Checksums and hash codes

Computer science problems

• Redundancy in numbers (ids, card numbers, etc.)
• Data integrity checking
• Authorization
• Hash tables

Redundancy in numbers

• How can we quickly check that the number is correct?
• תעודת זהות (כולל ספרת ביקורת)
  • 318892652*
  • 3188926525! Why 5?
• "Card number is invalid" How do they know?

Data integrity checking

• Quickly check that a download is complete
• Verify a document:
  

Authorization

Hash tables (dictionaries)

dictionaries in Python

>>> tel = {'jack': 4098, 'sape': 4139}
>>> tel['guido'] = 4127
>>> tel
{'jack': 4098, 'sape': 4139, 'guido': 4127}
>>> tel['jack']
4098
>>> del tel['sape']
>>> tel['irv'] = 4127
>>> tel
{'jack': 4098, 'guido': 4127, 'irv': 4127}
>>> list(tel)
['jack', 'guido', 'irv']
>>> sorted(tel)
['guido', 'irv', 'jack']
>>> 'guido' in tel
True
>>> 'jack' not in tel
False
						

Modular arithmetics

$$0 \le m < b$$

For example: $$n \% 1 = 0$$ $$10 \% 3 = 1$$ $$3 \% 2 = 1$$ $$5 \% 3 = 2$$

Modular equivalence

$$a \equiv b \pmod c$$ Means that $$a \% c = b \% c$$

For example: $$9 \equiv 5 \pmod 4$$ $$4 \equiv 2 \pmod 2$$ $$3 \equiv 5 \pmod 2$$

Properties

• Reflexivity: $a \equiv a \pmod n$
• Symmetry: $a \equiv b \pmod n \Leftrightarrow b\equiv a \pmod n$
• Transitivity: $a = b \pmod n \land b \equiv c \pmod n \Rightarrow a \equiv c \pmod n$

More properties

If a1 ≡ b1 (mod n) and a2 ≡ b2 (mod n), or if a ≡ b (mod n), then:

• a + k ≡ b + k (mod n)
• k a ≡ k b (mod n) for any integer k
• a1 + a2 ≡ b1 + b2 (mod n)
• a1 – a2 ≡ b1 – b2 (mod n)
• a1 · a2 ≡ b1 · b2 (mod n)

We can compute with remainders.

Example

• As: 1000 integers sent
• Ar: 1000 integers received
• Is S=R?

						def checksum(A):
							cksum = 0
							for a in A: 
								cksum += a
							return cksum % 101
						

Will overflow if integers are big

Example

• As: 1000 integers sent
• Ar: 1000 integers received
• Is S=R?

						def checksum(A):
							cksum = 0
							for a in A: 
								cksum += a % 101
							return cksum % 101
						

Will NOT overflow even if integers are big.

Example

• As: 1000 integers sent
• Ar: 1000 integers received
• Is S=R?

						def checksum(A):
							cksum = 0
							for a in A: 
								cksum += a % 101
								cksum %= 101
							return cksum
						

Will NOT overflow even if integers are big.