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<ol class="chapter"><li class="chapter-item affix "><a href="index.html">Introduction</a></li><li class="chapter-item affix "><li class="part-title">Specifications</li><li class="chapter-item "><a href="representation.html"><strong aria-hidden="true">1.</strong> Big Integer Representation v1.0</a></li><li class="chapter-item expanded "><a href="arithmetic.html" class="active"><strong aria-hidden="true">2.</strong> Arithmetic Algorithms v1.0</a></li><li class="chapter-item "><a href="montgomery.html"><strong aria-hidden="true">3.</strong> Montgomery & Modular Exponentiation v1.0</a></li><li class="chapter-item "><a href="gas.html"><strong aria-hidden="true">4.</strong> Gas Schedule v1.0</a></li></ol>
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<h1 class="menu-title">ClockinChain Big Integer Specifications</h1>
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<h1 id="arithmetic-algorithm-specification-v10"><a class="header" href="#arithmetic-algorithm-specification-v10">Arithmetic Algorithm Specification v1.0</a></h1>
<h2 id="overview"><a class="header" href="#overview">Overview</a></h2>
<p>This specification defines the normative algorithms for all big integer arithmetic operations in the ClockinChain ecosystem. All algorithms must execute in constant time and produce deterministic results.</p>
<h2 id="1-common-conventions"><a class="header" href="#1-common-conventions">1. Common Conventions</a></h2>
<h3 id="11-limb-operations"><a class="header" href="#11-limb-operations">1.1 Limb Operations</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>// Basic limb arithmetic with carry/borrow
fn limb_add_carry(a: Limb, b: Limb, carry: Limb) -> (Limb, Limb) {
let sum = a as u128 + b as u128 + carry as u128;
(sum as Limb, (sum >> 64) as Limb)
}
fn limb_sub_borrow(a: Limb, b: Limb, borrow: Limb) -> (Limb, Limb) {
let diff = a as i128 - b as i128 - borrow as i128;
(diff as Limb, if diff < 0 { 1 } else { 0 })
}
fn limb_mul_wide(a: Limb, b: Limb) -> (Limb, Limb) {
let product = a as u128 * b as u128;
(product as Limb, (product >> 64) as Limb)
}
<span class="boring">}</span></code></pre></pre>
<h3 id="12-fixed-loop-rule"><a class="header" href="#12-fixed-loop-rule">1.2 Fixed Loop Rule</a></h3>
<p><strong>All algorithms MUST iterate over full declared limb length:</strong></p>
<ul>
<li>Never exit loops early</li>
<li>Mask carries instead of branching</li>
<li>Use constant-time condition selection</li>
<li>Fixed execution paths regardless of input values</li>
</ul>
<h3 id="13-algorithm-selection"><a class="header" href="#13-algorithm-selection">1.3 Algorithm Selection</a></h3>
<p>Operations choose algorithms based on <strong>public parameters only</strong> (limb length), never secret values.</p>
<h2 id="2-addition-algorithm"><a class="header" href="#2-addition-algorithm">2. Addition Algorithm</a></h2>
<h3 id="21-function-signature"><a class="header" href="#21-function-signature">2.1 Function Signature</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn add(a: &BigInt, b: &BigInt) -> Result<BigInt>
<span class="boring">}</span></code></pre></pre>
<h3 id="22-precondition"><a class="header" href="#22-precondition">2.2 Precondition</a></h3>
<ul>
<li><code>a</code> and <code>b</code> have equal limb length <code>n</code></li>
<li>If dynamic-length, caller ensures sufficient capacity</li>
</ul>
<h3 id="23-algorithm"><a class="header" href="#23-algorithm">2.3 Algorithm</a></h3>
<pre><code>For i = 0 .. n-1:
(sum_i, carry) = adc(a.limbs[i], b.limbs[i], carry)
c.limbs[i] = sum_i
After final limb:
If carry = 1:
If dynamic-length → append new limb = 1
If fixed-length → overflow trap
</code></pre>
<h3 id="24-constant-time-adc"><a class="header" href="#24-constant-time-adc">2.4 Constant-Time adc</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>sum = x + y + carry
carry_out = (sum < x) OR ((carry == 1) AND (sum == x))
<span class="boring">}</span></code></pre></pre>
<p>No branches allowed.</p>
<h3 id="25-sign-handling"><a class="header" href="#25-sign-handling">2.5 Sign Handling</a></h3>
<ul>
<li>Same sign: <code>sign(c) = sign(a)</code></li>
<li>Different signs: invoke subtraction algorithm</li>
</ul>
<h3 id="26-gas-cost"><a class="header" href="#26-gas-cost">2.6 Gas Cost</a></h3>
<p><code>G_add(n) = 3n</code></p>
<h2 id="3-subtraction-algorithm"><a class="header" href="#3-subtraction-algorithm">3. Subtraction Algorithm</a></h2>
<h3 id="31-function-signature"><a class="header" href="#31-function-signature">3.1 Function Signature</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn sub(a: &BigInt, b: &BigInt) -> Result<BigInt>
<span class="boring">}</span></code></pre></pre>
<h3 id="32-algorithm-magnitude"><a class="header" href="#32-algorithm-magnitude">3.2 Algorithm (Magnitude)</a></h3>
<pre><code>For i = 0 .. n-1:
(diff_i, borrow) = sbb(a.limbs[i], b.limbs[i], borrow)
c.limbs[i] = diff_i
If final borrow = 1 → result negative:
Compute two's complement of magnitude
Flip sign bit
</code></pre>
<h3 id="33-constant-time-sbb"><a class="header" href="#33-constant-time-sbb">3.3 Constant-Time sbb</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>diff = x - y - borrow
borrow_out = (x < y) OR ((borrow == 1) AND (x == y))
<span class="boring">}</span></code></pre></pre>
<p>No branches.</p>
<h3 id="34-sign-rule"><a class="header" href="#34-sign-rule">3.4 Sign Rule</a></h3>
<p><code>sign(c) = sign(a) XOR sign(b)</code></p>
<h3 id="35-gas-cost"><a class="header" href="#35-gas-cost">3.5 Gas Cost</a></h3>
<p><code>G_sub(n) = 3n</code></p>
<h2 id="4-multiplication-algorithm"><a class="header" href="#4-multiplication-algorithm">4. Multiplication Algorithm</a></h2>
<h3 id="41-function-signature"><a class="header" href="#41-function-signature">4.1 Function Signature</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn mul(a: &BigInt, b: &BigInt) -> Result<BigInt>
<span class="boring">}</span></code></pre></pre>
<h3 id="42-comba-method-small-operands"><a class="header" href="#42-comba-method-small-operands">4.2 Comba Method (Small Operands)</a></h3>
<p>For operands ≤ 16 limbs:</p>
<pre><code>Let n = limb count
Initialize c limbs length = 2n all zero
for i = 0 .. n-1:
carry = 0
for j = 0 .. n-1:
(lo, hi) = mul64(a[i], b[j])
(c[i+j], carry) = mac(c[i+j], lo, carry)
carry += hi
c[i+n] = carry
</code></pre>
<h3 id="43-karatsuba-method-large-operands"><a class="header" href="#43-karatsuba-method-large-operands">4.3 Karatsuba Method (Large Operands)</a></h3>
<p>For operands > 16 limbs:</p>
<pre><code>fn karatsuba_mul(a: &[Limb], b: &[Limb]) -> Vec<Limb> {
let n = a.len();
let m = n / 2;
// Split operands
let (a0, a1) = a.split_at(m);
let (b0, b1) = b.split_at(m);
// Compute z0 = a0*b0
let z0 = karatsuba_mul(a0, b0);
// Compute z2 = a1*b1
let z2 = karatsuba_mul(a1, b1);
// Compute z1 = (a0+a1)*(b0+b1) - z0 - z2
let a01 = add_limbs(a0, a1);
let b01 = add_limbs(b0, b1);
let z1 = karatsuba_mul(&a01, &b01);
sub_limbs_inplace(&mut z1, &z0);
sub_limbs_inplace(&mut z1, &z2);
// Combine results
return combine_karatsuba_results(z0, z1, z2, m);
}
</code></pre>
<h3 id="44-threshold-selection"><a class="header" href="#44-threshold-selection">4.4 Threshold Selection</a></h3>
<ul>
<li>Threshold MUST be fixed constant: <code>KARATSUBA_THRESHOLD = 16</code></li>
<li>Execution path depends only on public limb length</li>
<li>Never on operand values</li>
</ul>
<h3 id="45-sign-rule"><a class="header" href="#45-sign-rule">4.5 Sign Rule</a></h3>
<p><code>sign(c) = sign(a) XOR sign(b)</code></p>
<h3 id="46-gas-cost"><a class="header" href="#46-gas-cost">4.6 Gas Cost</a></h3>
<p><code>G_mul(n) = 2n²</code></p>
<h2 id="5-division-algorithm"><a class="header" href="#5-division-algorithm">5. Division Algorithm</a></h2>
<h3 id="51-function-signatures"><a class="header" href="#51-function-signatures">5.1 Function Signatures</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn div_rem(a: &BigInt, b: &BigInt) -> Result<(BigInt, BigInt)>
fn div(a: &BigInt, b: &BigInt) -> Result<BigInt>
fn rem(a: &BigInt, b: &BigInt) -> Result<BigInt>
<span class="boring">}</span></code></pre></pre>
<h3 id="52-knuth-algorithm-d"><a class="header" href="#52-knuth-algorithm-d">5.2 Knuth Algorithm D</a></h3>
<ol>
<li><strong>Normalize divisor</strong>: Left-shift so highest bit set</li>
<li><strong>Left-shift dividend</strong> equally</li>
<li><strong>Main loop</strong>:</li>
</ol>
<pre><code>for i from high_limb downto low_limb:
// Estimate quotient digit q̂
q̂ = estimate_quotient_digit(dividend, divisor, i)
// Multiply and subtract
(partial_product, borrow) = mul_sub(divisor, q̂, dividend_slice)
// Correct if overestimated
while borrow != 0:
q̂ -= 1
add_back(divisor, dividend_slice)
borrow = check_borrow(dividend_slice)
</code></pre>
<h3 id="53-constant-time-corrections"><a class="header" href="#53-constant-time-corrections">5.3 Constant-Time Corrections</a></h3>
<ul>
<li>All correction steps use <strong>masked subtraction</strong></li>
<li>No branches based on borrow values</li>
<li>Fixed iteration count based on limb length</li>
</ul>
<h3 id="54-division-by-zero"><a class="header" href="#54-division-by-zero">5.4 Division by Zero</a></h3>
<p>Must return deterministic <code>DivisionByZero</code> error.</p>
<h3 id="55-sign-rules"><a class="header" href="#55-sign-rules">5.5 Sign Rules</a></h3>
<pre><code>quotient_sign = sign(a) XOR sign(b)
remainder_sign = sign(a)
</code></pre>
<h3 id="56-gas-cost"><a class="header" href="#56-gas-cost">5.6 Gas Cost</a></h3>
<p><code>G_div(n) = 4n²</code></p>
<h2 id="6-squaring-algorithm"><a class="header" href="#6-squaring-algorithm">6. Squaring Algorithm</a></h2>
<h3 id="61-function-signature"><a class="header" href="#61-function-signature">6.1 Function Signature</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn sqr(a: &BigInt) -> Result<BigInt>
<span class="boring">}</span></code></pre></pre>
<h3 id="62-specialized-algorithm"><a class="header" href="#62-specialized-algorithm">6.2 Specialized Algorithm</a></h3>
<p>Squaring uses optimized algorithm for <code>a²</code>:</p>
<ul>
<li>Exploits symmetry: <code>a² = (a0 + a1*B)² = a0² + 2*a0*a1*B + a1²*B²</code></li>
<li>Fewer multiplications than general multiplication</li>
<li>Same constant-time properties</li>
</ul>
<h3 id="63-gas-cost"><a class="header" href="#63-gas-cost">6.3 Gas Cost</a></h3>
<p><code>G_sqr(n) = 1.6n²</code> (cheaper than multiplication)</p>
<h2 id="7-bitwise-operations"><a class="header" href="#7-bitwise-operations">7. Bitwise Operations</a></h2>
<h3 id="71-bit-shift"><a class="header" href="#71-bit-shift">7.1 Bit Shift</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn shift_left(a: &BigInt, bits: usize) -> Result<BigInt>
fn shift_right(a: &BigInt, bits: usize) -> Result<BigInt>
<span class="boring">}</span></code></pre></pre>
<h3 id="72-algorithm"><a class="header" href="#72-algorithm">7.2 Algorithm</a></h3>
<ul>
<li>Convert bit shifts to limb shifts + intra-limb shifts</li>
<li>Handle carry/borrow across limb boundaries</li>
<li>Constant-time regardless of shift amount</li>
</ul>
<h3 id="73-gas-cost"><a class="header" href="#73-gas-cost">7.3 Gas Cost</a></h3>
<p><code>G_shift(n) = 2n</code></p>
<h2 id="8-comparison-operations"><a class="header" href="#8-comparison-operations">8. Comparison Operations</a></h2>
<h3 id="81-comparison-functions"><a class="header" href="#81-comparison-functions">8.1 Comparison Functions</a></h3>
<pre><pre class="playground"><code class="language-rust"><span class="boring">#![allow(unused)]
</span><span class="boring">fn main() {
</span>fn cmp(a: &BigInt, b: &BigInt) -> Ordering
fn eq(a: &BigInt, b: &BigInt) -> bool
fn lt(a: &BigInt, b: &BigInt) -> bool
<span class="boring">}</span></code></pre></pre>
<h3 id="82-constant-time-implementation"><a class="header" href="#82-constant-time-implementation">8.2 Constant-Time Implementation</a></h3>
<ul>
<li>Compare magnitudes first</li>
<li>Then compare signs if magnitudes equal</li>
<li>No early exit on first difference</li>
</ul>
<h3 id="83-gas-cost"><a class="header" href="#83-gas-cost">8.3 Gas Cost</a></h3>
<p><code>G_cmp(n) = 2n</code></p>
<h2 id="9-error-conditions"><a class="header" href="#9-error-conditions">9. Error Conditions</a></h2>
<div class="table-wrapper"><table><thead><tr><th>Condition</th><th>Result</th></tr></thead><tbody>
<tr><td>Overflow (fixed)</td><td>VM Trap</td></tr>
<tr><td>Overflow (dynamic beyond max)</td><td>Overflow error</td></tr>
<tr><td>Division by zero</td><td>Deterministic error</td></tr>
<tr><td>Non-canonical input</td><td>Decode rejection</td></tr>
</tbody></table>
</div>
<h2 id="10-determinism-guarantee"><a class="header" href="#10-determinism-guarantee">10. Determinism Guarantee</a></h2>
<p>Given identical inputs, all implementations MUST:</p>
<ul>
<li>Produce identical limb outputs</li>
<li>Execute identical iteration counts</li>
<li>Consume identical gas cost</li>
<li>Never branch on secret data</li>
</ul>
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