RSA in one lesson

02The pair you can publish is (n, e)

The phrase x mod n means the remainder when x is divided by n. The toy encryption raises m to e and keeps that remainder; the inverse operation uses d. These powers explain roles but omit the encoded-message processing that real RSA schemes require.

RFC 8017 separates key types, primitives and encryption schemes such as RSAES-OAEP. A scheme specifies more than n=p×q. The earlier 2026-10-12 course record reports toy-rsa.py printing worked n=33, activity n=15 and round trip 4. Those are historical toy outputs, not a new key-generation run or proof of usable security.

03Multiply the activity primes

The guided case asks only for a product, with p=3 and q=5. No encryption, private-exponent search or real key is required.

Label: toy hand arithmetic. Historical 2026-10-12 QA separately recorded toy-rsa.py printing worked n 33, activity n 15 and round trip 4.
Public pair on the worked card: n=33, e=3. Private exponent on that card: d=7.
Message m=4. Cipher c = 4^3 mod 33 = 64 mod 33 = 31.
Return: 31^7 mod 33.
31^2 = 961 = 29*33 + 4
31^4 = 16 mod 33.
31^5 = 1 mod 33.
31^7 = (31^5)*(31^2) = 4 mod 33.
31^7 = 4 mod 33.
Predicted return: 4.

Activity primes: p=3, q=5.
Predicted modulus: 15.

"""Print the worked and activity moduli. Authoring did not run this file.

The numbers are a classroom size. They are not a deployable key.
The file does not contain the field primes. Do not use the result as a real key.
"""


def main():
    print("worked n", 3 * 11)
    print("activity n", 3 * 5)
    print("worked round trip", pow(31, 7, 33))


if __name__ == "__main__":
    main()

1. Write n=p×q using the two activity primes rather than the worked pair.
2. Multiply 3×5 and compare your result with the activity modulus 15.
3. Write one limit beside the result: a product this small is educational and must not be deployed as a key.
4. Stop after the calculation. The later checkpoint supplies its own prime pair; leave it for that independent question.

04Optional challenge

For about ten optional minutes, make a three-row vocabulary note: modulus n, public exponent e and private exponent d. Then add a separate line for a complete encryption scheme’s encoding/padding requirements. The rubric is correct roles and an explicit warning that the toy arithmetic is incomplete for real protection.

05Fresh cases for the checks

Use these fictional cases for the checks below. They have different inputs from the worked examples. Read them here or make notes; no installation or external service is needed.

Worksheet primes: p=11, q=13.
Rule: n=p×q. Only the modulus is requested; this is not a usable key.

Toy parameters: n=35; public exponent e=5; corresponding private exponent d=5.
Message m=2. Forward rule: c=m^e mod n. Here ^ means a mathematical power.
Supplied arithmetic: 2^5=32; 32=35×0+32.
For a separate reverse check: 32^5=33554432; 33554432=35×958698+2.
These are raw arithmetic statements, not production RSA encryption.

For this tiny two-prime model: n sets the remainder arithmetic; e is the public exponent; d is the corresponding private exponent.
The public key pair is (n,e). The private exponent is kept separate.
Real encryption schemes require additional encoding/padding and appropriate security parameters.

06Explain the evidence and its limit

Optional self-check: state modulus, public pair and private exponent, then connect one raw forward and reverse remainder trace.

Review it: keep powers distinct from remainders, do not copy mathematical ^ into Python as exponentiation, and explain that tiny raw RSA omits required production scheme details.