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{SECT 0 {EXCHG {PARA 256 "" 0 "" {TEXT -1 25 "1. An RSA public key has
 " }{TEXT 256 9 "n = 25021" }{TEXT -1 5 " and " }{TEXT 257 6 "a = 29" 
}{TEXT -1 38 ".  Find the corresponding private key." }}}{EXCHG {PARA 
0 "> " 0 "" {MPLTEXT 1 0 20 "n := 25021; a := 29;" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"nG\"&@]#" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"aG
\"#H" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "ifactor(n);" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#*&-%!G6#\"$J\"\"\"\"-F%6#\"$\">F(" }}}
{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "p := 131; q := 191;" }}
{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"pG\"$J\"" }}{PARA 11 "" 1 "" 
{XPPMATH 20 "6#>%\"qG\"$\">" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 
19 "phi := (p-1)*(q-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$phiG\"&+
Z#" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "x := modp(1/a, phi);
" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"xG\"%p$*" }}}{EXCHG {PARA 0 "
" 0 "" {TEXT -1 30 "Therefore the private key has " }{TEXT 266 9 "n = \+
25021" }{TEXT -1 5 " and " }{TEXT 267 8 "x = 9369" }{TEXT -1 1 "." }}}
{EXCHG {PARA 257 "" 0 "" {TEXT -1 22 "2. Encode the integer " }
{XPPEDIT 18 0 "beta;" "6#%%betaG" }{TEXT -1 1 " " }{TEXT 264 5 "= 271
" }{TEXT -1 97 " using the public key.  What do you get? Decode this u
sing the private key you found. Do you get " }{TEXT 265 3 "271" }
{TEXT -1 6 " back?" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 12 "beta \+
:= 271;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%%betaG\"$r#" }}}{EXCHG 
{PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "m := modp(beta &^ a, n);" }}{PARA 
11 "" 1 "" {XPPMATH 20 "6#>%\"mG\"%Jf" }}}{EXCHG {PARA 0 "> " 0 "" 
{MPLTEXT 1 0 16 "modp(m &^ x, n);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#
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