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In Python 3

def string_multiplication(a, b):
    result = 0
    for i in range(len(a)):
        for j in range(len(b)):
            result += (ord(a[i])-ord('0'))*(10**(len(a)-i-1))*(ord(b[j])-ord('0'))*(10**(len(b)-j-1))
    return result

number_one = 193283492420348904832902348908239048823480823
number_two = 3248234890238902348823940990234
print(number_one*number_two)
print(string_multiplication(str(number_one), str(number_two)))

decimal multiplication on string does it, although converting the string to binary and doing a binary multiplication would obviously be more effective

in python 3

def string_multiplication(a, b):
    result = []
    result_start_idx = 0    
    for ai in map(lambda x: ord(x) - ord('0'), reversed(a)):
        overflow = 0
        result_idx = result_start_idx
        for b_idx in range(len(b) - 1, -1, -1):
            m = (ord(b[b_idx]) - ord('0')) * ai + overflow
            if result_idx < len(result): 
                m += result[result_idx]
                result[result_idx] = m % 10
            else :
                result.append(m % 10)
            overflow = m // 10
            result_idx += 1
        if overflow > 0:
            result.append(overflow)
        result_start_idx += 1
    return ''.join(map(lambda x: chr(ord('0') + x), reversed(result)))

number_one = 193283492420348904832902348908239048823480823
number_two = 3248234890238902348823940990234
print(number_one*number_two)
print(string_multiplication(str(number_one), str(number_two)))

#the product of ith index of A and jth index of B would be going to i+j and i+j+1 th position
def string_multiplication(a, b):
def string_multiplication(a, b):
    result=[0]*(len(a)+len(b))
    for i in range(len(a)-1,-1,-1):
        for j in range(len(b)-1,-1,-1):
            mul=int(a[i])*int(b[j])
            pos1,pos2=i+j,i+j+1
            sum_all=mul+result[pos2]
            result[pos1]+=int(sum_all/10)
            result[pos2]=int(sum_all%10)
    return result
print(string_multiplication('8', '100'))

def string_multiplication(a, b):
    result=[0]*(len(a)+len(b))
    for i in range(len(a)-1,-1,-1):
        for j in range(len(b)-1,-1,-1):
            mul=int(a[i])*int(b[j])
            pos1,pos2=i+j,i+j+1
            sum_all=mul+result[pos2]
            result[pos1]+=int(sum_all/10)
            result[pos2]=int(sum_all%10)
    return result
print(string_multiplication('8', '100'))

def string_multiplication(a, b):
    result=[0]*(len(a)+len(b))
    for i in range(len(a)-1,-1,-1):
        for j in range(len(b)-1,-1,-1):
            mul=int(a[i])*int(b[j])
            pos1,pos2=i+j,i+j+1
            sum_all=mul+result[pos2]
            result[pos1]+=int(sum_all/10)
            result[pos2]=int(sum_all%10)
    return result
print(string_multiplication('8', '100'))

def multiplystrings(s1,s2):
    totalsum,sum = 0,0
    for i in range(len(s1)):
        sum = 0
        for j in range(len(s2)):
            sum *= 10
            sum += int(s1[i]) * int(s2[j])
        totalsum *= 10
        totalsum += sum
    print(totalsum)

def multiplystrings(s1,s2):
totalsum,sum = 0,0
for i in range(len(s1)):
sum = 0
for j in range(len(s2)):
sum *= 10
sum += int(s1[i]) * int(s2[j])
totalsum *= 10
totalsum += sum
print(totalsum)

Same solution as given by Florent but in Java:

import java.util.*;
class Main {
    public static void main(String[] args) {
        String n1 = "3922445243";
        String n2 = "3248234";
       
        double result=0;
        for(int i=0;i<n1.length();i++){
           for(int j=0;j<n2.length();j++){
                double num1 = (n2.charAt(j)-'0')*(Math.pow(10,n2.length()-j-1));
                double num2 = (n1.charAt(i)-'0')*(Math.pow(10,n1.length()-i-1));
                result +=(num1*num2);
            }
        }
        System.out.println(result);
    }
}

def string_multiplication(a, b):

result = [0] * (len(a) + len(b))
for i in reversed(range(len(a))):
for j in reversed(range(len(b))):
result[i+j+1] += (ord(a[i]) - ord('0')) * (ord(b[j]) - ord('0'))
result[i+j] += result[i+j+1] / 10
result[i+j+1] %= 10

return ''.join( map(str, result[ next( idx for idx, digit in enumerate(result) if digit != 0) : ]) )

number_one = 193283492420348904832902348908239048823480823
number_two = 3248234890238902348823940990234

print(number_one*number_two)
print(string_multiplication(str(number_one), str(number_two)))

def string_multiplication(a, b):

    result = [0] * (len(a) + len(b))
    for i in reversed(range(len(a))):
        for j in reversed(range(len(b))):
            result[i+j+1] += (ord(a[i]) - ord('0')) * (ord(b[j]) - ord('0'))
            result[i+j]   += result[i+j+1] / 10
            result[i+j+1] %= 10

    return ''.join( map(str, result[ next( idx for idx, digit in enumerate(result) if digit != 0) : ]) )

number_one = 193283492420348904832902348908239048823480823
number_two = 3248234890238902348823940990234

print(number_one*number_two)
print(string_multiplication(str(number_one), str(number_two)))

def string_multiplication(a, b):

    result = [0] * (len(a) + len(b))
    for i in reversed(range(len(a))):
        for j in reversed(range(len(b))):
            result[i+j+1] += (ord(a[i]) - ord('0')) * (ord(b[j]) - ord('0'))
            result[i+j]   += result[i+j+1] / 10
            result[i+j+1] %= 10

    return ''.join( map(str, result[ next( idx for idx, digit in enumerate(result) if digit != 0) : ]) )

number_one = 193283492420348904832902348908239048823480823
number_two = 3248234890238902348823940990234

print(number_one*number_two)
print(string_multiplication(str(number_one), str(number_two)))

def string_multiplication(a, b):
    result = [0] * (len(a) + len(b))
    for i in reversed(range(len(a))):
        for j in reversed(range(len(b))):
            result[i+j+1] += (ord(a[i]) - ord('0')) * (ord(b[j]) - ord('0'))
            result[i+j]   += result[i+j+1] / 10
            result[i+j+1] %= 10

    return ''.join( map(str, result[ next( idx for idx, digit in enumerate(result) if digit != 0) : ]) )

number_one = 193283492420348904832902348908239048823480823
number_two = 3248234890238902348823940990234
print(number_one*number_two)
print(string_multiplication(str(number_one), str(number_two)))

def multiplication(s1, s2):
    product = []
    counter = 0
    for i in range(len(s2)-1,-1,-1):
        product.append(int(s2[i])*int(s1)*(10**counter))
        counter+=1
    return sum(product)

As an alternative solution, the code below uses recursion and dictionaries. It's more lines of code, but it also accounts for the negative edge case.

def multiplyStrings(str1, str2):
    
    if(str1[0]=="-"):
        initStr1 = str1[1:]
        multStr1 = -1
    else:
        initStr1 = str1[0:]
        multStr1 = 1
    
    if(str2[0]=="-"):
        initStr2 = str2[1:]
        multStr2 = -1
    else:
        initStr2 = str2[0:]
        multStr2 = 1
        
    return multStr1*multiplyString1(initStr1)*multStr2*multiplyString2(initStr2)
    

def multiplyString1(str1):
    
    if(len(str1)==1):
        return findNumber(str1)
    else:
        return findNumber(str1[0])*(10**(len(str1)-1))+multiplyString1(str1[1:])
    

def multiplyString2(str2):
    if(len(str2)==1):
        return findNumber(str2)
    else:
        return findNumber(str2[0])*(10**(len(str2)-1))+multiplyString2(str2[1:])


def findNumber(string):
    numbers = {
            "0" : 0,
            "1" : 1,
            "2" : 2,
            "3" : 3,
            "4" : 4,
            "5" : 5,
            "6" : 6,
            "7" : 7,
            "8" : 8,
            "9" : 9
            }
    
    return numbers[string]

num1 = 193283492420348904832902348908239048823480823
num2 = 3248234890238902348823940990234

print(num1*num2)

print(multiplyStrings(str(num1),str(num2)))

def multiplyStrings(str1, str2):
    
    if(str1[0]=="-"):
        initStr1 = str1[1:]
        multStr1 = -1
    else:
        initStr1 = str1[0:]
        multStr1 = 1
    
    if(str2[0]=="-"):
        initStr2 = str2[1:]
        multStr2 = -1
    else:
        initStr2 = str2[0:]
        multStr2 = 1
        
    return multStr1*multiplyString1(initStr1)*multStr2*multiplyString2(initStr2)
    

def multiplyString1(str1):
    
    if(len(str1)==1):
        return findNumber(str1)
    else:
        return findNumber(str1[0])*(10**(len(str1)-1))+multiplyString1(str1[1:])
    

def multiplyString2(str2):
    if(len(str2)==1):
        return findNumber(str2)
    else:
        return findNumber(str2[0])*(10**(len(str2)-1))+multiplyString2(str2[1:])


def findNumber(string):
    numbers = {
            "0" : 0,
            "1" : 1,
            "2" : 2,
            "3" : 3,
            "4" : 4,
            "5" : 5,
            "6" : 6,
            "7" : 7,
            "8" : 8,
            "9" : 9
            }
    
    return numbers[string]

num1 = 193283492420348904832902348908239048823480823
num2 = 3248234890238902348823940990234

print(num1*num2)

print(multiplyStrings(str(num1),str(num2)))