Cryptographic Applications - Introduction to Cryptography with Java Applets - page 322

9356433725290818322063237477659539960679619018282576265414797224176810239
4461794720293192639

e=10738975467716313016226411982616063375096935280096973378820148373901000
0377470313772198875565203421763473641679025697618732906800200949388901714
4337965064160547094535922354431198904536464821168309068889027430457499760
1091047270120113110989646463412317973823604744394241847030235719949845268
584917833408071651

d=34120427819749165542258873630680243843885206912393395291704730575235416
8499010091359026204518569891972265492221968149078409284214574471096638649
6508925391380739896819274937783290922865761442395151744157771365693061239
9916291158832430035510226908257536825863178385774457780384766899733604450
9358887363459579267

Doubly enciphered signed message:

Ü¡… ??-au¡?(dRc(µß¶s«ÙáàóR.K>; ¸> ÿÍSE-ÌC9B ¿cd_øT»%ìß?{ ŒW9 Q
n 1 ⁄ 2 ?Á?‡ü¿/CH30ôÜ

Áµ?)ÙjÁ™<R-1@øA
£µ ù¯{èüÜ\±by|© -R,ÙèYpL0â?Sú Ò Æôs22!H 3 ˆÕ a KP Ë)´ ã¬- 1 ⁄ 4

áœ1«%Q ¶v 2 y E=ùu-¸Zcê™7ß- ëkû Õ ŸÛf¨--P ú pR MY9@/+

MÓíË x a =z}©xÅG âñ a B Yå ´ YÀè Ñqp ?± _?W 1

⁄ 2 ÍúŸæœ p

†™1
Doubly deciphered signed message:
Little Willy Willy won't GO HOME!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!

Signing with Rabin

Rabin signatures are simple to produce. Suppose individual A

P

wants to send a message to individual B using Rabin in such a way so that B knows the

message could only have come from A. Suppose A uses the Rabin modulus

n pq

= , while

* . (The primes involved now are all congruent to 3 modulo 4, as required by
Rabin.) Of course, neither party knows the other's private key. A does the following:
1.

n

* =

p

*

q

B uses

A computes a square root of P modulo n (if P has such a square root); that is, she com-
putes a value, say C 1 , such that

C 1 2

P (mod n ).
This transformation can produce up to four roots; it doesn't matter which root she chooses.
(No one else can do this if A is protecting her primes p and q , as computing a square root
modulo n without its prime factorization is an intractable problem.)
Note that a particular message may not have a square root modulo n . The sender
must salt the message (or salt each block) in such a way that the salted result has a square
root modulo n . This isn't difficult, because the odds of some random integer having a
square root modulo n is quite likely. The amount of salt to use for this purpose is agreed
on beforehand.