and Extended Steins (ES) were developed to eliminate the need for multiplication and division, replacing them with bit shifts and addition/subtraction. Computer scientists have utilized Stein's algorithms to more efficiently compute Bėzout's constants but the need for increasingly large keys, more frequent computations, and increased computing power has made the software implementation of Stein's algorithm less practical. If software implementations for calculating Bėzout's constants have reached a plateau the possibility of calculating these constants using hardware offers the opportunity to do so in a quicker, more efficient manner. This presentation will analyze a hardware implementation of ES using a traditional control/datapath implementation, its strengths and limitations as well as potential paths for both faster and more efficient hardware implementations of Extended GCD algorithms.
Recommended Citation
Massie, Ryan, "Hardware Implementation of Extended Stein’s Algorithm" (2024). College of Engineering Student Research Colloquium. 7.
https://digitalcommons.onu.edu/eng_student_research_colloquium/2024/Presentations/7
Hardware Implementation of Extended Stein’s Algorithm
Ada, Ohio
Bėzout's Constant’s are an important mathematical concept for the field of cryptography. The development of cryptographic systems utilizing asymmetric algorithms requires the calculation of Bėzout’s constants. Bėzout’s Identity states that if a and b are integers with greatest common divisor d. Then there exist integers x and y such that ax + by = d. Moreover, the integers of the form az + bt are exactly the multiples of d. Bėzout’s constants are traditionally computed using Extended Euclid’s Algorithm (EEA) due to its simplicity and minimal number of iterations. However, Euclid's heavy use of multiplication and division operations presents problems for computer scientists/engineers. Stein’s and Extended Steins (ES) were developed to eliminate the need for multiplication and division, replacing them with bit shifts and addition/subtraction. Computer scientists have utilized Stein's algorithms to more efficiently compute Bėzout's constants but the need for increasingly large keys, more frequent computations, and increased computing power has made the software implementation of Stein's algorithm less practical. If software implementations for calculating Bėzout's constants have reached a plateau the possibility of calculating these constants using hardware offers the opportunity to do so in a quicker, more efficient manner. This presentation will analyze a hardware implementation of ES using a traditional control/datapath implementation, its strengths and limitations as well as potential paths for both faster and more efficient hardware implementations of Extended GCD algorithms.