DALOS Elliptic Curve
Elliptic curve cryptography (ECC) secures data by leveraging the Elliptic Curve Discrete Logarithm Problem (ECDLP), which is a computationally difficult problem. In ECC, security is built around a key pair: a private key, which is a large, randomly chosen number, and a corresponding public key, derived by multiplying a base point (called the generator) on the elliptic curve by the private key.
The private key acts as the multiplier and is critical because it's the secret value only known to the user. The security comes from the fact that, while it's easy to compute the public key from the private key, it's nearly impossible to reverse the process and compute the private key from the public key. This provides strong security for encryption, digital signatures, and authentication.
With the DALOS Ellipse, the elliptic curve is defined over a prime field that allows for the generation of 2^1600 distinct private keys—far exceeding the 256-bit private key space typically used in other blockchains. This vast key space significantly enhances the level of security, as it offers an exponentially greater number of unique private key possibilities.
Curve Equation:
The elliptic curve employs a Twisted Edwards Curve, given by the formula Below, the DALOS Ellipse, having the following specific formula
Curve Form: The DALOS Ellipse is a twisted Edwards curve, commonly used for cryptographic purposes due to its efficiency and strong security properties. In this case, the parameters (a = 1, d = -26) define the shape of the curve.
Twisted Edwards Curve: Twisted Edwards curves are an alternate form of elliptic curves, and they are often used in cryptography due to their improved speed and security over standard Weierstrass-form elliptic curves.
Prime Field (P):
The elliptic curve is defined over a prime field, its Prime Number being:
Decimal Form:
1422797327267009427840538050096555658106215504388410858221383139000117858232612871647513042437567021359550512676513580279843425806569087338124320534451609923826002722515853311255382582347042785249149784360802682472492979074093381556885921145244739647732065354535899210048956069171919597180088946524547104950094769024061289963739599006148764550591932282504503081289479047974485591701434070586194072480318729612468538986301952269327160804624381735897223889905247070633477140043668654347
This prime number defines the field FP, over which all calculations on the elliptic curve are carried out. The size of Field ensures that the group formed by the curve is large enough for cryptographic security.
Base-Point Prime Order (Q):
The order of the base point (The Generator) GGG is given by the prime prime number Q:
1258387060301909514024042379046449850251725029634697115619073843890705481440046740552204199635883885272944914904655483501916023678206167596650367826811846862157534952990004386839463386963494516862067933899764941962204635259228497801901380413
Decimal Form:
355699331816752356960134512524138914526553876097102714555345784750029464558153217911878260609391755339887628169128395069960856451642271834531080133612902480956500680628963327813845645586760696312287446090200670618123244768523345389221480286312443298993318248147998844891285467143231624324656933746755850081414397737455369231487103951173075022920927985530781253824285785671827565522008885473360364982237217356107139133414951454295284718018163367874070914438516402917597782812818543421
This prime number Q defines the size of the subgroup generated by the base point G, which ensures a secure environment for cryptographic operations, such as key pair generation.
Trace (T):
The trace of Frobenius, T, is used to calculate the number of points on the elliptic curve.
T=-5033548241207638056096169516185799401006900118538788462476295375562821925760186962208816798543535541091779659618621934007664094712824670386601471307247387448630139811960017547357853547853978067448271735599059767848818541036913991207605519336
Cofactor (R):
The cofactor R of the elliptic curve is:
R=4
The cofactor is typically a small integer that represents the ratio between the total number of points on the elliptic curve and the order of the base point G. In this case, the cofactor of 4 indicates that the full group of points is four times the size of the subgroup generated by GGG.
Safe Scalar Size (S=1600):
In elliptic curve cryptography, the safe scalar size refers to the maximum number of distinct private keys that can be securely generated, considering the structure of the curve and its cofactor. For the DALOS Ellipse, the safe scalar is 1600 bits, meaning we can create 2^1600 unique private keys.
Each private key is represented as a 1600-bit binary number. To ensure security, this number is "clamped" based on the curve's cofactor, meaning certain conditions are applied to the key structure. Specifically, the first bit is always set to 1 (ensuring the number is sufficiently large), and the last two bits are set to 00 (as required for the curve's properties). This results in a 1603-bit scalar, where the essential middle portion remains a 1600-bit number.
This clamping method ensures that the 2^1600 private keys are unique and safe for cryptographic operations. These private keys can also be expressed in different number bases, such as base 10 or base 49, but their security and uniqueness stem from the 1600-bit size of the scalar itself. This structure is key to the vast, secure key space that the DALOS Ellipse provides, vastly expanding the cryptographic strength compared to conventional elliptic curves used in most blockchains today.
Base-Point Coordinates (G):
The generator point GGG (base point) has the following coordinates:
Base-Point X Coordinate G(x):
G(x) = 2Base-Point Y Coordinate G(y):
G(y)=479577721234741891316129314062096440203224800598561362604776518993348406897758651324205216647014453759416735508511915279509434960064559686580741767201752370055871770203009254182472722342456597752506165983884867351649283353392919401537107130232654743719219329990067668637876645065665284755295099198801899803461121192253205447281506198423683290960014859350933836516450524873032454015597501532988405894858561193893921904896724509904622632232182531698393484411082218273681226753590907472
These are the coordinates of the generator point on the curve, and they define the starting point for cryptographic operations such as key generation.
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