مروری بر عملیات معکوس در میدان‌های متناهی دودویی و اول
الموضوعات :
رضا روح قلندری
1
(دانشگاه صنعتی شریف)
حاتمه مثنایی بورانی
2
(دانشگاه صنعتی شریف)
سیاوش بیات سرمدی
3
(دانشگاه صنعتي شریف)
الکلمات المفتاحية: الگوریتم‌های معکوسمیدان‌های متناهیقضیه کوچک فِرماالگوریتم اقلیدسی تعمیم‌یافته,
ملخص المقالة :
رمزنگاری کلید عمومی یکی از روش‌های مرسوم رمزنگاری است که به دلیل عدم نیاز به تبادل کلید، در سال‌های اخیر بسیار مورد توجه قرار گرفته است. روش‌هایی مانند توان‌رسانی، جفت‌سازی، رمزنگاری خم بیضوی و همگونی در این دسته قرار می‌گیرند که تاکنون پژوهش‌های زیادی برای کاهش پیچیدگی زمانی و مساحت این روش‌ها صورت گرفته است. عملیات معکوس به عنوان یکی از اصلی‌ترین اعمال موجود در روش‌های رمزنگاری کلید عمومی است که بخش زیادی از پیچیدگی محاسباتی و زمانی را در این پردازنده‌های رمزنگاری به خود اختصاص می‌دهد. بنابراین برای افزایش کارایی و سرعت پردازنده‌های رمزنگاری کلید عمومی، بهبود سرعت و مساحت عملیات معکوس در میدان‌های متناهی بسیار ضروری به نظر می‌رسد. در این مقاله، روش‌های موجود برای انجام عملیات معکوس بر روی میدان‌های دودویی و اول مورد بررسی قرار گرفته است. در سامانه‌های رمزنگاری امروزی، میدان‌های دودویی به دلیل سازگاری با سخت‌افزار، بسیار پرکاربرد هستند. در ابتدای این نوشتار، روش‌های موجود برای انجام عملیات معکوس در میدان‌های دودویی بررسی، الگوریتم‌های موجود بیان و از نظر پیچیدگی زمانی و منابع مورد نیاز با هم مقایسه شده است. سپس بهترین پیاده‌سازی‌های موجود روی بستر FPGA و به صورت مدار مجتمع معرفی می‌گردد. هدف اصلی در این روش‌ها کاهش تعداد ضرب مورد نیاز برای انجام عملیات معکوس در میدان دودویی و افزایش امکان توازی برای پیاده‌سازی هرچه بهتر این روش‌ها است. میدان‌های اول به دلیل پیچیدگی ساختاری و محاسباتی بیشتر نسبت به میدان‌های دودویی، تنوع و گستردگی کمتری دارند ولی در سالیان اخیر به دلیل ظهور کاربردهای جدیدی در رمزنگاری نظیر روش همگونی، مورد توجه بیشتری قرار گرفته‌اند. محققان این حوزه در تلاش هستند تا ضمن از بین بردن وابستگی زمان اجرای روش‌های موجود به مقدار ورودی، پیچیدگی زمانی و مساحت الگوریتم‌های محاسبه معکوس را تا حد امکان کاهش دهند. ارائه ساختارهایی نظیر آرایه ضربانی در همین راستا صورت گرفته که این مقاله به بررسی این روش‌ها می‌پردازد و در انتها روش‌های انجام عملیات معکوس را در میدان‌های اول از نظر پیچیدگی زمانی و محاسباتی با هم مقایسه می‌کند.
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