RSA Encryption

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Alexander Corner 11 months, 3 weeks ago

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Alexander Corner 11 months, 3 weeks ago

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Alexander Corner 11 months, 3 weeks ago

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Alexander Corner 1 year ago

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Question statement

In this question you will be asked to encrypt a number using the provided RSA public key. You will also be asked to factor the public key and provide the decryption key.

The public key is given by $(n,e) = (\var{modulus},\var{key})$ .

Give any introductory information the student needs.

			Name	Type	Generated Value

prime1	integer	11
prime2	integer	19
modulus	integer	209
totient	integer	180
key	integer	7
e_check	boolean	true
d_key	integer	103
message	integer	25
encrypted	number	9
decrypted	number	10
decrypt_check	boolean	false
bin_d_key	string	1100111
binlength	number	7
d_key8	string	01100111
key_check	integer	1
m_to_e	number	6103515625
sqrt_mod	number	14.5

Automatically regenerate variables?

Name

Data type

Value

xxxxxxxxxx
 
random(7,11,13)

JME function reference

Description

Describe what this variable represents, and list any assumptions made about its value.

Can an exam override the value of this variable?

Generated value: `integer`

→ Used by:

d_key
modulus
totient

"Part b)" → "p" - Minimum accepted value
"Part b)" → "p" - Maximum accepted value

Advice

For larger message sizes and larger encryption exponents, you may need to use some technology to help you, such as the Excel spreadsheet we made in order to repeatedly square and reduce numbers in order to obtain large powers.

In this question we had the message value of $m = \var{message}$ , meaning that we needed to find $m^e = \var{message}^\var{key} \operatorname{mod} \var{modulus}$ . Now $m^e = \var{message}^\var{key} = \var{m_to_e} = \var{encrypted} \operatorname{mod} \var{modulus}$ .

To find the private key, we can start by factoring the modulus $n = \var{modulus}$ into two primes. This should not be difficult since the numbers involved are so low. If we can't spot a sensible factor, then we just have to try dividing the modulus $n$ by prime numbers up to $\sqrt{n} = \sqrt{\var{modulus}} = \var{sqrt_mod}$ until we find one that divides it exactly: this will be the lower of the two primes. In order to find the decryption exponent $d$ , we must use the extended Euclidean algorithm in order to find the value of $d$ such that $de \equiv 1 \operatorname{mod} \varphi(n)$ , i.e., $d \times \var{key} \equiv 1 \operatorname{mod} \var{totient}$ . This works out to be $d = \var{d_key}$ .

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Taxonomy: mathcentre

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Taxonomy: Context

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Checkpoint description

Alexander Corner 11 months, 3 weeks ago

Alexander Corner 11 months, 3 weeks ago

Alexander Corner 11 months, 3 weeks ago

Alexander Corner 1 year ago

Error in variable testing condition

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Generated value: `integer`

Parts

Number entry

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RSA Encryption

Metadata

Taxonomy: mathcentre

Taxonomy: Kind of activity

Taxonomy: Context

Contributors

Feedback

History

Comment

Checkpoint description

Alexander Corner 11 months, 3 weeks ago

Alexander Corner 11 months, 3 weeks ago

Alexander Corner 11 months, 3 weeks ago

Alexander Corner 1 year ago

Error in variable testing condition

Error

Warning

Generated value: integer

Parts

Number entry

Advanced settings

Generated value: `integer`