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diff --git a/docs/IMPLEMENTATIONS.md b/docs/IMPLEMENTATIONS.md index b4a4ea8..d333ed5 100644 --- a/docs/IMPLEMENTATIONS.md +++ b/docs/IMPLEMENTATIONS.md @@ -249,7 +249,7 @@ Uses binary addition chain. INPUT: k = (k_{t-1}, ..., k_1, k_0)_2, P ∈ E(F_q). OUTPUT: [k]P. 1. Q ← ∞. - 2. For i from t - 1 downto 0 do + 2. For i from 0 to t-1 do 2.1 If k_i = 1 then Q ← Q + P. 2.2 P ← 2P. 3. Return(Q). @@ -432,7 +432,7 @@ The same name, Montgomery ladder, is used both for the general ladder idea of ex INPUT: k = (k_{t-1}, ..., k_1, k_0)_2, P ∈ E(F_q). OUTPUT: [k]P . 1. P_1 ← P and P_2 ← [2]P - 2. For i = t − 2 downto 0 do + 2. For i = t − 1 downto 0 do 2.1 If k_i = 0 then P_1 ← [2]P_1; P_2 ← P_1 + P_2. Else @@ -443,13 +443,13 @@ The same name, Montgomery ladder, is used both for the general ladder idea of ex INPUT: G ∈ E(F_q), k = (1, k_{t−2}, ..., k_0)2 OUTPUT: Y = kG - R0 ← G; R1 ← [2]G - for j = t − 2 downto 0 do - if (k_j = 0) then - R1 ← R0 + R1; R0 ← [2]R0 - else [if (kj = 1)] - R0 ← R0 + R1; R1 ← [2]R1 - return R0 + 1. R0 ← G; R1 ← [2]G + 2. for j = t − 2 downto 0 do + 2.1 if (k_j = 0) then + R1 ← R0 + R1; R0 ← [2]R0 + else [if (kj = 1)] + R0 ← R0 + R1; R1 ← [2]R1 + 3. return R0 Montgomery addition formulas (Projective coordinates/XZ coordinates):[^2] @@ -536,17 +536,27 @@ y_n &= \frac{(x_n + x_1)((x_n + x_1)(x_{n+1} + x_1) + x_1^2 + y_1)}{x_1} + y_1 ## References [^1]: HANKERSON, Darrel; MENEZES, Alfred J.; VANSTONE, Scott. Guide to Elliptic Curve Cryptography. New York, USA: Springer, 2004. ISBN 9780387218465. Available from DOI: [10.1007/b97644](https://dx.doi.org/10.1007/b97644). -[^2]: COHEN, Henri; FREY, Gerhard; AVANZI, Roberto M.; DOCHE, Christophe; LANGE, -Tanja; NGUYEN, Kim; VERCAUTEREN, Frederik. Handbook of Elliptic and Hyper- -elliptic Curve Cryptography. CRC Press, 2005-07-19. Discrete Mathematics and It’s Applications, no. 34. ISBN 9781584885184. + +[^2]: COHEN, Henri; FREY, Gerhard; AVANZI, Roberto M.; DOCHE, Christophe; LANGE, Tanja; NGUYEN, Kim; VERCAUTEREN, Frederik. Handbook of Elliptic and Hyper-elliptic Curve Cryptography. CRC Press, 2005-07-19. Discrete Mathematics and It’s Applications, no. 34. ISBN 9781584885184. + [^3]: BERNSTEIN, Daniel J.; LANGE, Tanja. Explicit Formulas Database, <https://www.hyperelliptic.org/EFD/> + [^4]: <http://point-at-infinity.org/ecc/> + [^5]: KNUTH, Donald: The Art of Computer Programming, Volume 2: Seminumerical algorithms + [^6]: GORDON, Daniel M.: A survey of fast exponentiation methods. + [^7]: MORAIN, Francois; OLIVOS, Jorge: Speeding up the computations on an elliptic curve using addition-subtraction chains. + [^8]: JOYE, Marc; YEN, Sung-Ming: The Montgomery Powering Ladder. + [^9]: MOLLER, Bodo: Securing Elliptic Curve Point Multiplication against Side-Channel Attacks. + [^10]: MOLLER, Bodo: Improved Techniques for Fast Exponentiation. + [^11]: MOLLER, Bodo: Fractional Windows Revisited: Improved Signed-Digit Representations for Efficient Exponentiation. + [^12]: KOYAMA, Kenji; TSURUOKA, Yukio: Speeding up Elliptic Cryptosystems by Using a Signed Binary Window Method. + [^13]: GALLANT, Robert P.; LAMBERT, Robert J.; VANSTONE, Scott A.: Faster point multiplication on elliptic curves with efficient endomorphisms.
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