[{"data":1,"prerenderedAt":-1},["ShallowReactive",2],{"doc-detail-86203-en":3,"doc-seo-86203-105":28,"detail-sidebar-cat-0-en-105":90},{"code":4,"msg":5,"data":6},0,"success",{"doc_id":7,"user_id":8,"nickname":9,"user_avatar":10,"doc_module":4,"category_id":11,"category_name":12,"doc_title":13,"doc_description":14,"doc_content":15,"file_id":16,"file_url":17,"file_type":18,"file_size":19,"view_count":20,"is_deleted":4,"is_public":20,"is_downloadable":20,"audit_status":20,"page_count":11,"language":21,"language_code":22,"site_id":23,"html_lang":22,"table_of_contents":24,"faqs":25,"seo_title":13,"seo_description":14,"update_tm":26,"read_time":27},86203,1374391974564,"Clementine","https://ap-avatar.wpscdn.com/avatar/14000253aa45c000a9e?x-image-process=image/resize,m_fixed,w_180,h_180&k=1779874745381141002",8,"Research & Report","Low latency data-flow graphs for simultaneous modular inversion of many inputs","Low latency data-flow graph constructions accelerate Montgomery’s trick for simultaneous modular inversion of N inputs by transforming auxiliary complement-product multiplications from a typical linear schedule into a maximally parallelizable computation. By scheduling x complement products into idle multiplier slots during accumulation of the full product Q and the shared inversion, the post-inversion phase adds exactly one multiplication layer of latency, independent of N. The resulting critical path is ⌈log2 N⌉ multiply layers, one inversion, and one final parallel multiplication layer, with constant-time suitability for MPC batch workloads.","arXiv :2607 . 1 1337v 1 [ cs .DS] 13 Jul 2026  \nLow latency data-flow graphs for simultaneous modular inversion of many inputs  \nTam´as Visegr´ady  \nRipple Labs Research  \n[tvisegrady@ripple. com](tvisegrady@ripple. com)  \n2026-07-14  \nAbstract  \nMontgomery’s trick accelerates simultaneous modular inversion of N inputs by amortizing a single shared inversion, but auxiliary multiplications for complement products are typically scheduled in a linear, serial form. We construct a maximally parallelizable data-flow graph (DFG) that computes all x complement products by scheduling auxiliary multiplications into idle multiplier slots during accumulation of the product of all inputs, and that of the shared inversion. This scheduling ensures the post-inversion phase adds exactly one multiplication layer of latency regardless of N, yielding a critical path latency of ⌈log2 N⌉ multiply layers, one inversion, and one final parallel multiply layer.  \nMeasurements on an AMD Ryzen AI PRO 350 using libsecp256k1 primitives yield 1 .27× single-inversion latency for N = 8 and 1 .34× for N = 16, with peak multiplier load (> N/2 simultaneous multiplications) lasting only 80 ns out of roughly 1500–1600 ns total—considerably less than replicating N inversions would require. The construction suits latency-sensitive batch workloads such as those related to MPC presignature generation; it supports constant-time execution when the underlying primitives are constant-time.  \n1 Overview  \nSimultaneous modular inversions using the same modulus may be accelerated using “Montgomery’s trick,” adding modular multiplications to amortize the high cost of a single shared inversion [LMN10, 4.1] . This batching technique, extrapolated from a few inputs, is typically implemented with linear sequences of multiplications [Tom26] . Significant latency improvements have been demonstrated for single-input inversion [BCH+26, 1.2]; we consider latency reduction specific to the multi-input case.  \nFor inputs a and b, Montgomery’s trick calculates Q = a · b, 1/Q = 1/(a · b), 1/a = b · 1/Q, and 1/b = a · 1/Q, saving an inversion while adding three multiplications. Straightforward extrapolation to a higher number of inputs tends to be implemented in an inherently serial form. For many simultaneous inversions, one may construct an equivalent, parallelizable dataflow graph (DFG) which decreases overall latency at the cost of adding many simultaneous multiplications. We use x—“notX” in figures—to denote the product of all inputs except x, so 1/x = x ·1/Q (Q is the full product) . We schedule auxiliary multiplications for x in idle periods  \nfor multipliers, such that post-inversion steps add only one multiplication’s latency, regardless of the number of outputs—if sufficient multipliers are available.  \nSimultaneous inversion necessarily computes Q and all its partial products on the critical path. With two-input modular multipliers as building blocks, the critical path includes ⌈log2 N⌉ layers of multiplications [Arn24] for N inputs. The constant single-multiplication latency of final 1/x = x · 1/Q products is an improvement over straightforward serial or even tree-structured reconstruction [Arn24], made possible by computing products while inversion is running.  \nParallel computation of x terms is suitable for cases where bursts of peak load may be tolerated, and low latency is at a premium. Certain forms of batched presignature generation for MPC protocols [KU24, 6], or inverting k nonces for batch-generated ECDSA signatures are potential use cases where one may consider multi-input inversion. While we may use up to N multipliers simultaneously for minimal latency, since we parallelize only multiplications—but not the longer-running inversion—we generate shorter periods of peak load than replicating Ninversions would do.  \n1.1 Scheduling multiplications not on the critical path  \nUnlike Q, not all multiplications of x are on the critical path. There are several reasons the sche","cbCaiauq5B8x8oGe","https://ap.wps.com/l/cbCaiauq5B8x8oGe","pdf",237534,1,"English","en",105,"# Overview\n## Scheduling multiplications not on the critical path\n# Examples for N = 8 and N = 16 inputs","[{\"question\":\"What problem does the document address?\",\"answer\":\"It addresses latency inefficiency when using Montgomery’s trick for simultaneous modular inversion of many inputs, especially the typically linear, serial scheduling of auxiliary complement-product multiplications.\"},{\"question\":\"How does the proposed data-flow graph reduce latency?\",\"answer\":\"It schedules the auxiliary multiplications for complement products during idle multiplier periods while accumulating the full product Q and running the shared inversion, so the post-inversion stage costs only one multiplication layer regardless of N.\"},{\"question\":\"What are the key performance and suitability claims?\",\"answer\":\"Measurements on AMD Ryzen AI PRO 350 using libsecp256k1 show improved latency scaling (e.g., 1.27× for N=8 and 1.34× for N=16). The construction is intended for latency-sensitive batch workloads such as MPC presignature generation and supports constant-time execution when underlying primitives are constant-time.\"}]",1784209399,20,{"code":4,"msg":29,"data":30},"ok",{"site_id":23,"language":22,"slug":31,"title":13,"keywords":32,"description":14,"schema_data":33,"social_meta":85,"head_meta":87,"extra_data":89,"updated_unix":26},"low-latency-data-flow-graphs-for-simultaneous-modular-inversion-of-many-inputs","",{"@graph":34,"@context":84},[35,52,67],{"@type":36,"itemListElement":37},"BreadcrumbList",[38,42,46,49],{"item":39,"name":40,"@type":41,"position":20},"https://docshare.wps.com","Home","ListItem",{"item":43,"name":44,"@type":41,"position":45},"https://docshare.wps.com/document/","Document",2,{"item":47,"name":12,"@type":41,"position":48},"https://docshare.wps.com/document/research-report/",3,{"item":50,"name":13,"@type":41,"position":51},"https://docshare.wps.com/document/low-latency-data-flow-graphs-for-simultaneous-modular-inversion-of-many-inputs/86203/",4,{"url":50,"name":13,"@type":53,"author":54,"headline":13,"publisher":56,"fileFormat":59,"inLanguage":22,"description":14,"dateModified":60,"datePublished":61,"encodingFormat":59,"isAccessibleForFree":62,"interactionStatistic":63},"DigitalDocument",{"name":9,"@type":55},"Person",{"url":39,"name":57,"@type":58},"DocShare","Organization","application/pdf","2026-07-17","2026-07-16",true,{"@type":64,"interactionType":65,"userInteractionCount":20},"InteractionCounter",{"@type":66},"ViewAction",{"@type":68,"mainEntity":69},"FAQPage",[70,76,80],{"name":71,"@type":72,"acceptedAnswer":73},"What problem does the document address?","Question",{"text":74,"@type":75},"It addresses latency inefficiency when using Montgomery’s trick for simultaneous modular inversion of many inputs, especially the typically linear, serial scheduling of auxiliary complement-product multiplications.","Answer",{"name":77,"@type":72,"acceptedAnswer":78},"How does the proposed data-flow graph reduce latency?",{"text":79,"@type":75},"It schedules the auxiliary multiplications for complement products during idle multiplier periods while accumulating the full product Q and running the shared inversion, so the post-inversion stage costs only one multiplication layer regardless of N.",{"name":81,"@type":72,"acceptedAnswer":82},"What are the key performance and suitability claims?",{"text":83,"@type":75},"Measurements on AMD Ryzen AI PRO 350 using libsecp256k1 show improved latency scaling (e.g., 1.27× for N=8 and 1.34× for N=16). The construction is intended for latency-sensitive batch workloads such as MPC presignature generation and supports constant-time execution when underlying primitives are constant-time.","https://schema.org",{"og:url":50,"og:type":86,"og:title":13,"og:site_name":57,"og:description":14},"article",{"robots":88,"canonical":50},"index,follow",{"doc_id":7,"site_id":23},{"code":4,"msg":5,"data":91},[92,96,100,104,109,114,119,122,126,129,133],{"id":20,"doc_module":4,"doc_module_name":44,"category_name":93,"show_sort_weight":94,"slug":95},"Story & Novel",90,"story-novel",{"id":45,"doc_module":4,"doc_module_name":44,"category_name":97,"show_sort_weight":98,"slug":99},"Literature",80,"literature",{"id":51,"doc_module":4,"doc_module_name":44,"category_name":101,"show_sort_weight":102,"slug":103},"Exam",70,"exam",{"id":105,"doc_module":4,"doc_module_name":44,"category_name":106,"show_sort_weight":107,"slug":108},5,"Comic",60,"comic",{"id":110,"doc_module":4,"doc_module_name":44,"category_name":111,"show_sort_weight":112,"slug":113},6,"Technology",50,"technology",{"id":115,"doc_module":4,"doc_module_name":44,"category_name":116,"show_sort_weight":117,"slug":118},7,"Healthcare",40,"healthcare",{"id":11,"doc_module":4,"doc_module_name":44,"category_name":12,"show_sort_weight":120,"slug":121},30,"research-report",{"id":123,"doc_module":4,"doc_module_name":44,"category_name":124,"show_sort_weight":27,"slug":125},9,"Religion & Spirituality","religion-spirituality",{"id":27,"doc_module":4,"doc_module_name":44,"category_name":127,"show_sort_weight":27,"slug":128},"World Cup","world-cup",{"id":130,"doc_module":4,"doc_module_name":44,"category_name":131,"show_sort_weight":130,"slug":132},10,"Lifestyle","lifestyle",{"id":134,"doc_module":4,"doc_module_name":44,"category_name":135,"show_sort_weight":105,"slug":136},19,"General","general"]