When Pow Is Good

In cryptography we deal with large numbers … in fact very large numbers. While are processors have registers that hold 64 bits, and our integer values in programs are stored as 64-bit values, in crypto land these values are normally constrained only by the length of a large prime number. And, for example, if we operate on two 64-bit values, the result will be 128 bits in length, and we do not allow it to overflow (as we would in our normal operations).

A 2,048 bit value (and which is often used in RSA) is something of the form (2²⁰⁴⁸):

32,317,006,071,311,007,300,714,876,688,669,951,960,444,102,669,715,484,032,130,345,427,524,655,138,867,890,893,197,201,411,522,9 13,463,688,717,960,921,898,019,494,119,559,150,490,921,095,088,152,386,448,283,120,630,877,367,300,996,091,750,197,750,389,652,1 06,796,057,638,384,067,568,276,792,218,642,619,756,161,838,094,338,476,170,470,581,645,852,036,305,042,887,575,891,541,065,808,6 07,552,399,123,930,385,521,914,333,389,668,342,420,684,974,786,564,569,494,856,176,035,326,322,058,077,805,659,331,026,192,708,4 60,314,150,258,592,864,177,116,725,943,603,718,461,857,357,598,351,152,301,645,904,403,697,613,233,287,231,227,125,684,710,820,2 09,725,157,101,726,931,323,469,678,542,580,656,697,935,045,997,268,352,998,638,215,525,166,389,437,335,543,602,135,433,229,604,6 45,318,478,604,952,148,193,555,853,611,059,596,230,656