Polynomial Inverse
One of the important steps of computing point addition over elliptic curves is a division of two polynomials.
One Clock Cycle Polynomial Math
Error correction codes and cryptographic computations are most easily performed working with GF(2^n)
Elliptic Curve Cryptography
Secure online communications require encryption. One standard is AES (Advanced Encryption Standard) from NIST. But for this to work, both sides need the same key for encryption and decryption. This is called Private Key encryption.
Polynomial Math
This post walks through squaring and inversion in a tiny finite field to make ECC math tangible. Using GF(2^5) with primitive polynomial beta^5 + beta^2 + 1 it shows why squaring cancels cross terms so you only need half the lookup table, and how Fermat exponentiation computes inverses via repeated squarings and multiplies. It also demonstrates the Extended Euclid polynomial inverse and compares FPGA and CPU tradeoffs.
Ancient History
Technology moves fast, and the tools, platforms, and assumptions you rely on can become outdated almost overnight. In this reflective post, the author contrasts the rapid evolution of embedded development with the much slower pace of social change, from programming turnaround times to the underrepresentation of women in engineering. It is a reminder to keep learning, but also to think about how we work and who gets included.
Running Average
This post shows a lightweight running average variant that cleans up noisy wheel-encoder timing using a shift-based divide-by-8 and a small rounding constant. The filter updates as M1 = M0 + (t0 - A0), where A0 is estimated from the previous average, so each new measurement nudges the estimate rather than immediately replacing it. It trades exact oldest-sample subtraction for low resource cost and strong noise suppression.
Elliptic Curve Key Exchange
Elliptic Curve key exchange gives a fresh secret for every session so past messages stay safe even if one key is discovered. This post walks through an ElGamal-style ephemeral exchange and the MQV protocol, showing how MQV mixes static and random keys to provide mutual authentication and forward secrecy. It also explains how MQV can be implemented using only curve operations to save FPGA area and why erasing ephemeral values matters.
Elliptic Curve Digital Signatures
Elliptic curve digital signatures deliver compact, strong message authentication by combining a hash of the message with elliptic curve point math. This post walks through the standard sign and verify equations, showing why recomputing a point R' yields the same x coordinate only when the hash matches. It also explains the Nyberg-Rueppel alternative that removes modular inversion and an FPGA-friendly trick of transmitting point D to avoid integer modular arithmetic.
Elliptic Curve Cryptography - Security Considerations
The security of elliptic curve cryptography is determined by the elliptic curve discrete log problem. This article explains what that means. A comparison with real number logarithm and modular arithmetic gives context for why it is called a log problem.
Elliptic Curve Key Exchange
Elliptic Curve key exchange gives a fresh secret for every session so past messages stay safe even if one key is discovered. This post walks through an ElGamal-style ephemeral exchange and the MQV protocol, showing how MQV mixes static and random keys to provide mutual authentication and forward secrecy. It also explains how MQV can be implemented using only curve operations to save FPGA area and why erasing ephemeral values matters.
Polynomial Math
This post walks through squaring and inversion in a tiny finite field to make ECC math tangible. Using GF(2^5) with primitive polynomial beta^5 + beta^2 + 1 it shows why squaring cancels cross terms so you only need half the lookup table, and how Fermat exponentiation computes inverses via repeated squarings and multiplies. It also demonstrates the Extended Euclid polynomial inverse and compares FPGA and CPU tradeoffs.
Elliptic Curve Digital Signatures
Elliptic curve digital signatures deliver compact, strong message authentication by combining a hash of the message with elliptic curve point math. This post walks through the standard sign and verify equations, showing why recomputing a point R' yields the same x coordinate only when the hash matches. It also explains the Nyberg-Rueppel alternative that removes modular inversion and an FPGA-friendly trick of transmitting point D to avoid integer modular arithmetic.
