اميد رياضي نرخ پوشش براي ماتريس‌هاي هلمن

Expected coverage rate for the Hellman matrices

مصالحه حافظه‌زمان يك الگوريتم احتمالاتي براي وارون‌كردن توابع يك‌طرفه، با استفاده از داده‌هاي از پيش محاسبه‌شده است. هلمن در سال 1980 اين روش را معرفي كرد و يك كران پايين براي احتمال موفقيت آن به‌دست آورد. پس از آن نيز تحليل‌هاي پژوهش‌گران براي بررسي احتمال موفقيت، بر اساس همين كران پايين بوده است. در اين مقاله، ابتدا به بررسي اميد رياضي نرخ پوشش يك ماتريس هلمن مي‌پردازيم؛ سپس، اين نرخ را براي ماتريس‌هاي هلمني كه فقط از يك زنجيره تشكيل شده‌اند، محاسبه كرده‌ايم. نشان داده‌ايم كه اميد رياضي نرخ پوشش براي چنين ماتريس‌هايي بيشينه و برابر با 85/0 است. در ادامه، روش‌هايي براي تخمين دقيق‌تر اين نرخ ارائه، و با روش هلمن مقايسه و در‌نهايت، ماتريس‌هاي هلمني را معرفي كرده‌ايم كه فقط از يك زنجيره تشكيل شده‌اند؛ ولي طول اين زنجيره مقداري ثابت نيست و تا رسيدن به نخستين تكرار ادامه مي‌يابد. به‌صورت نظري و عملي نشان داده‌ايم كه احتمال موفقيت براي چنين ماتريس‌هايي بيشتر از روش هلمن است.

Hellman’s time-memory trade-off is a probabilistic method for inverting one-way functions, using pre-computed data. Hellman introduced this method in 1980 and obtained a lower bound for the success probability of his algorithm. After that, all further analyses of researchers are based on this lower bound. In this paper, we first studied the expected coverage rate (ECR) of the Hellman matrices, which are constructed by a single chain. We showed that the ECR of such matrices is maximum and equal to 0.85. In this process, we find out that there exists a gap between the Hellman’s lower bound and experimental coverage rate of a Hellman matrix. Specifically, this gap is larger, when considering the Hellman matrices constructed with one single chain. So, we are investigated to obtain an accurate formula for the ECR of a Hellman matrix. Subsequently, we presented a new formula that estimate the ECR of a Hellman matrix more accurately than the Hellman’s lower bound. We showed that the given formula is closely match experimental data. In the last, we introduced a new method to construct matrices which have much more ECR than Hellman matrices. In fact, each matrix in this new method is constructed with one single chain, which is non-repeating trajectory from a random point. So, this approach result in a number of matrices that each one contains a chain with variable length. The main advantage of this method is that we have more probability of success than Hellman method, however online time and memory requirements are increased. We have also verified theory of this new method with experimental results.