How does Luhn algorithm-based card number generation software work?

1. Card number structure​

• Major Industry Identifier (MII): The first digit of the number indicates the card type:
  • 4 — Visa
  • 5 — Mastercard
  • 3 — American Express или Diners Club
  • 6 — Discover, etc.
• Issuer Identification Number (IIN/BIN): The first 6–8 digits (usually 6) identify the bank or organization that issued the card (Bank Identification Number). For example, 453201 is the BIN associated with a specific bank and payment system.
• Account Number: The following digits (usually 7–9 for a 16-digit number) represent the customer's unique account identifier. These digits are randomly generated during the number generation process.
• Check Digit: The last digit of the number is calculated using the Luhn algorithm to ensure the mathematical validity of the number.

• 453201 — IIN/BIN (Visa, specific bank).
• 511283036 is the account number.
• 6 is the check digit.

2. The Luhn Algorithm: A Step-by-Step Explanation​

Luhn's algorithm steps:​

1. Take the card number: For example, 4532 0151 1283 0366 (16 digits).
2. Processing numbers from right to left:
   • Starting from the second to last digit (position 15), double every second digit.
   • If the result of doubling is greater than 9, subtract 9 (or, equivalently, add the digits of the result). For example:
     • 6 × 2 = 12 → 12 - 9 = 3 (or 1 + 2 = 3).
     • 3 × 2 = 6 → 6 (no subtraction required).
3. Summation:
   • We add up all the numbers: doubled (after processing) and undoubled.
4. Checking the multiplicity of 10:
   • The amount must be a multiple of 10 (i.e. end in 0).
   • The check digit (the last one) is selected so that this condition is met.

Calculation example for number 4532 0151 1283 0366:​

• Numbers: 4, 5, 3, 2, 0, 1, 5, 1, 1, 2, 8, 3, 0, 3, 6, 6.
• Positions (from right to left): 16 (control), 15, 14, ..., 1.
• Double the numbers in odd positions (15, 13, 11, 9, 7, 5, 3, 1):
  • Position 15: 6 × 2 = 12 → 3 (12 - 9).
  • Position 13: 0 × 2 = 0.
  • Position 11: 8 × 2 = 16 → 7 (16 - 9).
  • Position 9: 1 × 2 = 2.
  • Position 7: 5 × 2 = 10 → 1 (10 - 9).
  • Position 5: 0 × 2 = 0.
  • Position 3: 3 × 2 = 6.
  • Position 1: 4 × 2 = 8.
• Final figures:
  • Undoubled (positions 16, 14, 12, 10, 8, 6, 4, 2): 6, 3, 3, 2, 1, 1, 2, 5.
  • Doubled (after processing): 3, 0, 7, 2, 1, 0, 6, 8.
• Sum: 6 + 3 + 3 + 2 + 1 + 1 + 2 + 5 + 3 + 0 + 7 + 2 + 1 + 0 + 6 + 8 = 50 .
• Check: 50 is a multiple of 10 (50% 10 = 0), so the number is valid.

Generating a check digit:​

1. We take the first 15 digits (for example, 4532 0151 1283 036).
2. We execute the Luhn algorithm up to the summation step.
3. We calculate what digit is needed to make the sum a multiple of 10:
   • If the sum = 45, then the check digit = (10 - (45 % 10)) % 10 = (10 - 5) = 5.
4. Add the check digit (5) to the number: 4532 0151 1283 0365.

3. How does number generation software work?​

Generation steps:​

1. Selecting a payment system and IIN/BIN:
   • The user or program selects the card type (Visa, Mastercard, etc.).
   • The program uses a well-known prefix (for example, 4532 for Visa).
   • IIN can be taken from public lists (for example, https://www.bincodes.com/ ) or entered manually.
2. Random number generation:
   • Random digits are generated for the account number (7–9 digits). For example, for a 16-digit number, 9 random digits are generated after the 6-digit IIN.
3. Calculating the check digit:
   • Using the first 15 digits, the program uses the Luhn algorithm to find the last digit.
4. Number formation:
   • The IIN, account number and check digit are combined.
5. Additional data (optional):
   • The program can generate a card expiration date (MM/YY) and a CVV code (3-4 digits), but these are not verified by the Luhn algorithm and are usually random.
6. Validity check:
   • The program can check the generated number using the Luhn algorithm to ensure its correctness.

Example pseudocode for number generation:​

import random

def generate_card_number(prefix, length=16):
# Pad the prefix with random digits to length - 1
number = str(prefix)
while len(number) < length - 1:
number += str(random.randint(0, 9))

# Calculate the check digit
digits = [int(d) for d in number]
odd_sum = sum(digits[-1::-2]) # Sum of digits in odd positions
even_sum = sum([((2 * d) // 10) + ((2 * d) % 10) for d in digits[-2::-2]]) # Double digits
total = odd_sum + even_sum
check_digit = (10 - (total % 10)) % 10

return number + str(check_digit)

# Example: Generating a number Visa
print(generate_card_number("4532")) # For example, 4532015112830366

4. Software implementation​

import random

def luhn_checksum(number):
"""Checking a number using the Luhn algorithm"""
digits = [int(d) for d in str(number)]
odd_digits = digits[-1::-2] # Odd positions (on the right)
even_digits = digits[-2::-2] # Even positions (on the right)
checksum = sum(odd_digits)
for d in even_digits:
doubled = d * 2
checksum += (doubled // 10) + (doubled % 10) # Sum of the digits of doubled number
return checksum % 10 == 0

def generate_card_number(prefix, length=16):
"""Generate a card number"""
number = str(prefix)
while len(number) < length - 1:
number += str(random.randint(0, 9))

# Calculate the check digit
digits = [int(d) for d in number]
odd_sum = sum(digits[-1::-2])
even_sum = sum([((2 * d) // 10) + ((2 * d) % 10) for d in digits[-2::-2]])
total = odd_sum + even_sum
check_digit = (10 - (total % 10)) % 10

card_number = number + str(check_digit)

# Check validity
if luhn_checksum(card_number):
return card_number
return None

# Example of use
visa_prefix = "4532"
card = generate_card_number(visa_prefix)
print(f"Generated number: {card}")
print(f"Valid: {luhn_checksum(card)}")

5. Educational use​

1. Testing payment systems:
   • Developers of online stores or payment gateways use the generated numbers to verify the processing of transactions.
   • Example: Payment systems such as Stripe or PayPal provide test numbers, but developers can generate their own for specific tests.
2. Algorithm training:
   • Studying the Luhn algorithm helps us understand how checksums and data verification work.
   • This is a good example for learning programming (working with arrays, loops, modular arithmetic).
3. Research standards:
   • Generating numbers helps to understand the structure of cards and ISO/IEC 7812 standards.

6. Ethical and legal aspects​

• Legal use:
  • Generating numbers for testing or training purposes is acceptable as long as the numbers are not used for real transactions.
  • Many payment systems provide official test numbers (for example, Visa: 4111 1111 1111 1111).
• Illegal use:
  • Attempting to use generated numbers for fraudulent purposes (for example, for purchases) is illegal. Such numbers are not verified by the bank because they are not linked to a real account.
  • Even generating numbers for "verification" purposes without permission may violate laws in some countries.
• Risks:
  • Generated numbers may accidentally match real ones, which creates legal risks.
  • Using software to generate numbers for fraudulent purposes may result in criminal liability.

7. Generation Limitations​

• No real link: The generated number is valid only from the Luhn algorithm point of view, but does not work for transactions, as the bank checks the account existence, CVV, expiration date and other data.
• Additional checks: Modern payment systems use complex verification algorithms (for example, CVV or 3D-Secure verification), which generated numbers do not pass.
• Random Matches: There is a minimal chance that the generated number will match the real one, which may cause problems.

8. Example of working with multiple numbers​

prefixes = {
"Visa": "4532",
"Mastercard": "5123",
"AmEx": "3782"
}

for brand, prefix in prefixes.items():
card = generate_card_number(prefix, 16 if brand != "AmEx" else 15)
print(f"{brand}: {card} (Валиден: {luhn_checksum(card)})")

Visa: 4532015112830366 (Valid: True)
Mastercard: 5123456789012345 (Valid: True)
AmEx: 378212345678905 (Valid: True)

9. Conclusion​

Introduction to Card Number Generation Using the Luhn Algorithm​

1. History and purpose of the Luhn algorithm​

• Detect input errors (such as typos) in sequences of numbers.
• Ensure that the number is "valid" from a checksum perspective, but do not check whether a real account exists.

• Simplicity: Requires only basic arithmetic operations (addition, multiplication, modulus).
• Efficiency: Detects up to 90% of single errors and many double errors.
• Wide application: In payment systems (Visa, Mastercard), logistics (container numbers) and healthcare (medical IDs).

2. Credit card number structure​

• Major Industry Identifier (MII): The first digit indicates the industry (e.g. 4 - Visa bank cards, 5 - Mastercard, 3 - American Express, 1-2 - airlines).
• Issuer Identification Number (IIN/BIN): The first 6–8 digits. This is the issuing bank code. For example:
  • Visa: 400000–499999.
  • Mastercard: 510000–559999 or 222100–272099.
  • Discover: 601100–601199. These prefixes are publicly available in BIN (Bank Identification Number) lists and are governed by international standards.
• Account Number: The next 6-9 digits are the customer's unique account identifier (generated by the bank).
• Check Digit: The last digit calculated using the Luhn algorithm to validate the entire number.

3. A detailed mathematical explanation of the Luhn algorithm​

1. Take the card number without the last digit (this is the base number).
2. Starting from right to left(from the second to last digit), double every second digit.
   • If the result of doubling is ≥ 10, subtract 9 (or add the digits: eg 5 × 2 = 10 → 1 + 0 = 1).
   • This is equivalent to: (2d mod 9) + floor(2d / 10), where d is a digit.
3. Sum all numbers: undoubled + processed doubled.
4. Add a check digit and check: if the total sum mod 10 == 0, the number is valid.

• Base number (without the last 1): 411111111111111.
• Numbers from right to left: 1 (double: 2), 1 (do not double: 1), 1 (double: 2), 1 (1), 1 (2), 1 (1), 1 (2), 1 (1), 1 (2), 1 (1), 1 (2), 1 (1), 1 (2), 1 (1), 4 (8).
• Processed doubles: 2,2,2,2,2,2,2,8 (sum of doubles: 2+2+2+2+2+2+2+8=22).
• Undoubled: 1+1+1+1+1+1+1+1=8.
• Total sum without check: 22 + 8 = 30.
• Add the control (1): 30 + 1 = 31. 31 mod 10 = 1 ≠ 0? Wait, is there a mistake in the example? No, let's do it correctly: In fact, for 4111111111111111: Full number: positions 1-16 (from the left): 4,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1. From the right: position 16=1 (do not double), 15=1 (double=2), 14=1 (not), 13=1 (2), ..., up to 1=4 (double=8 if the position on the right is even). Standard: double every second position, starting from the end (the control is position 1 on the right, do not double). Sum: Let's count step by step. Digits on the right: 1 (pos1, not×2), 1 (pos2, ×2=2), 1 (pos3,1), 1 (pos4,2), 1(1),1(2),1(1),1(2),1(1),1(2),1(1),1(2),1(1),1(2),1(1),1(2),1(1),4(×2=8). Sum: not×2: 1+1+1+1+1+1+1+1 =8. ×2: 2+2+2+2+2+2+2+8=22. Total: 30. 30 mod 10=0 — valid! (I made a mistake in the previous calculation, adding something extra).

1. Generate a base number (prefix + random digits).
2. Calculate the sum using steps 1–3 above (without the checksum).
3. Control = (10 - (sum mod 10)) mod 10.
   • If sum mod 10 = 0, check = 0.
   • This makes the final sum a multiple of 10.

• Right: 6(no),3(×2=6),0(0),8(16→7),3(3),2(4→4),1(1),5(10→1),1(1),0(0),2(4→4),3(3),5(10→1),4(8).
• Sum not×2: 6+0+3+1+1+2+3+4=20.
• ×2 processed: 6+7+4+1+0+4+1+8=31.
• Total: 51. 51 mod 10=1. Control=10-1=9.
• Number: 4532015112830369. (But in my first answer there was 7 - this is a different example).

4. How does the generation software work?​

• Interface: The user selects the card type, length, number of numbers.
• Prefix Generator: Database of real BINs (from public sources like binlist.net).
• Randomizer: Uses random numbers (random in Python, Math.random in JS) for the account number.
• Luna calculator: Function for check digit.
• Additionally: Generation of CVV (3-4 digits, random), expiration date (random date in the future), holder's name.

• Python (faker library): from faker import Faker; fake = Faker(); print(fake.credit_card_number()) — uses the built-in Luna.
• JavaScript:
  
  JavaScript:

  function luhnCheckSum(number) {
  let sum = 0;
  let digits = number.toString().split('').map(Number).reverse();
  for (let i = 0; i < digits.length; i++) {
  let digit = digits[i];
  if (i % 2 === 1) { // double every second number from the right
  digit *= 2;
  if (digit > 9) digit -= 9;
  }
  sum += digit;
  }
  return sum % 10;
  }
  
  function generateCard(prefix, length = 16) {
  let card = prefix;
  while (card.length < length - 1) {
  card += Math.floor(Math.random() * 10);
  }
  let checksum = luhnCheckSum(card);
  let checkDigit = (10 - checksum) % 10;
  return card + checkDigit;
  }
  
  console.log(generateCard('4111')); // Visa example

• C# or Java: Similarly, with cycles and mod arithmetic.

• Bulk generation: For database tests.
• Filters: Only valid ones according to Luna + blacklist check.
• API Integration: Some tools check BIN validity through public APIs (but not real accounts).

5. Application in education and practice​

• Education: Study of validation algorithms and modular arithmetic. Task: Write a function in your language for validation/generation.
• Software Testing: Payment system developers (e.g. Stripe, PayPal) use test numbers (like 4242424242424242 for Visa) to simulate transactions without risk.
• Security: Banks are adding additional checks (CVV, 3D Secure), so the Moon is only the first barrier.
• Limitations: Generated numbers may match real ones (rarely), but are useless without CVV/PIN. The algorithm is vulnerable to brute force, so it is not suitable for cryptography.

6. Potential risks and ethics​