RSA Coding Continued

For homework you read RSA Coding. Let's use the same methods in order to analyze the coded message from the article that we discussed on Tuesday. The following were made public in the article:

Coded Message


Encoding Number e


Composite Number n

Gardner said that it would millions of years in order to decode the message at 1977 computer speeds. In order to decode the message, we need d since the decoded message would equal message ^d mod n. There is no way to solve for d unless we can first decompose n into its prime factors p and q. But, this takes a lot of computer power. One could work for years and years and still never find p and q. We need p and q so that we can solve e d mod (p-1)(q-1) = 1 for d. In 1993, Arjen Lenstra a Belcore formed an international group to factor n and break the code. With 600 volunteers using 1600 workstations, mainframes, and supercomputers they broke the code in 6 months by factoring the 129 digit number n. In April 26, 1994, the presented the results and collected their $100 prize.

Factors of n
p = 32769132993266709549961988190834461413177642967992942539798288533 and
q = 3490529510847650949147849619903898133417764638493387843990820577 . You can check to see that pq = n.

Decoding Number d Once we have p and q, we can solve for
d = 106698614368578024442868771328920154780709906633937862801226224496631063125911
I solved for d by using a computer algebra system called Maple.

Decoding the Message We then compute message ^ d mod n via the square and reduce algorithm, as computers can not compute this directly since the number message ^ d is too big. The decoded message is

Turning the Decoded Message into Words Finally, we change the decoded message into letters via the conversion
00->_   01->a   02->b   03->c   04->d   05->e   06->f   07->g   08->h   09->i   10->j   11->k   12->l   13->m   14->n   15->0   16->p   17->q   18->r   19->s   20->t   21->u   22->v   23->w   24->x   25->y   26->z.