Elliptic Curve Cryptography - Key Exchange and Signatures
Elliptic curve mathematics over finite fields helps solve the problem of exchanging secret keys for encrypted messages as well as proving a specific person signed a particular document. This article goes over simple algorithms for key exchange and digital signature using elliptic curve mathematics. These methods are the essence of elliptic curve cryptography (ECC) used in applications such as SSH, TLS and HTTPS.
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 Cryptography - Basic Math
An introduction to the math of elliptic curves for cryptography. Covers the basic equations of points on an elliptic curve and the concept of point addition as well as multiplication.
New book on Elliptic Curve Cryptography
New book on Elliptic Curve Cryptography now online. Deep discount for early purchase. Will really appreciate comments on how to improve the book because physical printing won't happen for a few more months. Check it out here: http://mng.bz/D9NA
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.
Ancient History
The other day I was downloading an IDE for a new (to me) OS. When I went to compile some sample code, it failed. I went onto a forum, where I was told "if you read the release notes you'd know that the peripheral libraries are in a legacy download". Well damn! Looking back at my previous versions I realized I must have done that and forgotten about it. Everything changes, and keeping up with it takes time and effort.
When I first started with microprocessors we...
Dealing With Fixed Point Fractions
Fixed-point fractional math is easy to botch, and this post lays out pragmatic ways to avoid those mistakes. It clarifies the difference between integer and fractional overflow, shows how Q notation helps track binary-point scaling, and explains why multiplies add sign bits that may require shifting. Read for concrete FPGA strategies: keeping bit growth, selective shifts, or aggressive normalization, plus testing tips.
Mathematics and Cryptography
Cryptographic math can look intimidating, but this roundup trims it to what FPGA engineers actually need. It groups concise articles on number theory and elliptic curves, focusing on polynomial math over Galois fields, FPGA-friendly inversion and one-clock-cycle techniques, and elliptic-curve key exchange and digital signatures. Read this to learn which subroutines to implement first and how to turn math into Verilog or VHDL.
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 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 Cryptography - Basic Math
An introduction to the math of elliptic curves for cryptography. Covers the basic equations of points on an elliptic curve and the concept of point addition as well as multiplication.
Dealing With Fixed Point Fractions
Fixed-point fractional math is easy to botch, and this post lays out pragmatic ways to avoid those mistakes. It clarifies the difference between integer and fractional overflow, shows how Q notation helps track binary-point scaling, and explains why multiplies add sign bits that may require shifting. Read for concrete FPGA strategies: keeping bit growth, selective shifts, or aggressive normalization, plus testing tips.
Elliptic Curve Cryptography - Key Exchange and Signatures
Elliptic curve mathematics over finite fields helps solve the problem of exchanging secret keys for encrypted messages as well as proving a specific person signed a particular document. This article goes over simple algorithms for key exchange and digital signature using elliptic curve mathematics. These methods are the essence of elliptic curve cryptography (ECC) used in applications such as SSH, TLS and HTTPS.
New book on Elliptic Curve Cryptography
New book on Elliptic Curve Cryptography now online. Deep discount for early purchase. Will really appreciate comments on how to improve the book because physical printing won't happen for a few more months. Check it out here: http://mng.bz/D9NA
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 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.
Ancient History
The other day I was downloading an IDE for a new (to me) OS. When I went to compile some sample code, it failed. I went onto a forum, where I was told "if you read the release notes you'd know that the peripheral libraries are in a legacy download". Well damn! Looking back at my previous versions I realized I must have done that and forgotten about it. Everything changes, and keeping up with it takes time and effort.
When I first started with microprocessors we...
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 Inverse
One of the important steps of computing point addition over elliptic curves is a division of two polynomials.
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 Cryptography - Basic Math
An introduction to the math of elliptic curves for cryptography. Covers the basic equations of points on an elliptic curve and the concept of point addition as well as multiplication.
Dealing With Fixed Point Fractions
Fixed-point fractional math is easy to botch, and this post lays out pragmatic ways to avoid those mistakes. It clarifies the difference between integer and fractional overflow, shows how Q notation helps track binary-point scaling, and explains why multiplies add sign bits that may require shifting. Read for concrete FPGA strategies: keeping bit growth, selective shifts, or aggressive normalization, plus testing tips.
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.
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.
Ancient History
The other day I was downloading an IDE for a new (to me) OS. When I went to compile some sample code, it failed. I went onto a forum, where I was told "if you read the release notes you'd know that the peripheral libraries are in a legacy download". Well damn! Looking back at my previous versions I realized I must have done that and forgotten about it. Everything changes, and keeping up with it takes time and effort.
When I first started with microprocessors we...
Elliptic Curve Cryptography - Key Exchange and Signatures
Elliptic curve mathematics over finite fields helps solve the problem of exchanging secret keys for encrypted messages as well as proving a specific person signed a particular document. This article goes over simple algorithms for key exchange and digital signature using elliptic curve mathematics. These methods are the essence of elliptic curve cryptography (ECC) used in applications such as SSH, TLS and HTTPS.
Mathematics and Cryptography
Cryptographic math can look intimidating, but this roundup trims it to what FPGA engineers actually need. It groups concise articles on number theory and elliptic curves, focusing on polynomial math over Galois fields, FPGA-friendly inversion and one-clock-cycle techniques, and elliptic-curve key exchange and digital signatures. Read this to learn which subroutines to implement first and how to turn math into Verilog or VHDL.
New book on Elliptic Curve Cryptography
New book on Elliptic Curve Cryptography now online. Deep discount for early purchase. Will really appreciate comments on how to improve the book because physical printing won't happen for a few more months. Check it out here: http://mng.bz/D9NA
One Clock Cycle Polynomial Math
Error correction codes and cryptographic computations are most easily performed working with GF(2^n)
Polynomial Inverse
One of the important steps of computing point addition over elliptic curves is a division of two polynomials.
