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  <title>Articles</title>
  <link>https://articles.sidneiteixeira.com/</link>
  <description>Latest articles</description>
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    <title><![CDATA[Subgroups, Generators, Orders, And Cofactors]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/subgroups-generators-orders-and-cofactors/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/subgroups-generators-orders-and-cofactors/</guid>
    <pubDate>Wed, 22 Jul 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core analogy: Treat the finite curve group as a circular route. Curve order $#E$: total number of stops in the complete group. Point order $\operatorname{ord}(P)$: lap length—the first positive $n$ with $nP=\mathcal{O}$. Generated subgroup $\langle P\rangle$: all stops reachable by repeatedly taking the $P$-step. Generator: a point whose multiples reach every element of the group being discussed. Base point $G$: the public generator selected by a parameter set. Subgroup order: $n=\operatorname{ord}(G)$. When the selected order is prime, pairing literature commonly names it $r$. Cofactor $h$: number of subgroup-sized cosets in the full group: $$ \#E=hn, \qquad h=\frac{#E}{n}. $$ The notation $a\mid b$ means “$a$ divides $b$,” so $b=ca$ for some integer $c$. Lagrange's theorem guarantees that a point's order divides the curve order: $$ \operatorname{ord}(P)\mid#E(\mathbb{F}\_p). $$ Running example $P=(1,5)$ takes a 14-stop lap and generates the whole curve. Choosing $G=2P=(3,3)$ skips every other stop, producing a prime-order subgroup with $$ \#E=14, \qquad n=r=7, \qquad h=2. $$ Base field and scalar field The curve coefficients and point coordinates live in the base field $\mathbb{F}p$. Here, $x\_G,y\_G\in\mathbb{F}{11}$. The point $G$ lives in the subgroup $\langle G\rangle\subset E(\mathbb{F}\_{11})$; it is not itself a field element. Scalars live in the scalar field $\mathbb{F}\_r$. Here, $k\in\mathbb{F}\_7$. Scalar multiplication maps a scalar and a subgroup point back to the subgroup: $$ k\in\mathbb{F}\_r, \quad G\in\langle G\rangle, \quad kG\in\langle G\rangle. $$ Scalars wrap around modulo $r$ because $rG=\mathcal{O}$, so $(k+r)G=kG$. Protocol bridge: A nonzero private scalar $k\in\mathbb{F}\_r$ produces the public point $Q=kG$; production systems make $r$ enormous.]]></description>
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    <title><![CDATA[Introduction To Elliptic Curves]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/introduction-to-elliptic-curves/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/introduction-to-elliptic-curves/</guid>
    <pubDate>Thu, 11 Jun 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core idea: ECC turns points on a curve into an arithmetic system. Repeating its point operation is easy; reversing a large repetition is hard. Progression $$ \text{real curve} \longrightarrow \text{point addition} \longrightarrow \text{scalar multiplication} \longrightarrow \text{finite-field point group} \longrightarrow \text{cryptographic subgroup} $$ Remember Real curves provide geometric intuition. Finite fields make the coordinates finite, discrete, and exact. Cryptographic systems publicly select a base point and work with its subgroup. This collection builds the mathematics; later collections build keys, signatures, and protocols from it.]]></description>
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    <title><![CDATA[Elliptic Curves Over Real Numbers]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/elliptic-curves-over-real-numbers/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/elliptic-curves-over-real-numbers/</guid>
    <pubDate>Thu, 11 Jun 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core idea: An elliptic curve is the set of points satisfying a smooth cubic equation—not merely the curve drawn on a graph. Essential facts $$ y^2=x^3+ax+b $$ Valid point: $P=(x,y)$ belongs to the curve only when its coordinates satisfy the equation. Symmetry: If $(x,y)$ is valid, then $(x,-y)$ is valid because $y^2=(-y)^2$. Parameters: $a$ and $b$ control the curve's shape. Smoothness: The curve must satisfy $$ 4a^3+27b^2\ne0. $$ Cusps and crossings make the point operation unreliable. Role of real numbers: They make the geometry visible; cryptography later replaces them with exact finite-field arithmetic. Next: Turn the curve's symmetry and geometry into point addition.]]></description>
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    <title><![CDATA[Point Addition And The Elliptic Curve Group]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/point-addition-and-the-elliptic-curve-group/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/point-addition-and-the-elliptic-curve-group/</guid>
    <pubDate>Thu, 11 Jun 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core idea: Point addition turns the curve's valid points into a consistent arithmetic system. Addition rules Different points: Draw the line through $P$ and $Q$, find its third curve intersection, then reflect that intersection across the x-axis to obtain $P+Q$. Doubling: When $P=Q$, use the tangent at $P$ instead of a line through two distinct points. Inverse: If $P=(x,y)$, then $-P=(x,-y)$ and $$ P+(-P)=\mathcal{O}. $$ Identity: The point at infinity acts like zero: $$ P+\mathcal{O}=P. $$ Group view $E(\mathbb{R})$ means all valid real-curve points together with $\mathcal{O}$. Under point addition they form an abelian group: the operation is closed, associative, has an identity and inverses, and is commutative. Next: Reuse point addition efficiently to compute $kP$.]]></description>
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    <title><![CDATA[Scalar Multiplication]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/scalar-multiplication/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/scalar-multiplication/</guid>
    <pubDate>Thu, 11 Jun 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core idea: Scalar multiplication repeats point addition: $$ kP=\underbrace{P+P+\cdots+P}\_{k\text{ additions}}. $$ It does not multiply the coordinates: $k(x,y)\ne(kx,ky)$. Efficient computation Build larger multiples from doubling and addition; for example, $$ 5P=4P+P, \qquad 4P=2P+2P. $$ Double-and-add reads the binary digits of $k$: double for every bit, then add $P$ when the bit is $1$. The work grows roughly as $\log\_2 k$, making enormous scalars practical. Every intermediate result remains in the same elliptic-curve group. Next: Replace approximate real-number coordinates with exact modular arithmetic.]]></description>
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    <title><![CDATA[Finite Fields And Modular Arithmetic]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/finite-fields-and-modular-arithmetic/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/finite-fields-and-modular-arithmetic/</guid>
    <pubDate>Thu, 11 Jun 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core idea: A prime field replaces continuous, approximate real arithmetic with a finite and exact number system. Prime-field arithmetic $$ \mathbb{F}\_p={0,1,\ldots,p-1}, \qquad p\text{ prime}. $$ Reduce every addition, subtraction, and multiplication modulo $p$. $a\equiv b\pmod p$ means $a$ and $b$ have the same remainder modulo $p$. Division uses a modular inverse: $$ b,b^{-1}\equiv1\pmod p, \qquad \frac ab\equiv a b^{-1}\pmod p. $$ Example in $\mathbb{F}\_{11}$: $3^{-1}=4$ because $3\cdot4\equiv1\pmod{11}$. A prime modulus ensures every nonzero element has an inverse; a composite modulus generally does not. The curve equation is now interpreted as $$ y^2\equiv x^3+ax+b\pmod p. $$ Next: Enumerate the coordinate pairs in $\mathbb{F}\_p^2$ that satisfy this equation.]]></description>
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    <title><![CDATA[Elliptic Curves Over Finite Fields]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/elliptic-curves-over-finite-fields/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/elliptic-curves-and-finite-fields/elliptic-curves-over-finite-fields/</guid>
    <pubDate>Thu, 11 Jun 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Core idea: A finite-field curve is computed directly from its modular equation. It is not a cropped, rounded, or wrapped image of the real curve. Construct the point group For each $x\in\mathbb{F}\_p$: Compute $r=x^3+ax+b\pmod p$. Find every $y$ with $y^2\equiv r\pmod p$. Collect each matching $(x,y)$, then add the identity $\mathcal{O}$. A nonzero quadratic residue normally gives the inverse pair $(x,y)$ and $(x,p-y)$; zero gives $(x,0)$; a non-residue gives no point. For $$ y^2\equiv x^3+x+1\pmod{11}, $$ there are $13$ affine points and $\mathcal{O}$, so $#E(\mathbb{F}\_{11})=14$. Add finite-field points Use the same group law algebraically. For $P=(x\_1,y\_1)$ and $Q=(x\_2,y\_2)$: $$ \lambda= \begin{cases} (y\_2-y\_1)(x\_2-x\_1)^{-1}, & P\ne Q,\\ (3x\_1^2+a)(2y\_1)^{-1}, & P=Q, \end{cases} \pmod p $$ $$ x\_3=\lambda^2-x\_1-x\_2, \qquad y\_3=\lambda(x\_1-x\_3)-y\_1 \pmod p. $$ Division means multiplication by a modular inverse. Also, $P+(-P)=\mathcal{O}$ and $2(x,0)=\mathcal{O}$. Validity: Require $4a^3+27b^2\not\equiv0\pmod p$. Production curves use the same construction and operation at an enormous scale. Next: Measure scalar cycles and select a prime-order subgroup.]]></description>
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    <title><![CDATA[Noise And Jitter]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/dsp-acoustic-fundamentals/noise-and-jitter/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/dsp-acoustic-fundamentals/noise-and-jitter/</guid>
    <pubDate>Sun, 22 Feb 2026 00:00:00 GMT</pubDate>
    <description><![CDATA[Noise Noise is random fluctuation added to a signal. The main difference between noise “colors” is how their energy is distributed across frequencies. | Noise Type | Frequency Energy Distribution | Memory / Correlation | Perceived Sound | Intuition | |---|---|---|---|---| | White Noise | Equal power per Hz | No memory (samples are independent) | Bright, hissy, static-like | Flat spectrum across frequencies | | Pink Noise | Equal energy per octave | Some correlation (more low-frequency content than white) | Balanced, natural | Middle ground between white and brown | | Brown Noise | Strongly weighted toward low frequencies | Strong memory (random walk behavior) | Deep, smooth, waterfall / heavy-rain-like | Accumulated noise, low-end heavy |]]></description>
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    <title><![CDATA[Threshold Encryption Made SImple]]></title>
    <link>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/threshold-encryption/threshold-encryption-made-simple/</link>
    <guid>https://articles.sidneiteixeira.com/notes/computer-science/cryptography/threshold-encryption/threshold-encryption-made-simple/</guid>
    <pubDate>Mon, 15 Dec 2025 00:00:00 GMT</pubDate>
    <description><![CDATA[Since ancient times, people have needed to keep important information hidden from prying eyes. Whether it was a king's secret orders or a merchant's trade routes, secrecy was often the difference...]]></description>
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