WorksheetsCryptography and Number Theory Worksheet
Total questions: 100
Worksheet time: 51mins
Determine the complete system of residues mod 7
1, 3, 2, 4,
5, 6, 7, 10
0, 1, 2, 3, 4, 5, 6
3, 4, 5, 8
1, 2, 3, 4, 5
Determine the reduced system of residues modulo 9
0, 1, 2, 3, 4, 5, 6, 7, 8, 9
1, 2, 3, 4, 5,
7, 8, 9
0, 1, 3, 5, 7, 9
1, 2, 4, 5, 7, 8
The strength of the RSA cryptosystem is based
on the complexity of factoring large numbers
on the difficulty of computing discrete logarithms
on the difficulty of computing polynomials
on the difficulty of solving n-degree equations
on the difficulty of finding prime numbers
The Diffie-Hellman system's cryptographic strength is based
on the complexity of factoring large numbers
on the difficulty of computing discrete logarithms
on the difficulty of computing polynomials
on the difficulty of solving n-degree equations
on the difficulty of finding prime numbers
The cryptographic strength of the system based on the "backpack problem" is based on
on the complexity of factoring large numbers
on the difficulty of computing discrete logarithms
on the difficulty of computing polynomials
on the difficulty of solving n-degree equations
on the difficulties of solving problems of finding elements, the sum of which is equal to a given number
Theorem "System of equations {x ≡ a(mod n) x ≡ b(mod m)} has a unique solution modulo m * n" is called ...
Fermat's theorem
Euler's theorem
Legendre's theorem
Jacobi's theorem
Chinese Remainder Theorem
Solve the comparison system {x ≡ 4(mod 7) x ≡ 3(mod 5)}
x ≡ 18(mod 35)
x ≡ 10(mod 35)
x ≡ 12(mod 35)
x ≡ 7(mod 35)
x ≡ 21(mod 35)
Expression of the form i=1∑naixi called
polynomial
binomial
monomial
the sum
work
If b= ax (mod p), where a is the primitive root modulo p, x ∈ [0,1,...,p-1], then x is called
antiderivative root
exponent
discrete logarithm
a random number
a constant number
Discrete logarithm is called differently
natural
decimal
index
simple
super complicated
Solve comparison 3x ≡ 15(mod 4)
x=5(mod 4)
x=7(mod 4)
x=3(mod 4)
x=4(mod 4)
x=12(mod 4)
Vernal's cipher is
table encryption
encryption substitution
permutation encryption
encryption replacement
bitwise encryption
Given equality 2≡32(mod 5) What is the discrete logarithm of 2 to base 5?
A) 2
B) 3
C) 5
D) 1
E) 4
The Vigenira system is
table cipher
public key system
a complex substitution cipher
permutation cipher
RSA
Ensuring authentication, integrity, non-traceability is solved with the help of
electronic signature
protocols
public keys
cryptosystems
A sequence of steps taken by two or more parties to jointly solve a certain problem is
protocol
electronic signature
ciphertext
coding
decoding
A renegade is
rework
fake
active interception
failure
repeat
Message modification is
rework
failure
fake
active interception
repeat
Imitation is
failure
fake
repeat
disguise
active interception
The simplest and most common electronic signature tool is the algorithm
RSA
CBA
ElGamal
backpack problem
Diffie-Hellman
A digital signature is
a sequence of random numbers
a series of natural numbers
a string of characters depending on the sender ID and the content of the message
set of primes
an increasing sequence of integers
True digital signature if
the signed message is sent through a third party
the signed message is sent directly from the sender to the recipient
the signed message is sent through some third party
the signed message is not sent to the recipient, but to the second person
the signed message is passed from the sender to the recipient's proxy
Key management is an informational process that includes
generating keys
accumulation of keys
key distribution
declassifying keys
selection of keys
Pseudo-random number sensors are used
to generate keys
to get primes
solving comparisons
receiving ciphers
the formation of number sequences
The organization of storage of keys, their accounting and deletion is called
generating keys
key distribution
accumulation of keys
deleting keys
generating keys
Promptness, secrecy and accuracy of key distribution is a requirement for
generating keys
accumulation of keys
key distribution
deleting keys
key storage
Galois field is denoted
GF(p)
G
F(p)
Dp
p
Field with elements {0,1,2, ... p-1} – all possible remainders from dividing integers by p and algebraic actions modulo p are called
by the Diffie field
Galois field
Caesar's field
by the Viginier field
by Shannon field
The smallest positive number k for which k*a≡0(mod p) for ∀ a ∈ GF(p) called
field characteristic
field boundary
field area
field length
the perimeter of the field
The trade-off between public key systems and conventional algorithms, which require the sender and recipient to have the same key, is
using "roaming keys"
using public keys
using private keys
use of voice communication
using voicemail
Square tables with sequential natural numbers inscribed in their cells, starting from 1, which add up the same number for each column, each row and each diagonal, are called
fairy squares
unreal squares
magic squares
The permutation cipher "skital" uses devices
conical shape
cylindrical
ball shape
combined form
truncated-conical shape
One of the first ciphers of simple replacement is the Polybian square, which was proposed by the Greek writer and historian Polybius for encryption purposes. It is a square size
2×2
3×3
4×4
5×5
7×7
The virtue of the Caesar encryption system
ease of encryption
ease of decryption
low durability
high durability
impossibility of breaking
Disadvantages of Caesar's encryption system
the frequency of appearance of the letters of the original text is not masked
the alphabetical order of the sequence of replacement letters is preserved
the number of possible keys is small and the cipher is easily revealed based on the analysis of the frequencies of occurrence of letters in the ciphertext
ease of encryption
ease of decryption
The process of imposing a cipher on open data according to a certain gamma law is called
gamming
encryption
decryption
permutation
tabling
Asymmetric cryptosystems include
RSA
El Gamal
Diffie-Hellman
Caesar
Vignera
The plaintext encryption procedure is as follows
1. selects 2 large primes p и q and calculate n=pq and m=(p-1)(q-1)
2. chooses an integer k such that K<m and (e,m.)=1
3. вычисляет число k, удовлетворяющее условию K*kº1 (mod m)
4. the subscriber's secret key is a triplet of numbers (p,q,k), open - a pair of numbers (n,K)
Such a cryptosystem is called
El Gamal
RSA
Vigniera
Diffie-Hellman
A function with the following properties: 1. function description H(k,x) should be open, and secret information should only be contained in the choice of the key k 2. function x argument H(k, x) can be an arbitrary length string, and the function value must be a fixed length string 3. for any given k and x, the calculation H(k, x) must be fast (in polynomial time) 4. for any given x it should be hard to guess the value H(k, x) with a probability greater than 1/2n, where n – the number of bits in the output string. It should be difficult to determine the key k even over a large number of pairs {x, H(k,x) } or calculate from this information H(k,x) for x≠x₁ called
exponential
hash function
unilateral
unidirectional
polynomial
Keyless hash functions are divided into 2 classes
weak one-way hash functions
strong one-way hash functions
weak two-way hash functions
strong two-way hash functions
weak two-way hash functions
Distributed algorithm, i.e. a set of algorithms for each of the participants, plus the specifications of the message formats sent between the participants, plus the specifications for synchronizing the actions of the participants, plus a description of actions in case of errors is called
by signature
protocol
cipher system
electronic signature
RSA algorithm
The difference between a protocol and a cryptosystem
ensuring confidentiality
ensuring integrity, ensuring non-traceability
interactivity, having more than two participants
participants do not trust each other; protection not only from the enemy, but also from dishonest actions of partners
providing special key storage
Solve comparison x+5=20(mod18)
x≡25(mod18)
x≡15(mod18)
x≡4(mod18)
x≡2(mod18)
x≡100(mod18)
Solve the comparison system 2x ≡ 5(mod 7) x ≡ 3(mod 10)
x≡4(mod70)
x≡8(mod70)
x≡13(mod70)
x≡9(mod70)
x≡18(mod70)
By system RSA known p=2, q=11, public key K=3, find the private key.
8
5
11
7
9
Fermat's theorem is expressed by the equality
ap≡1(mod p)
ap-2º1(mod p)
ap+1≡1(modp)
ap-1º1(mod p)
ap−2≡0(mod p)
Euler's theorem can be written as the equality
aϕ(m)≡1(modm)
am≡1(modϕ(m)
a2φ(m)≡1(modm)
aϕ(n)−1≡1(mod m)
aϕ(n)+1≡1(mod m)
If in the RSA cryptosystem K is the public key, k is the secret key, then from which equality can the private key be found, knowing the public key?
K+k ≡ 1 (mod φ(n))
K*k ≡ 1 (mod φ(n))
K-k ≡ 1 (mod φ(n))
K*k ≡ 1 (mod n)
K+k ≡ 1 (mod n)
In the RSA cryptosystem, the public key is K = 7. Find the private key k if n = 22.
5
7
4
11
3
Solve the comparison system {x ≡ 3 (mod 2) x ≡ 2 (mod 5)}
x ≡ 2(mod 5)
x ≡ 3(mod 2)
x ≡ 4(mod 7)
x ≡ 7(mod 10)
x ≡ 5(mod 10)
A and B are remote legitimate users of the protected information, then P –
illegal user
buddy A
buddy B
buddy A and B
outsider
the message is converted at once
the whole message is converted at once
the message is converted into blocks of a certain length
the message is converted piece by piece
the message is converted into blocks of different lengths
the message is transformed in halves
Solve comparison 2x+5=19(mod 13)
x=5(mod 13)
x=7(mod 13)
x=1(mod 13)
x=12(mod 13)
x=10(mod 13)
Solve comparison 2x=18(mod 11)
x=5(mod 11)
x=10(mod 11)
x=1(mod 11)
x=12(mod 11)
x=9(mod 11)
Block ciphers are
imposition on the source text of some pseudo-random sequence generated based on the key
a sequence (with possible repetition and alternation) of basic transformation methods applied to a block of ciphertext
replacing characters of the original text with others of the same alphabet
imposition on the source text of some pseudo-random sequence generated based on the key
Caesar cipher
Converting ciphertext characters to replacement ciphers
is transformed using a matrix
are rearranged according to some rule within some block of this text
are transformed according to some analytical rule
are replaced by characters of another or the same alphabet in accordance with the replacement scheme.
is transformed using an inverse key matrix
Convert cipher text characters to permutation ciphers
are replaced by symbols of a different or the same alphabet in accordance with the replacement scheme.
is transformed using an inverse key matrix
are transformed according to some analytical rule
is transformed using a matrix
are rearranged according to some rule within some block of this text.
Substitution types used in cryptography
Huffman, Caesar, Playfair
mono-alphabetic, homophonic, polyalphabetic
Markov, homophonic, Playfair
mono-alphabetic, homophonic, polyalphabetic and polygram
Vizhner, Caesar, Playfer, analytical
Principle of mono-alphabet substitution
each letter of the plaintext alphabet is assigned one letter of the ciphertext from the same alphabet.
each letter of the plaintext alphabet is associated with two letters of the ciphertext from the same alphabet.
each letter of the plaintext is assigned a letter with a shift
each letter of the plaintext alphabet is associated with one letter of the key
each plaintext letter is assigned a closedtext letter
The principle of polyalphabetic substitution
each letter of the plaintext alphabet is assigned one letter of the ciphertext from the same alphabet.
each letter of the plaintext alphabet is associated with two letters of the ciphertext from the same alphabet.
uses multiple ciphertext alphabets
maps several ciphertext characters to one plaintext character.
each letter of the plaintext alphabet is associated with one letter of the key
Cryptographic protocols must protect against the following violations and attackers:
renegade (rejection), repeat
modification (rework)
counterfeit, imitation, active interception
copy
mailing
Conversions that can be used as a one-way function
exponential transformations
sum of matrices
matrix difference
matrix zero multiplication
multiplication by the identity matrix
Gumming -
replacing characters of the original text with others from the other alphabet
Caesar cipher
replacement of symbols of the source text with others (of the same alphabet) according to a more or less complex rule
imposing on the source text some pseudo-random sequence generated based on the key
imposition on the source text of some pseudo-random sequence generated based on the key
For which function is the inverse operation computationally time consuming?
straight
bilateral
reverse
unilateral
terminal
What fact did the creators of RSA take advantage of?
finding large primes is computationally laborious, and factoring the product of two such numbers is practically impossible
finding large primes is computationally easy, but factoring the product of two such numbers is practically impossible
factoring the product of two numbers is easy
finding primes is computationally easy
finding prime divisors is computationally easy
Solve comparison x+2=17(mod18)
x=25(mod18)
x=15(mod18)
x=4(mod18)
x=2(mod18)
x=100(mod18)
Solve the comparison system 2x ≡ 5 (mod 7) 3x ≡ 9 (mod 10)
xº4(mod70)
xº8(mod70)
xº13(mod70)
xº9(mod70)
xº18(mod70)
According to the RSA system, the public key K=3, p=2, q=11. What will the private key be?
A) 8
B) 7
C) 11
D) 5
E) 9
Solve the comparison system x ≡ 3(mod 2) 2x ≡ 4(mod 5)
x ≡ 2(mod 5)
x ≡ 7(mod 10)
x ≡ 4(mod 7)
x ≡ 3(mod 2)
x ≡ 5(mod 10)
In RSA cryptosystem, the private key k=3, n=22. Find the public key K.
5
3
4
11
7
How is a message transformed in streaming cryptosystems?
the message is converted immediately
the message is converted into blocks of a certain length
the message is converted piece by piece
the message is converted into blocks of different lengths
the message is transformed in halves
In which cryptosystem the message is transformed into blocks of a certain length?
in a streaming cryptosystem
in a block cryptosystem
in the RSA system
in the El Gamal system
in a combined system
Solve comparison 2x ≡ 18(mod 13)
x ≡ 5(mod 13)
x ≡ 1(mod 13)
x ≡ 9(mod 13)
x ≡ 12(mod 13)
x ≡ 10(mod 13)
The RSA algorithm is named -
honoring the creators: Ron Rivest, Adi Shamir and Leonard Adelman
by the name of the algorithm - random number algorithm (Random Seek Acess)
by the name of the algorithm - speed algorithm
random selection of characters
by the name of the company that developed this method
Knowing the public key in the RSA cryptosystem, how can you find the private key?
K+k ≡ 1 (mod φ(n))
K*k ≡ 1 (mod φ(n))
K*k ≡ 1 (mod φ(n))
K*k ≡ 1 (mod n)
K+k ≡ 1 (mod n)
What Yi is - in the mono-alphabetic substitution formula: Yi=kl*Xi + k2 (mod N)
i-th character of the alphabet
constant;
i-th plaintext character;
alphabet length
key length
What Xi is - in the monoalphabetic substitution formula: Yi = k1 * Xi + k2 (mod N)
i-th character of the alphabet
constant;
i-th plaintext character;
the length of the alphabet used
key length
Gronsfeld cipher - cipher modification
Caesar's numeric key
Vigenere, for m = 2
Vernam's numeric key
DES numeric key
GOST symbolic key
Vigenère code: yi = xi + ki (mod n), where ki is
i-th letter of the key, which is used as a word or phrase
i-th letter of the alphabet
i-th letter of the key
key
alphabet
With Homophonic Replacement
each letter of the plaintext alphabet is associated with one letter of the ciphertext from the same alphabet
each letter of the plaintext alphabet is associated with two letters of the ciphertext from the same alphabet
each letter of the plaintext is assigned a letter with a shift
each letter of the plaintext alphabet is associated with one letter of the key
maps several ciphertext characters to one plaintext character
With polyalphabetic substitution
each letter of the plaintext alphabet is associated with one letter of the ciphertext from the same alphabet
each letter of the plaintext alphabet is associated with two letters of the ciphertext from the same alphabet
uses multiple ciphertext alphabets
each letter of the plaintext alphabet is associated with one letter of the key
maps several ciphertext characters to one plaintext character
In the problem of packing a knapsack, the ciphertext is obtained as
dot product
cross product
prime power
binary key decomposition
The creators of RSA took advantage of the fact that
finding large primes is computationally easy, but factoring the product of two such numbers is practically impossible
finding large primes is computationally laborious, and factoring the product of two such numbers is practically impossible
factoring the product of two numbers is easy
finding primes is computationally easy
finding prime divisors is computationally easy
For increased secrecy, the length of the password should be (when the user is forced to set passwords not with a word, but at least with a phrase)
must not be shorter than 12 characters
must not be shorter than 7 characters
must not be shorter than 5 characters
must not be shorter than 4 characters
must be at least 1024 characters
The alphanumeric key must be
at least 7 characters
at least 6 characters
at least 5 characters
at least 12 characters
maximum of 7 characters
What are the names of cryptoalgorithms in which the encoding unit is one bit, and the encoding result does not depend on the previously passed input stream:
Blocky
Wildcards
Streaming
Asymmetrical
Permutation
What are the names of the cryptoalgorithms in which the encoding unit is a block of several bytes (from 4 to 32), and the encoding result depends on all the original bytes of this block:
Blocky
Wildcards
Streaming
Asymmetrical
Permutation
The secret system includes
two statistical choices: message selection and key selection
three statistical choices: message choice and key choice, ciphertext choice
message selection
key selection
encryption algorithm
The only key is used in cryptosystems
symmetrical
with public key
asymmetric
with private key
Class D
Two keys are used in cryptosystems
symmetrical
with public key
asymmetric
with private key
Class D
Which group does the Caesar Cipher belong to?
mono-alphabetic
homophonic
polyalphabetic
polygram
packing the knapsack
Comparison is given as x≡b(mod m). Please enter an invalid equality.
2x≡2b(mod m)
3x≡b(mod m)
ax+c≡b+c(mod m)
ax≡b+m*k(mod m)
ax+m≡b(mod m)
The most popular public key encryption algorithms are
PGP и RSA
PGP и DES
RSA и DES
Gost и DES
Gost
Solve the comparison system 2x ≡ 6 (mod 2) x ≡ 2 (mod 5)
x≡1(mod 5)
x≡5(mod 10)
x≡4(mod 7)
x≡13(mod 2)
x≡7(mod 10)
The process of breaking a cipher without knowing the key (checking the strength of the cipher):
Code
Cryptography
Coding
Cryptanalysis
Key
The specific secret state of some parameters of the cryptographic data transformation algorithm, which ensues the selection of only one option out of all possible for this algorithm:
Code
Cryptography
Coding
Cryptanalysis
Key
The first stage in the work of any asymmetric cryptoalgorithm is:
Distributing the public key "worldwide"
Message encryption
Decrypting the message
Create a key pair: public and private
Message transmission
The basis for creating keys for any asymmetric cryptoalgorithm are:
Differentiating the message length value
Logarithm of the message length
Bijective mathematical functions
Third order equations
Prime numbers and operations on them
The main idea of asymmetric cryptoalgorithms:
Session key is used to encrypt plaintext
One key is used to encrypt a message, and another key is used when decrypting a message
The same key is used for both encrypting and decrypting the message
Knowing the public key, you can calculate the private key
The encryption procedure is reversible
Solve the comparison system (x = 6(mod 17)
x= 4(mod 11)
(x = -3(mod 8)
xº125(mod 1496)
xº125 (mod 1494)
xº125(mod 1498)
xº125(mod 1499)
xº125(mod 1490)
