Exercises: RSA Sign / Verify

In this exercise we shall sign messages and verify signatures using the PKCS#1 v.1.5 RSA signature algorithm with 4096-bit keys, following the technical specification from RFC 8017, using SHA3-512 for hashing the input message. The RSA-PKCS1 v1.5 digital signature algorithm can be found as library for the most programming languages.

The RSA private key will be given encoded in PEM format (RFC 7468, see the example). The corresponding RSA public key will also be given encoded in PEM format. The RSA signature is 4096-bit integer (1024 hex digits).

Sign a Message with RSA

Write a program to sign given text message with given 4096-bit private key, using the PKCS#1 v.1.5 RSA signature algorithm with SHA3-512 hashing for the input message. The input consists of 2 text lines. The first line holds the input message for signing. The next few lines holds the private key as hex string. Print the output as JSON document, holding the input message + the public key of the signer (as hex string) + the RSA digital signature (as hex string).

Sample input (text message + PEM-encoded RSA private key):

Message for RSA signing
-----BEGIN RSA PRIVATE KEY-----
MIIJJwIBAAKCAgEAoYjbzkMmss6BU/eCBYhs3bFRBVldM9DlaM7WhPZEgmYl7C1O
b0kmEPG2O/GaNoBOnwptHykzvpYS42rr/D8I0LeJ/2dz0uStQv6kQDIDmNRTthW9
spcuBsqW3e1bRyf0JrL8ot1X8M4k0NGU7w508htPzRZXRS2Zonv8G/N35ePDV5Xf
5mrcOPOebconndXHRqfLlyXCxCS5x0qlJGX1d/oe2Xop7VfBP/UhrrmNmstZsBEv
L3f33+a482xuCTvPIQcv62N3laR2wKOJN9CpFOpqjwR8cj9oLjbCRHd5T/+bgc6D
O/XVzr9DAPVaYLZvm8puMhhgckNIjgPOfWsVyI+/G1zuUH5caFMhMaS95M2a1XNI
wJ5sF25PFpsttIrFXC+b0JS325mk5qvzCgnu0DrWT2VPBNjw4rvmhBPnFb/JdXy5
e5ZIpxtGRpvKYcBxDk6yqgt9SC/e81+tqCrFSQ9cG1QCV/BnZw4cd1JHe3/AHot1
YeWjb8YpIFifJ7WxFb0g9GIYJZTh3dC9hrNOfSsIoOL+ZAyNFp43u0FJT7qtC7Ct
xPvJnt/h8ACE07ewgkq3M0OzmT2RNIf0HhySMjwdIhlY8QGGoV3+zDu9nTx63tvv
aBikPwtZuCz76ipWb5Kt6SA80fsN/QOtDflmBwjlT6UvoivgK8C7iIrwQGUCAwEA
AQKCAgAEuDBbD3672jG3DicpLzkMpYNvhTTZmzXxvJsvtC1V6MqUhPPRq9DQvGLl
JP2zfIsxpEss43YeVL+wD6EIkfTmRyTpt6FLYXXjIIDhJyPdB8yudP/0wR9FK1OC
W6yp8bf97IcEvwDfQPHPjcs5mTYqW3xBbM5xEdA7ihyLlj/hxI7VFNWuVrS6hvpM
93o6Xr3l2r2DZIDO5R/uJj/LztBEFVesiBezCXx/TbXOL9NV0I7SkR1uT6WQpUiw
iItOnDkaHDu/Jlr8ppJ/6RYzJLSlBG3Hcw/fEwz7RgLmBe90h0O8Su1/PwNCRgSc
6DBbw3nX6Bo3gzThotIUsKkufgZsW+eGIEeH5SFGFcIsnVU8AP/FZEQ3RMNXrxdx
U67asN7iQICVJujO419aZ/r+GiLnCp5+UIippYVBXf5RaHcZiurQ/SbuvppZskrA
8oDI4ykB2Y3DPxqFjygxSkxhwBNcBv5CNvAwpFTu94NDYCEgz2O3x4VaDvc52RNF
IlX8Hc9kvvpWBYO24i8jV1Unnb5ojEhlJNfAmPn8GMI/d+j5oVBC0QemieWdc1cp
OkiKliiVvbFPVSwZo1NClEPg8K9TLMynE/PqV+0Ud05qeXn4Y08b9sqrF02JiTDH
aEdYtvCM80FPjqXWi4zIDkr9Im4fuX2anzFU3T0od6Ye15OyPwKCAQEAyPQD5rQM
4yR6iUTTV0x00Se5noY2ztUL3JTi0qkm+9kBDPHdGYRIB7sN1RacWgXdqPRC4M0h
+1IWYg6lJj8IsjJjEszpRmSwLUpt2xKuh+beaWPQhg/bx8jGUrTFqNo6FPJ8RT/3
UaeRUEmqyj3vxw+lc7pZjk9GxVMD7uWbMU4GLE20h97oMr5cW4wvzzWoeofKMkyn
sDTORAC8siWZSAYKJEDlziDujiCfhVmU780igWiTODuAg4GXVqsbXC6FIMs6zGod
7D2+AUZGBj4H4o03miTMYsWCPDkPZMtvAbAKnNEp6UB4/j8ofApDUIYWF5fLBZgk
2Ri6x4XM/0TS9wKCAQEAzciWYXKjkQIVEtIXMlYGRE+irrsWL/wPmRBVFX13THqW
7Ul/rYNHSN1JaIeE9KJqgLUD1kQ+x9vHiSgKTVjrL2c/VyZrJbI3OUIAZmQ9Ahpu
qJ709waXs2VnRdZspAp/1rnf5H8u/1zWHTebzvfJEq6Ys9vRam6Jcr1wI1/9u6pJ
lwSOps67YVxAp2kAC6w2HuMlwQYTaphOHiySO+Jx6nk15r4Vgbv5mJvjQetS65AD
gJPhtep+9Sa1HvzUPQ3Gu7sz0uz5vjRJTv4RRgNSEGCLHjJflgBsl1epsBlH7Xrh
rotRox8AxVnbdXOVcONBB7GAAb7Z+bPCmXdSS2sUgwKCAQAyg3Q/l97tcgwDWXOu
rB9pPA0i1iYM0+0JY7uorLCJ+kCTWnDzqxbYKqMNf4OJ9ZOElvIAxE/Ydwf9WiUV
eh7bfGL/JNc2xLSsjdsTiJyquNQLtfWC3ZWnoMaJn7tX+JNFFLc8SRoIQpD6l6oA
8JTHex1h++PrK+5kR7vjX4AlYrGWjWnmBZhkuQlKUfDqq1hQhLXE8xPr7To0SeMk
/OKNAkemWVHrAMg2nei7gos3xF76HKl1Jy/k3ryGIrjb8S2x0qRTIhGngtWySFHt
28XrowfpDXr7ER7tuIIwGhsrV28zgDiC05wWfRXWKFZHdY00HQoBu/73O4ooAXBI
cqp9AoIBAHQHHYNkeAVS/z7VZm7jQjVSEZAjvKbhoInVQ6QSUim1FVRFlM/orVDQ
NIvTnYux6AsaBUfSwvM9YIxdHzHtaO4ZcQVajB99FNYb+M3CxwNgk/RPbB+8f9yO
2GPwOuFjaiFQPIVBkOY7Gh7vM9LGs4DtIPyIfNNd7/HaDlhjz1T49vVHhIdZGR4U
PgAmm/f46asQuEDVhC0eIy2wQ+OwEjr6jQHFO6siqeD6RHDulpprYQ4mU6WWym6/
nHAUbjbehadkLhxHsaklIhCAAI1RYfwJ82bbUDnrk07iBrNcDcpA9u7LbwRifrTH
rY3T1fcIq6oC0wIo8g5w5NBTDvunLLECggEASMBX4MfNUOYNqWoutQKNSAjjTdSy
MpSXD9MHI5uZnTEG2VEGbN74oQmMx1BtANIOhz17XrYy/fCGyRXiwIqtszMYoYDN
J3HXPQFCvUFtsb8f8fOCbnfRyxiA4jRlFYrobjtTB0kBol6ISsT7+YQIean96Ppk
PHdAA+K0/yDvhG8wC1CUwzaBwfayKM2bhGQgVIqM+ZCV/Ji0g1KKiMVXUF0cvgU6
FI+vQvLnxE7zbFOewbZmqqYGen7J7gOpNJyHvkSIia3KpJpc8IzWsrOH+8xOIX06
fi/wuMFXO7Zz9wyzEKuNXiFExo3FNRChSNonWCJj14Q8cuNW4S0J9hhvAg==
-----END RSA PRIVATE KEY-----

Sample output (message + hex-encoded signature + PEM-encoded RSA public key):

Message for RSA signing
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
-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----

Verify Message Signature with RSA

Write a program to verify RSA signature (calculated by using PKCS#1 v.1.5 + SHA3-512), created by the previous exercise. The input comes as signed message (first line) + RSA digital signature (second line) + 4096-bit RSA public key (all input lines to the end). Print as output a single word: "valid' or "invalid".

Sample input (correctly signed message):

Message for RSA signing
28ce01ec0f8511e92407496b7aa777a8b38fbc27a3656c9862830427bf8a006d59d19ac3b3441f174aee2fe6560b2b5f91aeb72ffa150dad4bd4e93a3b78eabfb48f8051b09cdfc5cbb63b014a7beb81cae4220fa01bedf5fa10aa3b6dcdd7ad229fd7ce2e8883a4893bef532c7b7822993408577a20a198387e073e15e53ffab8d7f3c46fa50347cd45d7a5d319b66d4ead658297a2e7127cc6a1dc5e104beb3ae2ba390d5ae90a66bd8a537e4df296c212186754206c5a0b155c226a3a580e943db33bf9e9829ea3e38f00aafa0e4839bf4673d83b14c74cd9ac359afc3fa56ca7f3a8391e6fb234561b9deca9ac3fa9147d70b0f1189db734b5ea96f287c604467dab0efbf5f1c78437920283dfe6fbcd56f4e859e24ee37552b3b8de5c4fa35e65cf9fe63ec7e2fd155d503fec2648303e495d86af97ac53c14b203bdc7233365d481fda08cac905c48e31bd389bea0f983e035225c4fca269a66009332bbba83542dcd484832f4d40ab14eee3609aa78c01539984758592de3355b528de3c8a669442cbd6f03808226ef0c13924b9553b7af7ff02a6916f106061a7bcbe093f795d9455709ab2b030ff485e9993eb978ae57cabeade4747e981b47c36128c39df970d1f29c4e4a3c225a8384d55995599c9ce3afe14068f4eb4edeb120beb534955b519a4424f82136d3511a0dafad46b010c3fa15bd233bdc85a7fd42a
-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----

Sample output:

valid

Sample input (tampered message):

Tampered message
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
-----BEGIN PUBLIC KEY-----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-----END PUBLIC KEY-----

Sample output:

invalid

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