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  <title>P3642R0: Carry-less product: std::clmul</title>
<style data-fill-with="stylesheet">/******************************************************************************
 *                   Style sheet for the W3C specifications                   *
 *
 * Special classes handled by this style sheet include:
 *
 * Indices
 *   - .toc for the Table of Contents (<ol class="toc">)
 *     + <span class="secno"> for the section numbers
 *   - #toc for the Table of Contents (<nav id="toc">)
 *   - ul.index for Indices (<a href="#ref">term</a><span>, in § N.M</span>)
 *   - table.index for Index Tables (e.g. for properties or elements)
 *
 * Structural Markup
 *   - table.data for general data tables
 *     -> use 'scope' attribute, <colgroup>, <thead>, and <tbody> for best results !
 *     -> use <table class='complex data'> for extra-complex tables
 *     -> use <td class='long'> for paragraph-length cell content
 *     -> use <td class='pre'> when manual line breaks/indentation would help readability
 *   - dl.switch for switch statements
 *   - ol.algorithm for algorithms (helps to visualize nesting)
 *   - .figure and .caption (HTML4) and figure and figcaption (HTML5)
 *     -> .sidefigure for right-floated figures
 *   - ins/del
 *     -> ins/del.c### for candidate and proposed changes (amendments)
 *
 * Code
 *   - pre and code
 *
 * Special Sections
 *   - .note       for informative notes             (div, p, span, aside, details)
 *   - .example    for informative examples          (div, p, pre, span)
 *   - .issue      for issues                        (div, p, span)
 *   - .advisement for loud normative statements     (div, p, strong)
 *   - .annoying-warning for spec obsoletion notices (div, aside, details)
 *   - .correction for "candidate corrections"       (div, aside, details, section)
 *   - .addition   for "candidate additions"         (div, aside, details, section)
 *   - .correction.proposed for "proposed corrections" (div, aside, details, section)
 *   - .addition.proposed   for "proposed additions"   (div, aside, details, section)
 *
 * Definition Boxes
 *   - pre.def   for WebIDL definitions
 *   - table.def for tables that define other entities (e.g. CSS properties)
 *   - dl.def    for definition lists that define other entitles (e.g. HTML elements)
 *
 * Numbering
 *   - .secno for section numbers in .toc and headings (<span class='secno'>3.2</span>)
 *   - .marker for source-inserted example/figure/issue numbers (<span class='marker'>Issue 4</span>)
 *   - ::before styled for CSS-generated issue/example/figure numbers:
 *     -> Documents wishing to use this only need to add
 *        figcaption::before,
 *        .caption::before { content: "Figure "  counter(figure) " ";  }
 *        .example::before { content: "Example " counter(example) " "; }
 *        .issue::before   { content: "Issue "   counter(issue) " ";   }
 *
 * Header Stuff (ignore, just don't conflict with these classes)
 *   - .head for the header
 *   - .copyright for the copyright
 *
 * Outdated warning for old specs
 *
 * Miscellaneous
 *   - .overlarge for things that should be as wide as possible, even if
 *     that overflows the body text area. This can be used on an item or
 *     on its container, depending on the effect desired.
 *     Note that this styling basically doesn't help at all when printing,
 *     since A4 paper isn't much wider than the max-width here.
 *     It's better to design things to fit into a narrower measure if possible.
 *
 *   - js-added ToC jump links (see fixup.js)
 *
 ******************************************************************************/

/* color variables included separately for reliability */

/******************************************************************************/
/*                                    Body                                    */
/******************************************************************************/

	html {
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	body {
		counter-reset: example figure issue;

		/* Layout */
		max-width: 50em;			  /* limit line length to 50em for readability   */
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		padding: 1.6em 1.5em 2em calc(26px + 1.5em); /* leave space for status flag    */

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		background: white top left fixed no-repeat;
		background: var(--bg) top left fixed no-repeat;
		background-size: 25px auto;
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/******************************************************************************/
/*                         Front Matter & Navigation                          */
/******************************************************************************/

/** Header ********************************************************************/

	div.head { margin-bottom: 1em; }
	div.head hr { border-style: solid; }

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	div.head h2 { margin-bottom: 1.5em;}

/** W3C Logo ******************************************************************/

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	.head a:hover > img[src*="logos/W3C"],
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	.head a:active > img[src*="logos/W3C"] {
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	/* see also additional rules in Link Styling section */

/** Copyright *****************************************************************/

	p.copyright,
	p.copyright small { font-size: small; }

/** Back to Top / ToC Toggle **************************************************/

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			display: none;
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	@media not print {
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			height: 1.33em;
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			box-shadow: 0 0 2px;
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		#toc-nav > #toc-jump {
			padding-bottom: 2em;
			margin-bottom: -1.9em;
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		/* statusbar gets in the way on keyboard focus; remove once browsers fix */
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		#toc-nav:not(:hover) > a:not(:focus) > span + span {
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/** ToC Sidebar ***************************************************************/

	/* Floating sidebar */
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			left: 0;
			width: 23.5em;
			max-width: 80%;
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			color: var(--tocsidebar-text);
			background: inherit;
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			box-shadow: -.1em 0 .25em var(--tocsidebar-shadow) inset;
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			margin-top: .8rem;
			font-variant: small-caps;
			font-variant: all-small-caps;
			text-transform: lowercase;
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			padding-left: calc(1em + 26px);
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			color: var(--tocsidebar-text);
			background: inherit;
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			box-shadow: -.1em 0 .25em var(--tocsidebar-shadow) inset;
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		body:not(.toc-inline) #toc h2 {
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			font-variant: small-caps;
			font-variant: all-small-caps;
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			color: gray;
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		body:not(.toc-inline) {
			padding-left: 29em;
		}
		/* See also Overflow section at the bottom */

		body:not(.toc-inline) #toc-jump:not(:focus) {
			width: 0;
			height: 0;
			padding: 0;
			position: absolute;
			overflow: hidden;
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		body:not(.toc-inline) {
			margin: 0 4em;
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/******************************************************************************/
/*                                Sectioning                                  */
/******************************************************************************/

/** Headings ******************************************************************/

	h1, h2, h3, h4, h5, h6, dt {
		page-break-after: avoid;
		page-break-inside: avoid;
		font: 100% sans-serif;   /* Reset all font styling to clear out UA styles */
		font-family: inherit;	/* Inherit the font family. */
		line-height: 1.2;		/* Keep wrapped headings compact */
		hyphens: manual;		/* Hyphenated headings look weird */
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	h2, h3, h4, h5, h6 {
		margin-top: 3rem;
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	h1, h2, h3 {
		color: #005A9C;
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	h1 { font-size: 170%; }
	h2 { font-size: 140%; }
	h3 { font-size: 120%; }
	h4 { font-weight: bold; }
	h5 { font-style: italic; }
	h6 { font-variant: small-caps; }
	dt { font-weight: bold; }

/** Subheadings ***************************************************************/

	h1 + h2,
	#profile-and-date {
		/* #profile-and-date is a subtitle in an H2 under the H1 */
		margin-top: 0;
	}
	h2 + h3,
	h3 + h4,
	h4 + h5,
	h5 + h6 {
		margin-top: 1.2em; /* = 1 x line-height */
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/** Section divider ***********************************************************/

	:not(.head) > :not(.head) + hr {
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		text-align: center;
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		height: auto;
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		border: transparent solid 0;
		background: transparent;
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	:not(.head) > hr::before {
		content: "\2727\2003\2003\2727\2003\2003\2727";
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/******************************************************************************/
/*                            Paragraphs and Lists                            */
/******************************************************************************/

	p {
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	dd > p:first-child,
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	ul, ol {
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		padding-left: 2em;
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	li {
		margin: 0.25em 0 0.5em;
		padding: 0;
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	dl dd {
		margin: 0 0 .5em 2em;
	}

	.head dd + dd { /* compact for header */
		margin-top: -.5em;
	}

	/* Style for algorithms */
	ol.algorithm ol:not(.algorithm),
	.algorithm > ol ol:not(.algorithm) {
	border-left: 0.5em solid #DEF;
	border-left: 0.5em solid var(--algo-border);
	}

	/* Put nice boxes around each algorithm. */
	[data-algorithm]:not(.heading) {
	 padding: .5em;
	 border: thin solid #ddd;
	 border: thin solid var(--algo-border);
	 border-radius: .5em;
	 margin: .5em calc(-0.5em - 1px);
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	[data-algorithm]:not(.heading) > :first-child {
	 margin-top: 0;
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	[data-algorithm]:not(.heading) > :last-child {
	 margin-bottom: 0;
	}

	/* Style for switch/case <dl>s */
	dl.switch > dd > ol.only,
	dl.switch > dd > .only > ol {
	margin-left: 0;
	}
	dl.switch > dd > ol.algorithm,
	dl.switch > dd > .algorithm > ol {
	margin-left: -2em;
	}
	dl.switch {
	padding-left: 2em;
	}
	dl.switch > dt {
	text-indent: -1.5em;
	margin-top: 1em;
	}
	dl.switch > dt + dt {
	margin-top: 0;
	}
	dl.switch > dt::before {
	content: '\21AA';
	padding: 0 0.5em 0 0;
	display: inline-block;
	width: 1em;
	text-align: right;
	line-height: 0.5em;
	}

/** Terminology Markup ********************************************************/


/******************************************************************************/
/*                                 Inline Markup                              */
/******************************************************************************/

/** Terminology Markup ********************************************************/
	dfn   { /* Defining instance */
		font-weight: bolder;
	}
	a > i { /* Instance of term */
		font-style: normal;
	}
	dt dfn code, code.idl {
		font-size: inherit;
	}
	dfn var {
		font-style: normal;
	}

/** Change Marking ************************************************************/

	del {
		color: #aa0000;
		color: var(--del-text);
		background: transparent;
		background: var(--del-bg);
		text-decoration: line-through;
	}
	ins {
		color: #006100;
		color: var(--ins-text);
		background: transparent;
		background: var(--ins-bg);
		text-decoration: underline;
	}

	/* for amendments (candidate/proposed changes) */

	.amendment ins, .correction ins, .addition ins,
	ins[class^=c] {
		text-decoration-style: dotted;
	}
	.amendment del, .correction del, .addition del,
	del[class^=c] {
		text-decoration-style: dotted;
	}
	.amendment.proposed ins, .correction.proposed ins, .addition.proposed ins,
	ins[class^=c].proposed {
		text-decoration-style: double;
	}
	.amendment.proposed del, .correction.proposed del, .addition.proposed del,
	del[class^=c].proposed {
		text-decoration-style: double;
	}

/** Miscellaneous improvements to inline formatting ***************************/

	sup {
		vertical-align: super;
		font-size: 80%
	}

/******************************************************************************/
/*                                    Code                                    */
/******************************************************************************/

/** General monospace/pre rules ***********************************************/

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		font-size: .9em;
		hyphens: none;
		text-transform: none;
		text-align: left;
		text-align: start;
		font-variant: normal;
		orphans: 3;
		widows: 3;
		page-break-before: avoid;
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	pre code,
	code code {
		font-size: 100%;
	}

	pre {
		margin-top: 1em;
		margin-bottom: 1em;
		overflow: auto;
	}

/** Inline Code fragments *****************************************************/

	/* Do something nice. */

/******************************************************************************/
/*                                    Links                                   */
/******************************************************************************/

/** General Hyperlinks ********************************************************/

	/* We hyperlink a lot, so make it less intrusive */
	a[href] {
		color: #034575;
		color: var(--a-normal-text);
		text-decoration: underline #707070;
		text-decoration: underline var(--a-normal-underline);
		text-decoration-skip-ink: none;
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	a:visited {
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		color: var(--a-visited-text);
		text-decoration-color: #bbb;
		text-decoration-color: var(--a-visited-underline);
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	/* Indicate interaction with the link */
	a[href]:focus,
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		text-decoration-thickness: 2px;
	}
	a[href]:active {
		color: #c00;
		color: var(--a-active-text);
		text-decoration-color: #c00;
		text-decoration-color: var(--a-active-underline);
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	/* Backout above styling for W3C logo */
	.head .logo,
	.head .logo a {
		border: none;
		text-decoration: none;
		background: transparent;
	}

/******************************************************************************/
/*                                    Images                                  */
/******************************************************************************/

	img {
		border-style: none;
	}

	img, svg {
		/* Intentionally not color-scheme aware. */
		background: white;
	}

	/* For autogen numbers, add
	  .caption::before, figcaption::before { content: "Figure " counter(figure) ". "; }
	*/

	figure, .figure, .sidefigure {
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		text-align: center;
		margin: 2.5em 0;
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	.figure img,	.sidefigure img,	figure img,
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		max-width: 100%;
		margin: auto;
		height: auto;
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	.figure pre, .sidefigure pre, figure pre {
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		display: table;
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	.figure table, figure table {
		margin: auto;
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		.sidefigure {
			float: right;
			width: 50%;
			margin: 0 0 0.5em 0.5em;
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	}
	.caption, figcaption, caption {
		font-style: italic;
		font-size: 90%;
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	.caption::before, figcaption::before, figcaption > .marker {
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	.caption, figcaption {
		counter-increment: figure;
	}

	/* DL list is indented 2em, but figure inside it is not */
	dd > .figure, dd > figure { margin-left: -2em; }

/******************************************************************************/
/*                             Colored Boxes                                  */
/******************************************************************************/

	.issue, .note, .example, .assertion, .advisement, blockquote,
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		margin: 1em auto;
		padding: .5em;
		border: .5em;
		border-left-style: solid;
		page-break-inside: avoid;
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	span.issue, span.note {
		padding: .1em .5em .15em;
		border-right-style: solid;
	}

	blockquote > :first-child,
	.note  > p:first-child,
	.issue > p:first-child,
	.amendment > p:first-child,
	.correction > p:first-child,
	.addition > p:first-child {
		margin-top: 0;
	}
	blockquote > :last-child,
	.note  > p:last-child,
	.issue > p:last-child,
	.amendment > p:last-child,
	.correction > p:last-child,
	.addition > p:last-child {
		margin-bottom: 0;
	}


	.issue::before, .issue > .marker,
	.example::before, .example > .marker,
	.note::before, .note > .marker,
	details.note > summary > .marker,
	.amendment::before, .amendment > .marker,
	details.amendment > summary > .marker,
	.addition::before, .addition > .marker,
	addition.amendment > summary > .marker,
	.correction::before, .correction > .marker,
	correction.amendment > summary > .marker
	{
		text-transform: uppercase;
		padding-right: 1em;
	}

	.example::before, .example > .marker {
		display: block;
		padding-right: 0em;
	}

/** Blockquotes ***************************************************************/

	blockquote {
		border-color: silver;
		border-color: var(--blockquote-border);
		background: transparent;
		background: var(--blockquote-bg);
		color: currentcolor;
		color: var(--blockquote-text);
	}

/** Open issue ****************************************************************/

	.issue {
		border-color: #e05252;
		border-color: var(--issue-border);
		background: #fbe9e9;
		background: var(--issue-bg);
		color: black;
		color: var(--issue-text);
		counter-increment: issue;
		overflow: auto;
	}
	.issue::before, .issue > .marker {
		color: #831616;
		color: var(--issueheading-text);
	}
	/* Add .issue::before { content: "Issue " counter(issue) " "; } for autogen numbers,
	  or use class="marker" to mark up the issue number in source. */

/** Example *******************************************************************/

	.example {
		border-color: #e0cb52;
		border-color: var(--example-border);
		background: #fcfaee;
		background: var(--example-bg);
		color: black;
		color: var(--example-text);
		counter-increment: example;
		overflow: auto;
		clear: both;
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	.example::before, .example > .marker {
		color: #574b0f;
		color: var(--exampleheading-text);
	}
	/* Add .example::before { content: "Example " counter(example) " "; } for autogen numbers,
	  or use class="marker" to mark up the example number in source. */

/** Non-normative Note ********************************************************/

	.note {
		border-color: #52e052;
		border-color: var(--note-border);
		background: #e9fbe9;
		background: var(--note-bg);
		color: black;
		color: var(--note-text);
		overflow: auto;
	}

	.note::before, .note > .marker,
	details.note > summary {
		color: hsl(120, 70%, 30%);
		color: var(--noteheading-text);
	}
	/* Add .note::before { content: "Note "; } for autogen label,
	  or use class="marker" to mark up the label in source. */

	details.note[open] > summary {
		border-bottom: 1px silver solid;
		border-bottom: 1px var(--notesummary-underline) solid;
	}

/** Assertion Box *************************************************************/
	/*  for assertions in algorithms */

	.assertion {
		border-color: #AAA;
		border-color: var(--assertion-border);
		background: #EEE;
		background: var(--assertion-bg);
		color: black;
		color: var(--assertion-text);
	}

/** Advisement Box ************************************************************/
	/*  for attention-grabbing normative statements */

	.advisement {
		border-color: orange;
		border-color: var(--advisement-border);
		border-style: none solid;
		background: #fec;
		background: var(--advisement-bg);
		color: black;
		color: var(--advisement-text);
	}
	strong.advisement {
		display: block;
		text-align: center;
	}
	.advisement::before, .advisement > .marker {
		color: #b35f00;
		color: var(--advisementheading-text);
	}

/** Amendment Box *************************************************************/

	.amendment, .correction, .addition {
		border-color: #330099;
		border-color: var(--amendment-border);
		background: #F5F0FF;
		background: var(--amendment-bg);
		color: black;
		color: var(--amendment-text);
	}
	.amendment.proposed, .correction.proposed, .addition.proposed {
		border-style: solid;
		border-block-width: 0.25em;
	}
	.amendment::before, .amendment > .marker,
	details.amendment > summary::before, details.amendment > summary > .marker,
	.correction::before, .correction > .marker,
	details.correction > summary::before, details.correction > summary > .marker,
	.addition::before, .addition > .marker,
	details.addition > summary::before, details.addition > summary > .marker {
		color: #220066;
		color: var(--amendmentheading-text);
	}
	.amendment.proposed::before, .amendment.proposed > .marker,
	details.amendment.proposed > summary::before, details.amendment.proposed > summary > .marker,
	.correction.proposed::before, .correction.proposed > .marker,
	details.correction.proposed > summary::before, details.correction.proposed > summary > .marker,
	.addition.proposed::before, .addition.proposed > .marker,
	details.addition.proposed > summary::before, details.addition.proposed > summary > .marker {
		font-weight: bold;
	}

/** Spec Obsoletion Notice ****************************************************/
	/* obnoxious obsoletion notice for older/abandoned specs. */

	details {
		display: block;
	}
	summary {
		font-weight: bolder;
	}

	.annoying-warning:not(details),
	details.annoying-warning:not([open]) > summary,
	details.annoying-warning[open] {
		background: hsla(40,100%,50%,0.95);
		background: var(--warning-bg);
		color: black;
		color: var(--warning-text);
		padding: .75em 1em;
		border: red;
		border: var(--warning-border);
		border-style: solid none;
		box-shadow: 0 2px 8px black;
		text-align: center;
	}
	.annoying-warning :last-child {
		margin-bottom: 0;
	}

@media not print {
	details.annoying-warning[open] {
		position: fixed;
		left: 0;
		right: 0;
		bottom: 2em;
		z-index: 1000;
	}
}

	details.annoying-warning:not([open]) > summary {
		text-align: center;
	}

/** Entity Definition Boxes ***************************************************/

	.def {
		padding: .5em 1em;
		background: #def;
		background: var(--def-bg);
		margin: 1.2em 0;
		border-left: 0.5em solid #8ccbf2;
		border-left: 0.5em solid var(--def-border);
		color: black;
		color: var(--def-text);
	}

/******************************************************************************/
/*                                    Tables                                  */
/******************************************************************************/

	th, td {
		text-align: left;
		text-align: start;
	}

/** Property/Descriptor Definition Tables *************************************/

	table.def {
		/* inherits .def box styling, see above */
		width: 100%;
		border-spacing: 0;
	}

	table.def td,
	table.def th {
		padding: 0.5em;
		vertical-align: baseline;
		border-bottom: 1px solid #bbd7e9;
		border-bottom: 1px solid var(--defrow-border);
	}

	table.def > tbody > tr:last-child th,
	table.def > tbody > tr:last-child td {
		border-bottom: 0;
	}

	table.def th {
		font-style: italic;
		font-weight: normal;
		padding-left: 1em;
		width: 3em;
	}

	/* For when values are extra-complex and need formatting for readability */
	table td.pre {
		white-space: pre-wrap;
	}

	/* A footnote at the bottom of a def table */
	table.def td.footnote {
		padding-top: 0.6em;
	}
	table.def td.footnote::before {
		content: " ";
		display: block;
		height: 0.6em;
		width: 4em;
		border-top: thin solid;
	}

/** Data tables (and properly marked-up index tables) *************************/
	/*
		<table class="data"> highlights structural relationships in a table
		when correct markup is used (e.g. thead/tbody, th vs. td, scope attribute)

		Use class="complex data" for particularly complicated tables --
		(This will draw more lines: busier, but clearer.)

		Use class="long" on table cells with paragraph-like contents
		(This will adjust text alignment accordingly.)
		Alternately use class="longlastcol" on tables, to have the last column assume "long".
	*/

	table {
		word-wrap: normal;
		overflow-wrap: normal;
		hyphens: manual;
	}

	table.data,
	table.index {
		margin: 1em auto;
		border-collapse: collapse;
		border: hidden;
		width: 100%;
	}
	table.data caption,
	table.index caption {
		max-width: 50em;
		margin: 0 auto 1em;
	}

	table.data td,  table.data th,
	table.index td, table.index th {
		padding: 0.5em 1em;
		border-width: 1px;
		border-color: silver;
		border-color: var(--datacell-border);
		border-top-style: solid;
	}

	table.data thead td:empty {
		padding: 0;
		border: 0;
	}

	table.data  thead,
	table.index thead,
	table.data  tbody,
	table.index tbody {
		border-bottom: 2px solid;
	}

	table.data colgroup,
	table.index colgroup {
		border-left: 2px solid;
	}

	table.data  tbody th:first-child,
	table.index tbody th:first-child  {
		border-right: 2px solid;
		border-top: 1px solid silver;
		border-top: 1px solid var(--datacell-border);
		padding-right: 1em;
	}

	table.data th[colspan],
	table.data td[colspan] {
		text-align: center;
	}

	table.complex.data th,
	table.complex.data td {
		border: 1px solid silver;
		border: 1px solid var(--datacell-border);
		text-align: center;
	}

	table.data.longlastcol td:last-child,
	table.data td.long {
		vertical-align: baseline;
		text-align: left;
	}

	table.data img {
		vertical-align: middle;
	}


/*
Alternate table alignment rules

	table.data,
	table.index {
		text-align: center;
	}

	table.data  thead th[scope="row"],
	table.index thead th[scope="row"] {
		text-align: right;
	}

	table.data  tbody th:first-child,
	table.index tbody th:first-child  {
		text-align: right;
	}

Possible extra rowspan handling

	table.data  tbody th[rowspan]:not([rowspan='1']),
	table.index tbody th[rowspan]:not([rowspan='1']),
	table.data  tbody td[rowspan]:not([rowspan='1']),
	table.index tbody td[rowspan]:not([rowspan='1']) {
		border-left: 1px solid silver;
	}

	table.data  tbody th[rowspan]:first-child,
	table.index tbody th[rowspan]:first-child,
	table.data  tbody td[rowspan]:first-child,
	table.index tbody td[rowspan]:first-child{
		border-left: 0;
		border-right: 1px solid silver;
	}
*/

/******************************************************************************/
/*                                  Indices                                   */
/******************************************************************************/


/** Table of Contents *********************************************************/

	.toc a {
		/* More spacing; use padding to make it part of the click target. */
		padding: 0.1rem 1px 0;
		/* Larger, more consistently-sized click target */
		display: block;
		/* Switch to using border-bottom for underlines */
		text-decoration: none;
		border-bottom: 1px solid;
		/* Reverse color scheme */
		color: black;
		color: var(--toclink-text);
		border-color: #3980b5;
		border-color: var(--toclink-underline);
	}
	.toc a:visited {
		color: black;
		color: var(--toclink-visited-text);
		border-color: #054572;
		border-color: var(--toclink-visited-underline);
	}
	.toc a:focus,
	.toc a:hover {
		background: rgba(75%, 75%, 75%, .25);
		background: var(--a-hover-bg);
		border-bottom-width: 3px;
		margin-bottom: -2px;
	}
	.toc a:not(:focus):not(:hover) {
		/* Allow colors to cascade through from link styling */
		border-bottom-color: transparent;
	}

	.toc, .toc ol, .toc ul, .toc li {
		list-style: none; /* Numbers must be inlined into source */
		/* because generated content isn't search/selectable and markers can't do multilevel yet */
		margin:  0;
		padding: 0;
	}
	.toc {
		line-height: 1.1em;
	}

	/* ToC not indented until third level, but font style & margins show hierarchy */
	.toc > li			{ font-weight: bold;   }
	.toc > li li		 { font-weight: normal; }
	.toc > li li li	  { font-size:   95%;	}
	.toc > li li li li	{ font-size:   90%;	}
	.toc > li li li li li { font-size:   85%;	}

	/* @supports not (display:grid) { */
		.toc > li			{ margin: 1.5rem 0;	}
		.toc > li li		 { margin: 0.3rem 0;	}
		.toc > li li li	  { margin-left: 2rem;   }

		/* Section numbers in a column of their own */
		.toc .secno {
			float: left;
			width: 4rem;
			white-space: nowrap;
		}
		.toc > li li li li .secno { font-size: 85%; }
		.toc > li li li li li .secno { font-size: 100%; }

		.toc li {
			clear: both;
		}

		:not(li) > .toc			 { margin-left:  5rem; }
		.toc .secno				 { margin-left: -5rem; }
		.toc > li li li .secno	  { margin-left: -7rem; }
		.toc > li li li li .secno	{ margin-left: -9rem; }
		.toc > li li li li li .secno { margin-left: -11rem; }

		/* Tighten up indentation in narrow ToCs */
		@media (max-width: 30em) {
			:not(li) > .toc			 { margin-left:  4rem; }
			.toc .secno				 { margin-left: -4rem; }
			.toc > li li li			 { margin-left:  1rem; }
			.toc > li li li .secno	  { margin-left: -5rem; }
			.toc > li li li li .secno	{ margin-left: -6rem; }
			.toc > li li li li li .secno { margin-left: -7rem; }
		}
		/* Loosen it on wide screens */
		@media screen and (min-width: 78em) {
			body:not(.toc-inline) :not(li) > .toc			 { margin-left:  4rem; }
			body:not(.toc-inline) .toc .secno				 { margin-left: -4rem; }
			body:not(.toc-inline) .toc > li li li			 { margin-left:  1rem; }
			body:not(.toc-inline) .toc > li li li .secno	  { margin-left: -5rem; }
			body:not(.toc-inline) .toc > li li li li .secno	{ margin-left: -6rem; }
			body:not(.toc-inline) .toc > li li li li li .secno { margin-left: -7rem; }
	}
	/* } */

	@supports (display:grid) and (display:contents) {
		/* Use #toc over .toc to override non-@supports rules. */
		#toc {
			display: grid;
			align-content: start;
			grid-template-columns: auto 1fr;
			grid-column-gap: 1rem;
			column-gap: 1rem;
			grid-row-gap: .6rem;
			row-gap: .6rem;
		}
		#toc h2 {
			grid-column: 1 / -1;
			margin-bottom: 0;
		}
		#toc ol,
		#toc li,
		#toc a {
			display: contents;
			/* Switch <a> to subgrid when supported */
		}
		#toc span {
			margin: 0;
		}
		#toc > .toc > li > a > span {
			/* The spans of the top-level list,
			  comprising the first items of each top-level section. */
			margin-top: 1.1rem;
		}
		#toc#toc .secno { /* Ugh, need more specificity to override base.css */
			grid-column: 1;
			width: auto;
			margin-left: 0;
		}
		#toc .content {
			grid-column: 2;
			width: auto;
			margin-right: 1rem;
			border-bottom: 3px solid transparent;
			margin-bottom: -3px;
		}
		#toc .content:hover,
		#toc .content:focus {
			background: rgba(75%, 75%, 75%, .25);
			background: var(--a-hover-bg);
			border-bottom-color: #054572;
			border-bottom-color: var(--toclink-underline);
		}
		#toc li li li .content {
			margin-left: 1rem;
		}
		#toc li li li li .content {
			margin-left: 2rem;
		}
	}


/** Index *********************************************************************/

	/* Index Lists: Layout */
	ul.index	  { margin-left: 0; columns: 15em; text-indent: 1em hanging; }
	ul.index li	{ margin-left: 0; list-style: none; break-inside: avoid; }
	ul.index li li { margin-left: 1em; }
	ul.index dl	{ margin-top: 0; }
	ul.index dt	{ margin: .2em 0 .2em 20px;}
	ul.index dd	{ margin: .2em 0 .2em 40px;}
	/* Index Lists: Typography */
	ul.index ul,
	ul.index dl { font-size: smaller; }
	@media not print {
		ul.index li a + span {
			white-space: nowrap;
			color: transparent; }
		ul.index li a:hover + span,
		ul.index li a:focus + span {
			color: #707070;
			color: var(--indexinfo-text);
		}
	}

/** Index Tables *****************************************************/
	/* See also the data table styling section, which this effectively subclasses */

	table.index {
		font-size: small;
		border-collapse: collapse;
		border-spacing: 0;
		text-align: left;
		margin: 1em 0;
	}

	table.index td,
	table.index th {
		padding: 0.4em;
	}

	table.index tr:hover td:not([rowspan]),
	table.index tr:hover th:not([rowspan]) {
		color: black;
		color: var(--indextable-hover-text);
		background: #f7f8f9;
		background: var(--indextable-hover-bg);
	}

	/* The link in the first column in the property table (formerly a TD) */
	table.index th:first-child a {
		font-weight: bold;
	}

/** Outdated warning **********************************************************/

.outdated-spec {
	color: black;
	color: var(--outdatedspec-text);
	background-color: rgba(0,0,0,0.5);
	background-color: var(--outdatedspec-bg);
}

.outdated-warning {
	position: fixed;
	bottom: 50%;
	left: 0;
	right: 0;
	margin: 0 auto;
	width: 50%;
	background: maroon;
	background: var(--outdated-bg);
	color: white;
	color: var(--outdated-text);
	border-radius: 1em;
	box-shadow: 0 0 1em red;
	box-shadow: 0 0 1em var(--outdated-shadow);
	padding: 2em;
	text-align: center;
	z-index: 2;
}

.outdated-warning a {
	color: currentcolor;
	background: transparent;
}

.edited-rec-warning {
	background: darkorange;
	background: var(--editedrec-bg);
	box-shadow: 0 0 1em;
}

.outdated-warning button {
	color: var(--outdated-text);
	border-radius: 1em;
	box-shadow: 0 0 1em red;
	box-shadow: 0 0 1em var(--outdated-shadow);
	padding: 2em;
	text-align: center;
	z-index: 2;
}

.outdated-warning a {
	color: currentcolor;
	background: transparent;
}

.edited-rec-warning {
	background: darkorange;
	background: var(--editedrec-bg);
	box-shadow: 0 0 1em;
}

.outdated-warning button {
	position: absolute;
	top: 0;
	right:0;
	margin: 0;
	border: 0;
	padding: 0.25em 0.5em;
	background: transparent;
	color: white;
	color: var(--outdated-text);
	font:1em sans-serif;
	text-align:center;
}

.outdated-warning span {
	display: block;
}

.outdated-collapsed {
	bottom: 0;
	border-radius: 0;
	width: 100%;
	padding: 0;
}

/******************************************************************************/
/*                                    Print                                   */
/******************************************************************************/

	@media print {
		/* Pages have their own margins. */
		html {
			margin: 0;
		}
		/* Serif for print. */
		body {
			font-family: serif;
		}

		.outdated-warning {
			position: absolute;
			border-style: solid;
			border-color: red;
		}

		.outdated-warning input {
			display: none;
		}
	}
	@page {
		margin: 1.5cm 1.1cm;
	}



/******************************************************************************/
/*                             Overflow Control                               */
/******************************************************************************/

	.figure .caption, .sidefigure .caption, figcaption {
		/* in case figure is overlarge, limit caption to 50em */
		max-width: 50rem;
		margin-left: auto;
		margin-right: auto;
	}
	.overlarge {
		/* Magic to create good item positioning:
		  "content column" is 50ems wide at max; less on smaller screens.
		  Extra space (after ToC + content) is empty on the right.

		  1. When item < content column, centers item in column.
		  2. When content < item < available, left-aligns.
		  3. When item > available, fills available + scroll bar.
		*/
		display: grid;
		grid-template-columns: minmax(0, 50em);
	}
	.overlarge > table {
		/* limit preferred width of table */
		max-width: 50em;
		margin-left: auto;
		margin-right: auto;
	}

	@media (min-width: 55em) {
		.overlarge {
			margin-right: calc(13px + 26.5rem - 50vw);
			max-width: none;
		}
	}
	@media screen and (min-width: 78em) {
		body:not(.toc-inline) .overlarge {
			/* 30.5em body padding 50em content area */
			margin-right: calc(40em - 50vw) !important;
		}
	}
	@media screen and (min-width: 90em) {
		body:not(.toc-inline) .overlarge {
			/* 4em html margin 30.5em body padding 50em content area */
			margin-right: calc(84.5em - 100vw) !important;
		}
	}

	@media not print {
		.overlarge {
			overflow-x: auto;
			/* See Lea Verou's explanation background-attachment:
			* http://lea.verou.me/2012/04/background-attachment-local/
			*
			background: top left  / 4em 100% linear-gradient(to right,  #ffffff, rgba(255, 255, 255, 0)) local,
						top right / 4em 100% linear-gradient(to left, #ffffff, rgba(255, 255, 255, 0)) local,
						top left  / 1em 100% linear-gradient(to right,  #c3c3c5, rgba(195, 195, 197, 0)) scroll,
						top right / 1em 100% linear-gradient(to left, #c3c3c5, rgba(195, 195, 197, 0)) scroll,
						white;
			background-repeat: no-repeat;
			*/
		}
	}
</style>
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 <body class="h-entry">
  <div class="head">
   <p data-fill-with="logo"></p>
   <h1 class="p-name no-ref" id="title">P3642R0<br>Carry-less product: <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>clmul</c-></code></h1>
   <h2 class="no-num no-toc no-ref heading settled" id="profile-and-date"><span class="content">Published Proposal, <time class="dt-updated" datetime="2025-03-13">2025-03-13</time></span></h2>
   <div data-fill-with="spec-metadata">
    <dl>
     <dt>This version:
     <dd><a class="u-url" href="https://eisenwave.github.io/cpp-proposals/clmul.html">https://eisenwave.github.io/cpp-proposals/clmul.html</a>
     <dt class="editor">Author:
     <dd class="editor p-author h-card vcard"><a class="p-name fn u-email email" href="mailto:janschultke@gmail.com">Jan Schultke</a>
     <dt>Audience:
     <dd>SG6, SG18
     <dt>Project:
     <dd>ISO/IEC 14882 Programming Languages — C++, ISO/IEC JTC1/SC22/WG21
     <dt>Source:
     <dd><a href="https://github.com/Eisenwave/cpp-proposals/blob/master/src/clmul.bs">eisenwave/cpp-proposals</a>
    </dl>
   </div>
   <div data-fill-with="warning"></div>
   <hr title="Separator for header">
  </div>
  <div class="p-summary" data-fill-with="abstract">
   <h2 class="no-num no-toc no-ref heading settled" id="abstract"><span class="content">Abstract</span></h2>
   <p>Add widening and non-widening carry-less multiplication functions.</p>
  </div>
  <nav data-fill-with="table-of-contents" id="toc">
   <h2 class="no-num no-toc no-ref" id="contents">Table of Contents</h2>
   <ol class="toc" role="directory">
    <li><a href="#introduction"><span class="secno">1</span> <span class="content">Introduction</span></a>
    <li>
     <a href="#motivation"><span class="secno">2</span> <span class="content">Motivation</span></a>
     <ol class="toc">
      <li><a href="#parity-computation"><span class="secno">2.1</span> <span class="content">Parity computation</span></a>
      <li><a href="#fast-space-filling-curves"><span class="secno">2.2</span> <span class="content">Fast space-filling curves</span></a>
     </ol>
    <li>
     <a href="#possible-implementation"><span class="secno">3</span> <span class="content">Possible implementation</span></a>
     <ol class="toc">
      <li><a href="#hardware-support"><span class="secno">3.1</span> <span class="content">Hardware support</span></a>
     </ol>
    <li>
     <a href="#design-considerations"><span class="secno">4</span> <span class="content">Design Considerations</span></a>
     <ol class="toc">
      <li><a href="#naming"><span class="secno">4.1</span> <span class="content">Naming</span></a>
      <li><a href="#simd"><span class="secno">4.2</span> <span class="content">SIMD</span></a>
     </ol>
    <li><a href="#proposed-wording"><span class="secno">5</span> <span class="content">Proposed wording</span></a>
    <li>
     <a href="#references"><span class="secno"></span> <span class="content">References</span></a>
     <ol class="toc">
      <li><a href="#normative"><span class="secno"></span> <span class="content">Normative References</span></a>
      <li><a href="#informative"><span class="secno"></span> <span class="content">Informative References</span></a>
     </ol>
   </ol>
  </nav>
  <main>
   <h2 class="heading settled" data-level="1" id="introduction"><span class="secno">1. </span><span class="content">Introduction</span><a class="self-link" href="#introduction"></a></h2>
   <p>Carry-less multiplication is a simple numerical operation on unsigned integers.
It can be a seen as a regular multiplication where <code class="highlight"><c- k>xor</c-></code> is being used as a reduction instead of <code class="highlight"><c- o>+</c-></code>.</p>
   <p>It is also known as "XOR multiplication" and "polynomial multiplication".
The latter name is used because mathematically, it is equivalent to performing a multiplication of
two polynomials in GF(2), where each bit is a coefficient.</p>
   <p>I propose a <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>clmul</c-></code> function to perform this operation:</p>
<pre class="language-cpp highlight"><c- k>template</c-><c- o>&lt;</c-><c- n>unsigned_integral</c-> <c- n>T</c-><c- o>></c-> <c- c1>// constraint is exposition-only</c->
<c- k>constexpr</c-> <c- n>T</c-> <c- n>clmul</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->
</pre>
   <p>I also propose a widening operation in the style of <a data-link-type="biblio" href="#biblio-p3161r2" title="Unified integer overflow arithmetic">[P3161R2]</a>, as follows:</p>
<pre class="language-cpp highlight"><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
<c- k>struct</c-> <c- nc>mul_wide_result</c-> <c- p>{</c->
    <c- n>T</c-> <c- n>low_bits</c-><c- p>;</c->
    <c- n>T</c-> <c- n>high_bits</c-><c- p>;</c->
<c- p>};</c->

<c- k>template</c-><c- o>&lt;</c-><c- n>unsigned_integral</c-> <c- n>T</c-><c- o>></c->
<c- k>constexpr</c-> <c- n>mul_wide_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c-> <c- n>clmul_wide</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->
</pre>
   <h2 class="heading settled" data-level="2" id="motivation"><span class="secno">2. </span><span class="content">Motivation</span><a class="self-link" href="#motivation"></a></h2>
   <p>Carry-less multiplication is an important operation in a number of use cases:</p>
   <ul>
    <li data-md>
     <p><strong>CRC Computation:</strong> While cyclic redundancy checks can theoretically be performed with a finite
field of any length, in practice, <a href="https://en.wikipedia.org/wiki/GF(2)">GF(2)[X]</a>,
the <em>polynomial ring</em> over the <em>Galois field</em> with two elements is used.
Polynomial addition in this ring can be implemented via <code class="highlight"><c- k>xor</c-></code>, and multiplication via <code class="highlight"><c- n>clmul</c-></code>,
which makes cyclic redundancy checks considerably faster.</p>
    <li data-md>
     <p><strong>Cryptography:</strong> <code class="highlight"><c- n>clmul</c-></code> may be used to implement AES-GCM. <a data-link-type="biblio" href="#biblio-intel1" title="Intel® Carry-Less Multiplication Instruction and its Usage for Computing the GCM Mode">[Intel1]</a> describes this process in great detail and motivates hardware support for
carry-less multiplication via the <code class="highlight"><c- n>pclmulqdq</c-></code> instruction.</p>
    <li data-md>
     <p><strong>Bit manipulation:</strong> <code class="highlight"><c- n>clmul</c-></code> performs a large amount of <code class="highlight"><c- o>&lt;&lt;</c-></code> and <code class="highlight"><c- k>xor</c-></code> operations in parallel.
This is utilized in the reference implementation <a data-link-type="biblio" href="#biblio-schultke1" title="C++26 Bit permutations reference implementation">[Schultke1]</a> of <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>bit_compressr</c-></code>,
proposed in <a data-link-type="biblio" href="#biblio-p3104r3" title="P3104R3 Bit permutations">[P3104R3]</a>.
For example, the form <code class="highlight"><c- n>clmul</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- mi>-1u</c-><c- p>)</c-></code> computes the bitwise inclusive parity for each bit of <code class="highlight"><c- n>x</c-></code> and the bits to its right.</p>
   </ul>
   <p>Carry-less multiplication is of such great utility that there is widespread hardware support,
some dating back more than a decade.
See below for motivating examples.</p>
   <h3 class="heading settled" data-level="2.1" id="parity-computation"><span class="secno">2.1. </span><span class="content">Parity computation</span><a class="self-link" href="#parity-computation"></a></h3>
   <p>The parity of an integer <code class="highlight"><c- n>x</c-></code> is <code class="highlight"><c- mi>0</c-></code> if the number of one-bits is even, and <code class="highlight"><c- mi>1</c-></code> if it is odd.
It is also equivalent to <code class="highlight"><c- n>popcount</c-><c- p>(</c-><c- n>x</c-><c- p>)</c-> <c- o>&amp;</c-> <c- mi>1</c-></code>.</p>
   <div class="example" id="example-955a87d2">
    <a class="self-link" href="#example-955a87d2"></a> The special form <code class="highlight"><c- n>clmul</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- mi>-1</c-><c- p>)</c-></code> computes the parity of each bit in <code class="highlight"><c- n>x</c-></code> and the bits to its right.
The most significant bit holds the parity of <code class="highlight"><c- n>x</c-></code> as a whole. 
<pre class="highlight line-numbered"><span class="line-no"></span><span class="line"><c- b>bool</c-> <c- nf>parity</c-><c- p>(</c-><c- b>uint32_t</c-> <c- n>x</c-><c- p>)</c-> <c- p>{</c-></span><span class="line-no highlight-line" data-line="2"></span><span class="line highlight-line">    <c- k>return</c-> <c- n>std</c-><c- o>::</c-><c- n>clmul</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- mi>-1u</c-><c- p>)</c-> <c- o>>></c-> <c- mi>31</c-><c- p>;</c-></span><span class="line-no"></span><span class="line"><c- p>}</c-></span></pre>
   </div>
   <h3 class="heading settled" data-level="2.2" id="fast-space-filling-curves"><span class="secno">2.2. </span><span class="content">Fast space-filling curves</span><a class="self-link" href="#fast-space-filling-curves"></a></h3>
   <p>The special form <code class="highlight"><c- n>clmul</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- mi>-1</c-><c- p>)</c-></code> can be used to accelerate the computation of Hilbert curves.
To properly understand this example, I will explain the basic notion of space-filling curves.</p>
   <p>We can fill space using a 2D curve by mapping the index <code class="highlight"><c- n>i</c-></code> on the curve
onto Cartesian coordinates <code class="highlight"><c- n>x</c-></code> and <code class="highlight"><c- n>y</c-></code>.
A naive curve that fills a 4x4 square can be computed as follows:</p>
<pre class="language-cpp highlight"><c- k>struct</c-> <c- nc>pos</c-> <c- p>{</c-> <c- b>uint32_t</c-> <c- n>x</c-><c- p>,</c-> <c- n>y</c-><c- p>;</c-> <c- p>};</c->

<c- n>pos</c-> <c- nf>naive_curve</c-><c- p>(</c-><c- b>uint32_t</c-> <c- n>i</c-><c- p>)</c-> <c- p>{</c-> <c- k>return</c-> <c- p>{</c-> <c- n>i</c-> <c- o>%</c-> <c- mi>4</c-><c- p>,</c-> <c- n>i</c-> <c- o>/</c-> <c- mi>4</c-> <c- p>};</c-> <c- p>}</c->
</pre>
   <p>When mapping the index <code class="highlight"><c- n>i</c-> <c- o>=</c-> <c- mi>0</c-><c- p>,</c-> <c- mi>1</c-><c- p>,</c-> <c- p>...,</c-> <c- mh>0xf</c-></code> onto the returned 2D coordinates,
we obtain the following pattern:</p>
<pre class="language-text highlight">0 1 2 3
4 5 6 7
8 9 a b
c d e f
</pre>
   <p>The problem with such a naive curve is that adjacent indices can be positioned very far apart
(the distance increases with row length).
For image processing, if we store pixels in this pattern, cache locality is bad;
two adjacent pixels can be very far apart in memory.</p>
   <p>A <a href="https://en.wikipedia.org/wiki/Hilbert_curve">Hilbert curve</a> is a family of space-filling curves where the distance between two adjacent
elements is <code class="highlight"><c- mi>1</c-></code>:</p>
<pre class="language-text highlight">0 1 e f
3 2 d c
4 7 8 b
5 6 9 a
</pre>
   <p>De-interleaving bits of <code class="highlight"><c- n>i</c-></code> into <code class="highlight"><c- n>x</c-></code> and <code class="highlight"><c- n>y</c-></code> yields a <a href="https://en.wikipedia.org/wiki/Z-order_curve">Z-order curve</a>,
and performing further transformations yields a <a href="https://en.wikipedia.org/wiki/Hilbert_curve">Hilbert curve</a>.</p>
   <div class="example" id="example-bf36f3b4">
    <a class="self-link" href="#example-bf36f3b4"></a> <code class="highlight"><c- n>clmul</c-></code> can be used to compute the bitwise parity for each bit and the bits to its right,
which is helpful for computing Hilbert curves.
Note that the following example uses the <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>bit_compress</c-></code> function from <a data-link-type="biblio" href="#biblio-p3104r3" title="P3104R3 Bit permutations">[P3104R3]</a>,
which may also be accelerated using <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>clmul</c-></code>. 
<pre class="highlight line-numbered"><span class="line-no"></span><span class="line"><c- n>pos</c-> <c- nf>hilbert_to_xy</c-><c- p>(</c-><c- b>uint32_t</c-> <c- n>i</c-><c- p>)</c-></span><span class="line-no"></span><span class="line"><c- p>{</c-></span><span class="line-no"></span><span class="line">    <c- c1>// De-interleave the bits of i.</c-></span><span class="line-no"></span><span class="line">    <c- b>uint32_t</c-> <c- n>i0</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>bit_compress</c-><c- p>(</c-><c- n>i</c-><c- p>,</c-> <c- mh>0x55555555u</c-><c- p>);</c-> <c- c1>// abcdefgh -> bdfh</c-></span><span class="line-no"></span><span class="line">    <c- b>uint32_t</c-> <c- n>i1</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>bit_compress</c-><c- p>(</c-><c- n>i</c-><c- p>,</c-> <c- mh>0xaaaaaaaau</c-><c- p>);</c-> <c- c1>// abcdefgh -> aceg</c-></span><span class="line-no"></span><span class="line">    </span><span class="line-no"></span><span class="line">    <c- c1>// Undo the permutation that Hilbert curves apply on top of Z-order curves.</c-></span><span class="line-no"></span><span class="line">    <c- b>uint32_t</c-> <c- n>A</c-> <c- o>=</c-> <c- n>i0</c-> <c- o>&amp;</c-> <c- n>i1</c-><c- p>;</c-></span><span class="line-no"></span><span class="line">    <c- b>uint32_t</c-> <c- n>B</c-> <c- o>=</c-> <c- n>i0</c-> <c- o>^</c-> <c- n>i1</c-> <c- o>^</c-> <c- mh>0xffffu</c-><c- p>;</c-></span><span class="line-no highlight-line" data-line="10"></span><span class="line highlight-line">    <c- b>uint32_t</c-> <c- n>C</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>clmul</c-><c- p>(</c-><c- n>A</c-><c- p>,</c-> <c- mi>-1u</c-><c- p>)</c-> <c- o>>></c-> <c- mi>16</c-><c- p>;</c-></span><span class="line-no highlight-line" data-line="11"></span><span class="line highlight-line">    <c- b>uint32_t</c-> <c- n>D</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>clmul</c-><c- p>(</c-><c- n>B</c-><c- p>,</c-> <c- mi>-1u</c-><c- p>)</c-> <c- o>>></c-> <c- mi>16</c-><c- p>;</c-></span><span class="line-no"></span><span class="line">    </span><span class="line-no"></span><span class="line">    <c- b>uint32_t</c-> <c- n>a</c-> <c- o>=</c-> <c- n>C</c-> <c- o>^</c-> <c- p>(</c-><c- n>i0</c-> <c- o>&amp;</c-> <c- n>D</c-><c- p>);</c-></span><span class="line-no"></span><span class="line">    <c- k>return</c-> <c- p>{</c-> <c- p>.</c-><c- n>x</c-> <c- o>=</c-> <c- n>a</c-> <c- o>^</c-> <c- n>i1</c-><c- p>,</c-> <c- p>.</c-><c- n>y</c-> <c- o>=</c-> <c- n>a</c-> <c- o>^</c-> <c- n>i0</c-> <c- o>^</c-> <c- n>i1</c-> <c- p>};</c-></span><span class="line-no"></span><span class="line"><c- p>}</c-></span></pre>
    <p>This specific example is taken from <a data-link-type="biblio" href="#biblio-rawrunprotected1" title="2D Hilbert curves in O(1)">[rawrunprotected1]</a>. <a data-link-type="biblio" href="#biblio-warren1" title="Hacker&apos;s Delight, Second Edition">[Warren1]</a> explains the basis behind this computation of Hilbert curves using bitwise operations.</p>
   </div>
   <p>When working with space-filling curves, the inverse operation is also common:
mapping the Cartesian coordinates onto an index on the curve.
In the case of Z-order curves aka. Morton curves,
this can be done by simply interleaving the bits of <code class="highlight"><c- n>x</c-></code> and <code class="highlight"><c- n>y</c-></code>.
A Z-order curve is laid out as follows:</p>
<pre class="language-text highlight">0 1 4 5
2 3 6 7
8 9 c d
a b e f
</pre>
   <div class="example" id="example-e67a5b9a">
    <a class="self-link" href="#example-e67a5b9a"></a> <code class="highlight"><c- n>clmul</c-></code> can be used to implement bit-interleaving in order to generate a <a href="https://en.wikipedia.org/wiki/Z-order_curve">Z-order curves</a>. 
<pre class="highlight line-numbered"><span class="line-no"></span><span class="line"><c- b>uint32_t</c-> <c- nf>xy_to_morton</c-><c- p>(</c-><c- b>uint32_t</c-> <c- n>x</c-><c- p>,</c-> <c- b>uint32_t</c-> <c- n>y</c-><c- p>)</c-></span><span class="line-no"></span><span class="line"><c- p>{</c-></span><span class="line-no highlight-line" data-line="3"></span><span class="line highlight-line">    <c- b>uint32_t</c-> <c- n>lo</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>clmul</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>x</c-><c- p>)</c-> <c- o>&lt;&lt;</c-> <c- mi>0</c-><c- p>;</c-> <c- c1>// abcd -> 0a0b0c0d</c-></span><span class="line-no highlight-line" data-line="4"></span><span class="line highlight-line">    <c- b>uint32_t</c-> <c- n>hi</c-> <c- o>=</c-> <c- n>std</c-><c- o>::</c-><c- n>clmul</c-><c- p>(</c-><c- n>y</c-><c- p>,</c-> <c- n>y</c-><c- p>)</c-> <c- o>&lt;&lt;</c-> <c- mi>1</c-><c- p>;</c-> <c- c1>// abcd -> a0b0c0d0</c-></span><span class="line-no"></span><span class="line">    <c- k>return</c-> <c- n>hi</c-> <c- o>|</c-> <c- n>lo</c-><c- p>;</c-></span><span class="line-no"></span><span class="line"><c- p>}</c-></span></pre>
   </div>
   <p class="note" role="note"><span class="marker">Note:</span> In the example above, <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>clmul</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>x</c-><c- p>)</c-></code> is equivalent to <a data-link-type="biblio" href="#biblio-p3104r3" title="P3104R3 Bit permutations">[P3104R3]</a>’s <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>bit_expand</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- mh>0x55555555u</c-><c- p>)</c-></code>.</p>
   <h2 class="heading settled" data-level="3" id="possible-implementation"><span class="secno">3. </span><span class="content">Possible implementation</span><a class="self-link" href="#possible-implementation"></a></h2>
   <p>A naive implementation looks as follows:</p>
<pre class="language-cpp highlight"><c- k>template</c-><c- o>&lt;</c-><c- n>unsigned_integral</c-> <c- n>T</c-><c- o>></c->
<c- k>constexpr</c-> <c- n>T</c-> <c- n>clmul</c-><c- p>(</c-><c- k>const</c-> <c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- k>const</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c->
<c- p>{</c->
    <c- n>T</c-> <c- n>result</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c->
    <c- k>for</c-> <c- p>(</c-><c- b>int</c-> <c- n>i</c-> <c- o>=</c-> <c- mi>0</c-><c- p>;</c-> <c- n>i</c-> <c- o>&lt;</c-> <c- n>numeric_limits</c-><c- o>&lt;</c-><c- n>T</c-><c- o>>::</c-><c- n>digits</c-><c- p>;</c-> <c- o>++</c-><c- n>i</c-><c- p>)</c-> <c- p>{</c->
        <c- n>result</c-> <c- o>^=</c-> <c- p>(</c-><c- n>x</c-> <c- o>&lt;&lt;</c-> <c- n>i</c-><c- p>)</c-> <c- o>*</c-> <c- p>((</c-><c- n>y</c-> <c- o>>></c-> <c- n>i</c-><c- p>)</c-> <c- o>&amp;</c-> <c- mi>1</c-><c- p>);</c->
    <c- p>}</c->
    <c- k>return</c-> <c- n>result</c-><c- p>;</c->
<c- p>}</c->
</pre>
   <h3 class="heading settled" data-level="3.1" id="hardware-support"><span class="secno">3.1. </span><span class="content">Hardware support</span><a class="self-link" href="#hardware-support"></a></h3>
   <p>The implementation difficulty lies mostly in utilizing available hardware instructions,
not in the naive fallback implementation.</p>
   <p>In the following table, let <code class="highlight"><c- n>uN</c-></code> denote <code class="highlight"><c- n>N</c-></code>-bit unsigned integer operands,
and <code class="highlight"><c- n>xN</c-></code> denote the amount of operands that are processed in parallel.</p>
   <table>
    <tbody>
     <tr>
      <th>Operation
      <th>x86_64
      <th>ARM
      <th>RV64
     <tr>
      <td><code class="highlight">clmul u64 -> u128</code>
      <td><code class="highlight"><c- n>pclmulqdq</c-></code><sup>PCLMULQDQ</sup>
      <td><code class="highlight"><c- n>pmull</c-></code>+<code class="highlight"><c- n>pmull2</c-></code><sup>Neon</sup>
      <td><code class="highlight"><c- n>clmul</c-></code>+<code class="highlight"><c- n>clmulh</c-></code><sup>Zbc, Zbkc</sup>
     <tr>
      <td><code class="highlight">clmul u64 -> u64</code>
      <td>
      <td><code class="highlight"><c- n>pmull</c-></code><sup>Neon</sup>
      <td><code class="highlight"><c- n>clmul</c-></code><sup>Zbc, Zbkc</sup>
     <tr>
      <td><code class="highlight">clmul u8x8 -> u16x8</code>
      <td>
      <td><code class="highlight"><c- n>pmull</c-></code><sup>Neon</sup>
      <td>
     <tr>
      <td><code class="highlight">clmul u8x8 -> u8x8</code>
      <td>
      <td><code class="highlight"><c- n>pmul</c-></code><sup>Neon</sup>
      <td>
   </table>
   <div class="example" id="example-efa3b485">
    <a class="self-link" href="#example-efa3b485"></a> A limited x86_64 implementation of <code class="highlight"><c- n>clmul_wide</c-></code> may look as follows: 
<pre class="language-cpp highlight"><c- cp>#include</c-> &lt;immintrin.h>
<c- cp>#include</c-> &lt;cstdint>

<c- n>mul_wide_result</c-><c- o>&lt;</c-><c- b>uint64_t</c-><c- o>></c-> <c- n>clmul_wide</c-><c- p>(</c-><c- b>uint64_t</c-> <c- n>x</c-><c- p>,</c-> <c- b>uint64_t</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c->
<c- p>{</c->
    <c- kr>__m128i</c-> <c- n>x_128</c-> <c- o>=</c-> <c- n>_mm_set_epi64x</c-><c- p>(</c-><c- mi>0</c-><c- p>,</c-> <c- n>x</c-><c- p>);</c->
    <c- kr>__m128i</c-> <c- n>y_128</c-> <c- o>=</c-> <c- n>_mm_set_epi64x</c-><c- p>(</c-><c- mi>0</c-><c- p>,</c-> <c- n>y</c-><c- p>);</c->
    <c- kr>__m128i</c-> <c- n>result_128</c-> <c- o>=</c-> <c- n>_mm_clmulepi64_si128</c-><c- p>(</c-><c- n>x_128</c-><c- p>,</c-> <c- n>y_128</c-><c- p>,</c-> <c- mi>0</c-><c- p>);</c->
    <c- k>return</c-> <c- p>{</c->
        <c- p>.</c-><c- n>low_bits</c->  <c- o>=</c-> <c- b>uint64_t</c-><c- p>(</c-><c- n>_mm_extract_epi64</c-><c- p>(</c-><c- n>result_128</c-><c- p>,</c-> <c- mi>0</c-><c- p>)),</c->
        <c- p>.</c-><c- n>high_bits</c-> <c- o>=</c-> <c- b>uint64_t</c-><c- p>(</c-><c- n>_mm_extract_epi64</c-><c- p>(</c-><c- n>result_128</c-><c- p>,</c-> <c- mi>1</c-><c- p>))</c->
    <c- p>};</c->
<c- p>}</c->
</pre>
   </div>
   <h2 class="heading settled" data-level="4" id="design-considerations"><span class="secno">4. </span><span class="content">Design Considerations</span><a class="self-link" href="#design-considerations"></a></h2>
   <p>Multiple design choices lean on <a data-link-type="biblio" href="#biblio-p0543r3" title="Saturation arithmetic">[P0543R3]</a> and <a data-link-type="biblio" href="#biblio-p3161r2" title="Unified integer overflow arithmetic">[P3161R2]</a>.
Specifically,</p>
   <ul>
    <li data-md>
     <p>the choice of header <code class="highlight"><c- o>&lt;</c-><c- n>numeric</c-><c- o>></c-></code>,</p>
    <li data-md>
     <p>the choice to have a widening operation,</p>
    <li data-md>
     <p>the <code class="highlight"><c- n>_wide</c-></code> naming scheme,</p>
    <li data-md>
     <p>the <code class="highlight"><c- n>mul_wide_result</c-></code> template, and</p>
    <li data-md>
     <p>the decision to have a <code class="highlight"><c- p>(</c-><c- n>T</c-><c- p>,</c-> <c- n>T</c-><c- p>)</c-></code> parameter list.</p>
   </ul>
   <h3 class="heading settled" data-level="4.1" id="naming"><span class="secno">4.1. </span><span class="content">Naming</span><a class="self-link" href="#naming"></a></h3>
   <p>Carry-less multiplication is also commonly called "Galois Field Multiplication" or "Polynomial Multiplication".</p>
   <p>The name <code class="highlight"><c- n>clmul</c-></code> was chosen because it carries no domain-specific connotation,
and because it is widespread:</p>
   <ul>
    <li data-md>
     <p>Intel refers to <code class="highlight"><c- n>PCLMULQDQ</c-></code> As "Carry-Less Multiplication Quadword" in its manual; see <a data-link-type="biblio" href="#biblio-intel2" title="Intel® 64 and IA-32 Architectures Software Developer&apos;s Manual">[Intel2]</a>.</p>
    <li data-md>
     <p>RISC-V refers to <code class="highlight"><c- n>clmul</c-></code> as carry-less multiplication, and this is obvious from the mnemonic.</p>
    <li data-md>
     <p>The Wikipedia article for this operation is titled "Carry-less product" <a data-link-type="biblio" href="#biblio-wikipedia1" title="Carry-less product">[Wikipedia1]</a>.</p>
   </ul>
   <h3 class="heading settled" data-level="4.2" id="simd"><span class="secno">4.2. </span><span class="content">SIMD</span><a class="self-link" href="#simd"></a></h3>
   <p>Many but not all use cases of carry-less multiplication operate on large blocks of data,
and SIMD is ideal for that purpose.
I do not yet propose carry-less multiplication for <code class="highlight"><c- n>std</c-><c- o>::</c-><c- n>simd</c-></code>,
but a future paper, or perhaps this paper could do so, if LEWG wants that.</p>
   <p class="note" role="note"><span class="marker">Note:</span> Only ARM supports eight 8-bit multiplications in parallel.
Other than that, hardware only supports scalar multiplications,
albeit using vector registers except for RISC-V;
see <a href="#hardware-support">§ 3.1 Hardware support</a>.</p>
   <h2 class="heading settled" data-level="5" id="proposed-wording"><span class="secno">5. </span><span class="content">Proposed wording</span><a class="self-link" href="#proposed-wording"></a></h2>
   <p>The proposed changes are relative to the working draft of the standard as of <a data-link-type="biblio" href="#biblio-n5001" title="Working Draft, Programming Languages — C++">[N5001]</a>,
with the changes in <a data-link-type="biblio" href="#biblio-p3161r2" title="Unified integer overflow arithmetic">[P3161R2]</a> applied.</p>
   <p>Update subclause <a href="https://eel.is/c++draft/version.syn">[version.syn]</a> paragraph 2 as follows:</p>
   <blockquote>
<pre class="highlight"><c- cp>#define __cpp_lib_clamp                             201603L </c-><c- c1>// also in &lt;algorithm></c->
<ins><c- cp>#define __cpp_lib_clmul                             20????L </c-><c- c1>// also in &lt;numeric></c-></ins>
<c- p>[...]</c->
<c- cp>#define __cpp_lib_overflow_arithmetic               </c-><del><c- cp>20????L</c-></del><c- cp> </c-><ins><c- cp>20????L</c-></ins><c- cp> </c-><c- c1>// also in &lt;numeric></c->
</pre>
   </blockquote>
   <p>In subclause <a href="https://eel.is/c++draft/numeric.ops.overview">[numeric.ops.overview]</a>,
update the synopsis as follows:</p>
   <blockquote>
<pre class="highlight"><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- p>,</c-> <c- k>class</c-> <c- nc>U</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>T</c-> <c- n>saturate_cast</c-><c- p>(</c-><c- n>U</c-> <c- n>x</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->              <c- c1>// freestanding</c->

<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
<c- k>struct</c-> <c- nc>add_carry_result</c-> <c- p>{</c->                               <c- c1>// freestanding</c->
  <c- n>T</c-> <c- n>low_bits</c-><c- p>;</c->
  <c- b>bool</c-> <c- n>overflow</c-><c- p>;</c->
<c- p>};</c->

<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
<c- k>using</c-> <c- n>sub_borrow_result</c-> <c- o>=</c-> <c- n>add_carry_result</c-><c- p>;</c->             <c- c1>// freestanding</c->

<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
<c- k>struct</c-> <c- nc>mul_wide_result</c-> <c- p>{</c->                                <c- c1>// freestanding</c->
  <c- n>T</c-> <c- n>low_bits</c-><c- p>;</c->
  <c- n>T</c-> <c- n>high_bits</c-><c- p>;</c->
<c- p>};</c->

<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
<c- k>struct</c-> <c- nc>div_result</c-> <c- p>{</c->                                     <c- c1>// freestanding</c->
  <c- n>T</c-> <c- n>quotient</c-><c- p>;</c->
  <c- n>T</c-> <c- n>remainder</c-><c- p>;</c->
<c- p>};</c->

<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>add_carry_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c->
    <c- n>add_carry</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>,</c-> <c- b>bool</c-> <c- n>carry</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->           <c- c1>// freestanding</c->
<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>sub_borrow_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c->
    <c- n>sub_borrow</c-><c- p>(</c-><c- n>T</c-> <c- n>left</c-><c- p>,</c-> <c- n>T</c-> <c- n>right</c-><c- p>,</c-> <c- b>bool</c-> <c- n>borrow</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->  <c- c1>// freestanding</c->
<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>mul_wide_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c->
    <c- n>mul_wide</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->                        <c- c1>// freestanding</c->
<ins><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>mul_wide_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c->
    <c- n>clmul_wide</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->                      <c- c1>// freestanding</c-></ins>
<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>div_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c->
    <c- n>div_wide</c-><c- p>(</c-><c- n>T</c-> <c- n>dividend_high</c-><c- p>,</c-> <c- n>T</c-> <c- n>dividend_low</c-><c- p>,</c->           <c- c1>// freestanding</c->
             <c- n>T</c-> <c- n>divisor</c-> <c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->

<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- b>bool</c->
    <c- n>is_div_defined</c-><c- p>(</c-><c- n>T</c-> <c- n>dividend</c-><c- p>,</c-> <c- n>T</c-> <c- n>divisor</c-><c- p>)</c-> <c- k>noexcept</c->      <c- c1>// freestanding</c->
<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- b>bool</c->
    <c- n>is_div_wide_defined</c-><c- p>(</c-><c- n>T</c-> <c- n>dividend_high</c-><c- p>,</c->                <c- c1>// freestanding</c->
                        <c- n>T</c-> <c- n>dividend_low</c-><c- p>,</c->
                        <c- n>T</c-> <c- n>divisor</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->

<ins>
<c- c1>// [numeric.clmul], carry-less product</c->
<c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
  <c- k>constexpr</c-> <c- n>T</c-> <c- n>clmul</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>U</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->                 <c- c1>// freestanding</c->
</ins></pre>
   </blockquote>
   <p>In subclause <a href="https://eel.is/c++draft/numeric.overflow">[numeric.overflow]</a> (known as <a href="https://eel.is/c++draft/numeric.sat">[numeric.sat]</a> at the time of writing),
insert the following, immediately following the description of <code class="highlight"><c- n>mul_wide</c-></code>:</p>
   <blockquote class="ins-block">
<pre class="highlight"><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
   <c- k>constexpr</c-> <c- n>mul_wide_result</c-><c- o>&lt;</c-><c- n>T</c-><c- o>></c-> <c- n>clmul_wide</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->
</pre>
    <p class="indent"> Let ⊕ denote the exclusive bitwise OR operation ([expr.xor]).
    Let <em>x</em><sub><em>n</em></sub> and <em>y</em><sub><em>n</em></sub> denote the <em>n</em><sup>th</sup> least significant bit
    in the base-2 representation of <code class="highlight"><c- n>x</c-></code> and <code class="highlight"><c- n>y</c-></code> respectively.
    Let <em>w</em> be the width of <code class="highlight"><c- n>T</c-></code>. </p>
    <p class="indent"> <em>Constraints:</em> <code class="highlight"><c- n>T</c-></code> is an unsigned integer type ([basic.fundamental]). </p>
    <p class="indent"> <em>Returns:</em> Let <em>c</em> be an integer,
    where the value of the <em>n</em><sup>th</sup> least significant bit in the base-2 representation of <em>c</em>, <em>c</em><sub><em>n</em></sub>, is given by Formula ?.? below.
    The result object is initialized so that </p>
    <ul class="indent">
     <li>the bits of <code class="highlight"><c- n>low_bits</c-></code> are the <em>w</em> least significant bits of <em>c</em>, and
     <li>the bits of <code class="highlight"><c- n>high_bits</c-></code> are subsequent bits of <em>c</em>.
    </ul>
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   </blockquote>
   <p class="note" role="note"><span class="marker">Note:</span> Italicized text such as <em>c</em> shall be in math font.</p>
   <p>In subclause <a href="https://eel.is/c++draft/numeric.ops">[numeric.ops]</a>,
append a subclause immediately following <a href="https://eel.is/c++draft/numeric.overflow">[numeric.overflow]</a> (known as <a href="https://eel.is/c++draft/numeric.sat">[numeric.sat]</a> at the time of writing):</p>
   <blockquote class="ins-block">
    <p> <b>Carry-less product [numeric.clmul]</b> </p>
<pre class="highlight"><c- k>template</c-><c- o>&lt;</c-><c- k>class</c-> <c- nc>T</c-><c- o>></c->
   <c- k>constexpr</c-> <c- n>T</c-> <c- n>clmul</c-><c- p>(</c-><c- n>T</c-> <c- n>x</c-><c- p>,</c-> <c- n>T</c-> <c- n>y</c-><c- p>)</c-> <c- k>noexcept</c-><c- p>;</c->
</pre>
    <p class="indent"> <em>Constraints:</em> <code class="highlight"><c- n>T</c-></code> is an unsigned integer type ([basic.fundamental]). </p>
    <p class="indent"> <em>Returns:</em> <code class="highlight"><c- n>clmul_wide</c-><c- p>(</c-><c- n>x</c-><c- p>,</c-> <c- n>y</c-><c- p>).</c-><c- n>low_bits</c-></code> ([numeric.overflow]). </p>
   </blockquote>
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  <h2 class="no-num no-ref heading settled" id="references"><span class="content">References</span><a class="self-link" href="#references"></a></h2>
  <h3 class="no-num no-ref heading settled" id="normative"><span class="content">Normative References</span><a class="self-link" href="#normative"></a></h3>
  <dl>
   <dt id="biblio-n5001">[N5001]
   <dd>Thomas Köppe. <a href="https://wg21.link/n5001"><cite>Working Draft, Programming Languages — C++</cite></a>. 17 December 2024. URL: <a href="https://wg21.link/n5001">https://wg21.link/n5001</a>
   <dt id="biblio-p3161r2">[P3161R2]
   <dd>Tiago Freire. <a href="https://wg21.link/p3161r2"><cite>Unified integer overflow arithmetic</cite></a>. 15 July 2024. URL: <a href="https://wg21.link/p3161r2">https://wg21.link/p3161r2</a>
  </dl>
  <h3 class="no-num no-ref heading settled" id="informative"><span class="content">Informative References</span><a class="self-link" href="#informative"></a></h3>
  <dl>
   <dt id="biblio-intel1">[Intel1]
   <dd>Shay Gueron; Michael E. Kounavis. <a href="https://www.intel.com/content/dam/develop/external/us/en/documents/clmul-wp-rev-2-02-2014-04-20.pdf"><cite>Intel® Carry-Less Multiplication Instruction and its Usage for Computing the GCM Mode</cite></a>. URL: <a href="https://www.intel.com/content/dam/develop/external/us/en/documents/clmul-wp-rev-2-02-2014-04-20.pdf">https://www.intel.com/content/dam/develop/external/us/en/documents/clmul-wp-rev-2-02-2014-04-20.pdf</a>
   <dt id="biblio-intel2">[Intel2]
   <dd>Intel Corporation. <a href="https://software.intel.com/en-us/download/intel-64-and-ia-32-architectures-sdm-combined-volumes-1-2a-2b-2c-2d-3a-3b-3c-3d-and-4"><cite>Intel® 64 and IA-32 Architectures Software Developer's Manual</cite></a>. URL: <a href="https://software.intel.com/en-us/download/intel-64-and-ia-32-architectures-sdm-combined-volumes-1-2a-2b-2c-2d-3a-3b-3c-3d-and-4">https://software.intel.com/en-us/download/intel-64-and-ia-32-architectures-sdm-combined-volumes-1-2a-2b-2c-2d-3a-3b-3c-3d-and-4</a>
   <dt id="biblio-p0543r3">[P0543R3]
   <dd>Jens Maurer. <a href="https://wg21.link/p0543r3"><cite>Saturation arithmetic</cite></a>. 19 July 2023. URL: <a href="https://wg21.link/p0543r3">https://wg21.link/p0543r3</a>
   <dt id="biblio-p3104r3">[P3104R3]
   <dd>Jan Schultke. <a href="https://wg21.link/p3104r3"><cite>P3104R3 Bit permutations</cite></a>. URL: <a href="https://wg21.link/p3104r3">https://wg21.link/p3104r3</a>
   <dt id="biblio-rawrunprotected1">[RAWRUNPROTECTED1]
   <dd>rawrunprotected. <a href="http://threadlocalmutex.com/?p=188"><cite>2D Hilbert curves in O(1)</cite></a>. URL: <a href="http://threadlocalmutex.com/?p=188">http://threadlocalmutex.com/?p=188</a>
   <dt id="biblio-schultke1">[Schultke1]
   <dd>Jan Schultke. <a href="https://github.com/Eisenwave/cxx26-bit-permutations"><cite>C++26 Bit permutations reference implementation</cite></a>. URL: <a href="https://github.com/Eisenwave/cxx26-bit-permutations">https://github.com/Eisenwave/cxx26-bit-permutations</a>
   <dt id="biblio-warren1">[Warren1]
   <dd>Henry S. Warren, Jr. <a href="https://doc.lagout.org/security/Hackers%20Delight.pdf"><cite>Hacker's Delight, Second Edition</cite></a>. URL: <a href="https://doc.lagout.org/security/Hackers%20Delight.pdf">https://doc.lagout.org/security/Hackers%20Delight.pdf</a>
   <dt id="biblio-wikipedia1">[Wikipedia1]
   <dd>Wikipedia community. <a href="https://en.wikipedia.org/wiki/Carry-less_product"><cite>Carry-less product</cite></a>. URL: <a href="https://en.wikipedia.org/wiki/Carry-less_product">https://en.wikipedia.org/wiki/Carry-less_product</a>
  </dl>